#  On the conservation laws and the structure of the nonlinearity for SQG and
its generalizations

Philip Isett, Andrew Ma Department of Mathematics, Caltech. ASUS Robotics and
AI Center.

###### Abstract

Using a new definition for the nonlinear term, we prove that all weak
solutions to the SQG equation (and mSQG) conserve the angular momentum. This
result is new for the weak solutions of [Resnick, ’95] and rules out the
possibility of anomalous dissipation of angular momentum. We also prove
conservation of the Hamiltonian under conjecturally optimal assumptions,
sharpening a well-known criterion of [Cheskidov-Constantin-Friedlander-
Shvydkoy, ’08]. Moreover, we show that our new estimate for the nonlinearity
is optimal and that it characterizes the mSQG nonlinearity uniquely among
active scalar nonlinearities with a scaling symmetry.

###### Contents

  1. 1 Introduction
    1. 1.1 Organization of the paper
    2. 1.2 Definitions of basic function spaces
  2. 2 Hamiltonian conservation: The test function step
  3. 3 The test function step in Euclidean space
  4. 4 Conservation of the Hamiltonian
    1. 4.1 Hamiltonian conservation in the Euclidean case
  5. 5 Deriving the bilinear form
  6. 6 Defining the nonlinearity
    1. 6.1 The nonperiodic case
    2. 6.2 Remark on generalized estimates of this type
  7. 7 Continuous extension of the operator
    1. 7.1 Extension to $\dot{W}^{2,\infty}$: Definition, boundedness and uniqueness
    2. 7.2 Continuity property of the extended operator
    3. 7.3 On the optimal constant of the extension
    4. 7.4 Special properties and conservation laws of mSQG
  8. 8 Sharpness of the estimate on the nonlinearity
  9. 9 Uniqueness of the main estimate for the mSQG multiplier
    1. 9.1 Reducing to the case of a smooth multiplier
    2. 9.2 The case of an odd multiplier
    3. 9.3 The case of an incompressible velocity field with even multiplier
    4. 9.4 The case of a more general even multiplier
  10. A Homogeneous function spaces
    1. A.1 The space $\dot{W}^{2,\infty}$
    2. A.2 The space $\dot{H}^{s}$

##  1 Introduction

Many important models in fluid dynamics are active scalar equations, which in
general have the form

$\displaystyle\partial_{t}\theta+T^{\ell}\theta\nabla_{\ell}\theta$
$\displaystyle=0,\qquad\widehat{T^{\ell}\theta}(\xi)=m^{\ell}(\xi)\hat{\theta}(\xi),$

where the velocity field advecting the scalar is given by a Fourier multiplier
applied to the scalar, and the summation convention is used to indicate a sum
over repeated indices. Of primary interest to this paper are the SQG equation
($T^{\ell}=-\epsilon^{\ell a}\nabla_{a}(-\Delta)^{-1/2}$), the 2D Euler
equation in vorticity form ($T^{\ell}=\epsilon^{\ell
a}\nabla_{a}\Delta^{-1}$), and the mSQG family, which interpolates between the
two $T^{\ell}=-\epsilon^{\ell a}\nabla_{a}(-\Delta)^{-1+\delta}$, $\delta\geq
0$. Here $\epsilon^{\ell a}$ is the Levi-Civita symbol, which is characterized
by $\epsilon^{j\ell}=-\epsilon^{\ell j}$, $\epsilon^{12}=1$. This family of
equations has been studied from the point of view of existence of weak
solutions [39, 34] well-posedness or ill-posedness of the equation and patch
dynamics [5, 44, 10, 12, 30, 45], constructing invariant measures [36],
constructing global solutions [11, 22], constructing blow up of patch
solutions [31], and for applications in a geophysical context (see [9, 37, 40,
41] and the references therein). When we consider more general equations, we
will assume that the multiplier $m$ is smooth away from the origin in
$\widehat{\mathbb{R}}^{n}$ unless otherwise stated, and that it has real-
symmetry $m(-\xi)=\overline{m(\xi)}$ so that $T^{\ell}\theta$ is real-valued
whenever $\theta$ is.

Fundamental to studying these equations are the conservation laws that are
satisfied by the solutions. When the velocity field is incompressible, which
is equivalent to the multiplier being angular ($m(\xi)\cdot\xi=0$),
sufficiently regular solutions conserve all $L^{p}$ norms as the measure of
the set $\\{x:a\leq\theta(x)\leq b\\}$ remains invariant under the evolution
of the equation for any $a,b\in\mathbb{R}$. For sufficient conditions for
$L^{2}$ norm conservation for mSQG we refer to [43].

Of equal importance is the Hamiltonian of the flow, which exists in 2
dimensions when the multiplier is angular and odd. For the mSQG equations, the
Hamiltonian is given up to a constant by

$\displaystyle H(t)$
$\displaystyle=\int|\Lambda^{-1+\delta}\theta(t,x)|^{2}dx,$ (1)

where $\Lambda$ is the Fourier multiplier $\Lambda=\sqrt{-\Delta}$. (For a
more general angular and odd multiplier $m$, the Hamiltonian exists but may
not be coercive; see [29] for a definition.) Note that the Hamiltonian is
well-defined for solutions that are square integrable in time taking values in
the Sobolev space $\dot{H}^{-1+\delta}$ (i.e. $\theta$ is of class
$L_{t}^{2}\dot{H}^{-1+\delta}$), which, as we shall soon elaborate, is the
lowest regularity in which the nonlinearity in the equation is well-defined.

Finally, solutions to the mSQG equations with sufficient regularity and decay
conserve the mean, impulse and angular momentum defined by

$\displaystyle
M(t)=\underbrace{\int\theta(t,x)dx}_{\textrm{mean}},\qquad\vec{I}(t)=\underbrace{\int_{\mathbb{R}^{2}}x\,\theta(t,x)dx}_{\textrm{impulse}},\qquad
A(t)=\underbrace{\int_{\mathbb{R}^{2}}|x|^{2}\theta(t,x)dx}_{\textrm{angular
momentum}}$ (2)

The first theorem we present concerns the conservation of these last three
quantities. We state a more precise version of this result in Theorem 7 below.

######  Theorem 1.

Every weak solution to the mSQG equation ($0\leq\delta\leq 1/2$) (of class
$L_{t}^{2}\dot{H}^{-1+\delta}(I\times\mathbb{R}^{2})$) conserves the mean,
impulse and angular momentum (defined in a generalized sense).

There appears to be no precedent for this type of criterion for angular
momentum conservation in the literature, and the proof is quite different from
the better-studied conservation of energy. The result is new for the SQG
equation, and in particular applies to the weak solutions constructed by
Resnick [39] and Marchand [34].

Since $L_{t}^{2}\dot{H}^{-1+\delta}$ is the lowest regularity in which the
nonlinearity is defined (as we will elaborate later), this result implies that
there are no weak solutions that violate conservation of the mean, impulse or
angular momentum among any class that can presently be defined. Thus, in
contrast to the situation for the Hamiltonian, anomalous dissipation of
angular momentum is not possible.

Our next main theorem concerns the optimal regularity under which one can show
the Hamiltonian is conserved. We refer to Section 1.2 below for the definition
of the Besov space.

######  Theorem 2 (Onsager singularity theorem).

Let $0\leq\delta<1/2$ and let $\theta$ be a weak solution to mSQG of class
$L_{t}^{3}\dot{H}^{-1+\delta}$ with $\Lambda^{-1+\delta}\theta$ of class
$L_{t}^{3}\dot{B}_{3,c(\mathbb{N})}^{(1+\delta)/3}$ on either $\mathbb{T}^{2}$
or on $\mathbb{R}^{2}$. Then the Hamiltonian $H(t)$ is almost everywhere
constant in time. In particular, $\theta\in
L_{t}^{\infty}\dot{H}^{-1+\delta}$. Furthermore, the same conservation law
holds true in the case of SQG ($\delta=1/2$) if we assume $\theta\in
L_{t,x}^{3}\cap L_{t}^{3}\dot{H}^{-1/2}$.

The problem of establishing criteria for energy conservation and other
conservation laws under optimal hypotheses has attracted a large amount of
attention since the basic results of [17, 8]. The literature is now too vast
to survey here, but we note the results [16, 7, 27, 2, 18, 14, 15, 42, 13],
which pertain specifically to the Euler equations. The paper [7] by Cheskidov,
Constantin, Friedlander and Shvydkoy establishes energy conservation in the
incompressible Euler equations for velocity fields of class
$L_{t}^{\infty}L_{x}^{2}\cap L_{t}^{3}B_{3,c(\mathbb{N})}^{1/3}$. Our theorem
expresses a slightly sharper result, as we assume only $L_{t}^{3}L_{x}^{2}\cap
L_{t}^{3}\dot{B}_{3,c(\mathbb{N})}^{1/3}$ control of the velocity field (which
satisfies $\|v\|_{L^{2}}\approx\|\Lambda^{-1}\theta\|_{L^{2}}$), while the
$L_{t}^{\infty}L_{x}^{2}$ regularity of the velocity field is a conclusion
that follows from the conservation of energy. This improvement is obtained by
a more careful study of what we call the “test function step” in the proof.

Our results are conjecturally sharp and our approach is novel in that Theorem
2 is the first proof of energy conservation at the critical regularity that
proceeds at the level of the vorticity equation. In this way, we obtain a
proof of energy conservation that unifies both the 2D Euler and SQG equations.
Previously, Hamiltonian conservation for SQG solutions in $L_{t,x}^{3}$ in a
periodic domain was proven in [29]. By the recent solution of the 2D Onsager
conjecture in [19], the regularity exponent $1/3$ in the 2D Euler result is
sharp, and by the generalized Onsager conjecture (see e.g. [32, 4]), the
assumption on the regularity is expected by analogy to be sharp for the full
range $0\leq\delta\leq 1/2$. The best known results on constructing solutions
to SQG that violate Hamiltonian conservation are obtained in [3, 26]; see also
[33] for the mSQG generalization.

In forthcoming work by the first author and S. Looi, which builds on the
methods of the present paper, the full Onsager conjecture for Hamiltonian
conservation in the SQG equation will be solved [25]. This proof constructs
solutions to SQG that fail to conserve the Hamiltonian with any regularity
$(-\Delta)^{-1/4}\theta\in C_{t}C_{x}^{1/2-\epsilon}$, and in particular uses
the double divergence form of the nonlinearity first obtained in the present
work. We also remark that Theorem 2 and its proof extend in a straightforward
way to prove Hamiltonian conservation under the same assumptions for any
active scalar equation whose multiplier $m$ is odd, angular and degree
$-1+2\delta$ homogeneous, $0<\delta\leq 1/2$.

Theorem 2 is the first to prove Hamiltonian conservation for mSQG in the range
$0<\delta<1/2$, which encounters new analytical difficulties. Namely, although
there is a simple proof of energy conservation available for Euler
($\delta=0$) and also a short, rather different proof of Hamiltonian
conservation for SQG ($\delta=1/2$), such a direct approach for the mSQG
family ($0<\delta<1/2$) does not seem to be available. Part of the difficulty
is to take advantage of the special cancellations in the nonlinearity that
allow the nonlinear term to be defined when the solution has negative
regularity.

In fact, when the multiplier is odd and homogeneous, the nonlinearity
satisfies the following bound.

######  Theorem 3.

Let $m$ be odd and degree $-1+2\delta$ homogeneous, $\delta>0$. Then there is
a constant depending on $m$ such that

$\displaystyle\left|\int\phi(x)\nabla_{\ell}[T^{\ell}[\theta]\theta]dx\right|$
$\displaystyle\lesssim_{m}\|\nabla^{2}\phi\|_{L^{\infty}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$
(3)

for all $\phi\in\SS(\mathbb{R}^{d})$ in the Schwartz class and for all

$\displaystyle\theta\in\SS_{0}(\mathbb{R}^{d})\coloneqq\left\\{f\in\SS(\mathbb{R}^{d})~{}:~{}\mathrm{supp}\,\hat{f}\mbox{
is a compact subset of }\widehat{\mathbb{R}^{d}}\setminus\\{0\\}\right\\}$

The same result holds also on $\mathbb{T}^{d}$, where we consider $\phi$ and
$\theta\in C^{\infty}(\mathbb{T}^{d})$.

Theorem 3 gives a new way of defining the nonlinearity in the active scalar
equation for solutions of class $\theta\in L_{t}^{2}\dot{H}^{-1+\delta}$ (see
equation (5) below). This ability to formulate the nonlinearity in a weak
function space is crucial to the proof of existence of weak solutions [39, 34]
to the SQG equation by compactness methods. Note that by scaling
considerations (counting derivatives), if we restrict to
$\theta\in\dot{H}^{-1+\delta}$, we cannot expect to use fewer than $2$
derivatives in $L^{\infty}$ for the norm on the test function $\phi$. The
method of proof also applies to a large class of more general nonlinearities
with similar structure (see e.g. Section 6.2).

The standard approach to approaching an estimate such as in Theorem 3 would be
based on the theory of Calderón commutators. That is, because the operator $T$
is anti-symmetric, we can rewrite the nonlinearity in a commutator form

$\displaystyle\int\nabla_{\ell}\phi T^{\ell}\theta~{}\theta dx$
$\displaystyle=-\int[T^{\ell},\nabla_{\ell}\phi]\theta
dx-\int\nabla_{\ell}\phi\theta T^{\ell}\theta dx$
$\displaystyle\int\nabla_{\ell}\phi T^{\ell}\theta~{}\theta dx$
$\displaystyle=-\frac{1}{2}\int[T^{\ell},\nabla_{\ell}\phi]\theta dx$

To prove Theorem 3 one then must show that the commutator
$[T^{\ell},\nabla_{\ell}\phi]$ has one degree of $L^{2}$ smoothing compared to
$T^{\ell}$, with a constant depending on the Lipschitz norm of $\nabla\phi$.
Although the previous definitions of weak solution to SQG require control of
at least three derivatives of $\phi$ in their definition (see [34, 6]), we do
believe that the machinery of Calderón commutators (e.g. the $1D$ analysis in
[35, Chapter 4]) can be generalized to prove the bound in (3), at least in the
case of $\mathbb{R}^{d}$, but we could not find a reference for this bound.

Our definition of the nonlinearity, however, is independent of the commutator
structure, and our method of estimating it extends easily to the periodic
setting and to any degree of homogeneity for the multiplier. Moreover, we
obtain the following improved estimate in the special case of mSQG.

######  Theorem 4 (mSQG bound).

For any $0\leq\delta<\infty$ and $T^{\ell}=\epsilon^{\ell
a}\nabla_{a}(-\Delta)^{-1+\delta}$, one has the bound

$\displaystyle\left|\int\phi(x)T^{\ell}\theta~{}\nabla_{\ell}\theta dx\right|$
$\displaystyle\lesssim_{\delta}\|\mathring{\nabla}^{2}\phi\|_{L^{\infty}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$
(4)

for all $\phi\in\SS(\mathbb{R}^{2})$, $\theta\in\SS_{0}(\mathbb{R}^{2})$,
where
$\mathring{\nabla}^{2}_{j\ell}\phi{=}\nabla_{j}\nabla_{\ell}\phi{-}\frac{1}{2}\Delta\phi\delta_{j\ell}$
is the trace-free part of the Hessian of $\phi$.

This theorem is easy to prove in the case of $2D$ Euler (write the
nonlinearity as $\mbox{div }\mbox{div }v\otimes v^{\perp}$,
$v^{\ell}=T^{\ell}\theta$), but for $\delta>0$ (in particular for SQG) it
apparently requires a more involved analysis.

We show in the following theorem that the norm
$\|\mathring{\nabla}^{2}\phi\|_{L^{\infty}}$ cannot be replaced by any weaker
norm.

######  Theorem 5.

Let $0\leq\delta<\infty$. If there exists a bound for the mSQG nonlinearity of
the form

$\displaystyle\left|\int\phi(x)\epsilon^{\ell
a}\nabla_{a}(-\Delta)^{-1+\delta}\theta\nabla_{\ell}\theta dx\right|$
$\displaystyle\lesssim_{\delta}\|\phi\|_{X}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$

for all $\phi\in C_{c}^{\infty}(\mathbb{R}^{2})$ and
$\theta\in\SS_{0}(\mathbb{R}^{2})$ for some semi-norm $\|\cdot\|_{X}$, then we
must have
$\|\mathring{\nabla}^{2}\phi\|_{L^{\infty}}\lesssim_{\delta}\|\phi\|_{X}$.

Our next theorem shows that the estimate of Theorem 4 completely characterizes
the mSQG nonlinearity among active scalar nonlinearities with similar
translation and scaling symmetries. This theorem shows that the special
estimate of Theorem 4 in terms of the trace-free part of the Hessian is quite
delicate and cannot be expected to hold for more general nonlinearities.

######  Theorem 6 (Characterization of the nonlinearity).

Let $0\leq\delta\leq 1/2$, suppose that $m$ is degree $-1+2\delta$
homogeneous, not necessarily smooth away from $0$, and assume that the bound

$\displaystyle\left|\int\phi T^{\ell}[\theta]\nabla_{\ell}\theta dx\right|$
$\displaystyle\lesssim\|\mathring{\nabla}^{2}\phi\|_{L^{\infty}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$

holds for $\phi\in C_{c}^{\infty}$ and $\theta\in\SS_{0}(\mathbb{R}^{2})$.
Then $m^{\ell}$ is a constant multiple of the mSQG multiplier.

Along the way to proving Theorem 6 we elaborate how the nonlinearity is not
bounded on any Sobolev space rougher than $\dot{H}^{-1+\delta}$. The theorem
actually applies to the range $0\leq\delta<\infty$ with $\delta\neq 1$, but
our method breaks down in the special case $\delta=1$ (which is also the case
where the mSQG nonlinearity vanishes) and we do not know if the theorem holds
in this case, although we are able to make some conclusions about the
structure of $m$ for a potential counterexample (in particular the velocity
field cannot be incompressible).

The special estimate (4) is intimately tied to the conservation of impulse and
angular momentum for the mSQG equation, but is not quite strong enough to
prove these conservation laws. Namely, the functions
$\\{1,x_{1},x_{2},|x|^{2}\\}$ that appear in the definitions of mean, impulse
and angular momentum (equation (2)) span precisely those functions for which
the trace-free part of the Hessian vanishes. It is tempting to try to prove
angular momentum conservation by applying the estimate in Theorem 4 to a
version of $|x|^{2}$ that has been cut off in space. However, the bound in
Theorem 4 is not strong enough for this strategy to succeed in proving angular
momentum conservation.

Nonetheless, what we will show in Theorem 7 below is that conservation of
angular momentum and impulse holds for all weak solutions in
$L_{t}^{2}\dot{H}^{-1+\delta}$, which, as we have noted before, is the lowest
regularity in which the nonlinearity makes sense. We now explain the precise
formulation.

Upon first sight, the definition of angular momentum seems to require that
$|x|^{2}|\theta(t)|\in L^{1}$ to make sense, but we can actually define this
quantity for a much rougher class of data.

######  Definition 1.1.

We say a distribution $f\in{\cal D}^{\prime}(\mathbb{R}^{2})$ is in the
angular momentum class if it can be written in the form
$f(x)=A(x)+\nabla_{j}B^{j}(x)+\nabla_{j}\nabla_{\ell}C^{j\ell}(x)$ where
$|x|^{2}|A|+|x||B|+|C|\in L^{1}(\mathbb{R}^{2}\setminus B_{R})$ for some ball
$B_{R}$ around the origin of radius $R$.

Angular momentum can be defined for any distribution in the angular momentum
class as follows.

######  Definition 1.2.

Let $f\in{\cal D}^{\prime}(\mathbb{R}^{2})$. We say $f$ has well-defined
angular momentum if, for any bump function $\chi\in
C_{c}^{\infty}(\mathbb{R}^{2})$ that is equal to $1$ in a neighborhood of $0$,
the limit

$\displaystyle\lim_{R\to\infty}\int{\chi\left(\frac{x}{R}\right)}|x|^{2}f(x)\textrm{`}dx\textrm{'}$

exists and is independent of the choice of $\chi$. Here ‘$dx$’ means that the
integral is interpreted in the sense of distributions.

Generalized impulse can be defined similarly. It is a simple exercise to show
that any distribution in the angular momentum class has well-defined angular
momentum and impulse111In fact, one can also have terms of the form
$\nabla_{i}\nabla_{j}\nabla_{\ell}D^{ij\ell}$ with $|x|^{-1}|D|\in
L^{1}(\mathbb{R}^{2}\setminus B_{R})$ to have well-defined angular momentum
and impulse.. (See for example the proof of Lemma 7.1 below.) Our theorem on
conservation of impulse and angular momentum is the following.

######  Theorem 7 (Conservation laws).

If $0\leq\delta<\infty$ and $\theta$ is an mSQG solution in
$L_{t}^{2}\dot{H}^{-1+\delta}(I\times\mathbb{R}^{2})$, $I$ an open interval,
then for any time $t_{0}\in\bar{I}$, $\theta(t_{0})$ has well-defined angular
momentum (impulse) if and only if $\theta(t)$ has well-defined angular
momentum (impulse) at every time $t\in I$ and the angular momentum (impulse)
is constant in time. Furthermore, $\theta(t)$ belongs to the angular momentum
class at a time $t_{0}\in\bar{I}$ if and only if $\theta(t)$ belongs to the
angular momentum class at every time $t\in\bar{I}$.

At this point we pause to state precisely our definition of weak solution. We
also expound the remark that, much like for other evolution equations, one can
define a unique restriction of an mSQG solution to every time slice in its
domain in such a way that the family of distributions $\theta(t)$ are
continuous in $t$ with respect to weak convergence in ${\cal D}^{\prime}$. To
make these remarks we need the definition of the fixed-time nonlinearity from
Theorem 3.

######  Definition 1.3.

We introduce the notation $B(\psi,\theta)$ to denote the unique extension of
the linear-quadratic form $B(\psi,\theta)=\int\psi
T^{\ell}[\theta]\nabla_{\ell}\theta dx$ to the space
$C_{c}^{\infty}\times\dot{H}^{-1+\delta}$, whose existence follows from
Theorem 3. We use $B(\psi,\theta_{1},\theta_{2})$ to denote the unique
trilinear form, symmetric in $\theta_{1}$, $\theta_{2}$, such that
$B(\psi,\theta,\theta)=B(\psi,\theta)$.

Specifically, we define $B(\psi,\theta)=\lim_{n\to\infty}B(\psi,\theta_{n})$
where each $\theta_{n}\in\SS_{0}(\mathbb{R}^{2})$ and $\theta_{n}\to\theta$ in
$\dot{H}^{-1+\delta}$.

In terms of this definition, a weak solution to an active scalar equation with
odd, angular multiplier is a distribution of class $\theta\in
L_{t}^{2}\dot{H}^{-1+\delta}(I\times\mathbb{R}^{2})$ such that for all
$\phi(t,x)\in C_{c}^{\infty}$ one has

$\displaystyle-\int\partial_{t}\phi(t,x)\,\theta(t,x)\,\textrm{`}dxdt\textrm{'}+\int
B(\phi(t),\theta(t))dt$ $\displaystyle=0,$ (5)

where again we use the single quotes to denote the pairing of a distribution
with a test function. The time-integral is well-defined by virtue of the
estimate of Theorem 3 and an approximation of $\theta(t)$ by simple functions
of time taking values in $\dot{H}^{-1+\delta}$. We define the restriction of
$\theta$ to a time slice $t_{0}\in\bar{I}$ by the equation
$\theta(t_{0},x)=\theta(t,x)-\int_{t_{0}}^{t}B(\cdot,\theta(s))ds$. At first
sight the distribution on the right hand side (call it $u$) appears to depend
on $t$, but the definition of a weak solution implies that $\partial_{t}u=0$
in the weak sense, so that $u$ can be identified with a unique time-
independent distribution on $\mathbb{R}^{2}$.

#### Optimality of the definition

With these definitions in hand, the final question we wish to address in this
work is: Is our definition of the nonlinearity the best possible definition?
To compare different definitions, there are four criteria one can consider:

  1. 1.

Which definition applies at the lowest possible regularity for $\theta$?

  2. 2.

Which definition yields the best possible estimate for the nonlinearity?

  3. 3.

Which definition yields the best possible continuity properties for the
nonlinearity?

  4. 4.

Which definition yields a continuous extension to the largest possible space
of test functions?

Regarding the first point, it will be shown in the course of proving Theorem 6
that the nonlinearity does not make sense for $\theta$ below
$\dot{H}^{-1+\delta}$, so that our definition is optimal in this regard.
Regarding the second point, Theorem 5 shows that the bound of Theorem 4 is the
best possible one. (Recall that in contrast the existing definitions [34, 6]
require at least $3$ derivatives of the test function.) Regarding points three
and four, we have the following theorems on continuous extensions of the
nonlinearity.

Our first theorem shows that the test function need not even be $C^{2}$ but
can be any distribution whose weak-Hessian is bounded. Moreover, the
continuity is stronger than what is implied by the boundedness of Theorem 3.

######  Theorem 8.

Let $m$ be odd, angular and degree $-1+2\delta$ homogeneous. Let
$\dot{W}^{2,\infty}$ be the (non-Hausdorff) space of distributions with weak
Hessian in $L^{\infty}$. Then there is a unique extension of the linear-
quadratic form $B(\psi,\theta)$ to
$\dot{W}^{2,\infty}\times\dot{H}^{-1+\delta}$ that satisfies

$\displaystyle B(\Psi,\Theta)$ $\displaystyle=\lim_{n}B(\Psi_{n},\Theta_{n})$

whenever $\nabla^{2}\Psi_{n}$ converges to $\nabla^{2}\Psi$ in $L^{\infty}$
weak-*, and $\Theta_{n}\to\Theta$ strongly in $\dot{H}^{-1+\delta}$. The
extended linear-quadratic form also satisfies a bound of the form
$|B(\Psi,\Theta)|\lesssim\|\nabla^{2}\Psi\|_{L^{\infty}}\|\Theta\|_{\dot{H}^{-1+\delta}}^{2}$,
(perhaps with a different optimal constant).

We must stress that the continuous extension guaranteed by this theorem does
not follow from the bound of Theorem 3 by a simple density argument since test
functions are not norm dense in $\dot{W}^{2,\infty}$. The content of the
theorem is that the nonlinearity has a stronger property that allows
continuity in a weaker topology.

For the mSQG nonlinearity there is again an improvement, which is that we can
handle any test function whose weak Hessian has bounded trace-free part. We
let $\mathring{W}^{2,\infty}$ denote this non-Hausdorff space of
distributions.

######  Theorem 9.

Let $m$ be the mSQG multiplier, $0\leq\delta<\infty$. Let
$\mathring{W}^{2,\infty}$ be the (non-Hausdorff) space of distributions whose
weak Hessian has trace-free part in $L^{\infty}$. Then there is a unique
extension of the linear-quadratic form $B(\psi,\theta)$ to
$\mathring{W}^{2,\infty}\times\dot{H}^{-1+\delta}$ that satisfies

$\displaystyle B(\Psi,\Theta)$
$\displaystyle=\lim_{m}\lim_{n}B(\Psi_{n},\Theta_{m})$ (6)

whenever $\psi_{n}$ is a sequence in $\SS(\mathbb{R}^{2})$ converging in
$\SS^{\prime}(\mathbb{R}^{2})$ to $\Psi$, and $\Theta_{m}$ is a sequence in
$\SS_{0}(\mathbb{R}^{2})$ converging strongly in $\dot{H}^{-1+\delta}$ to
$\Theta$. This extended linear-quadratic form satisfies an estimate of the
form
$|B(\Psi,\Theta)|\lesssim\|\mathring{\nabla}^{2}\Psi\|_{L^{\infty}}\|\Theta\|_{\dot{H}^{-1+\delta}}^{2}$,
(perhaps with a different optimal constant).

The existence of the inner limit in (6) is immediate, while the existence of
the outer limit follows from the boundedness in Theorem 4. We remark that the
extended operator has an additional continuity analogous to Theorem 8. namely
that $B(\Psi_{n},\Theta_{n})\to B(\Psi,\Theta)$ if
$\mathring{\nabla}^{2}\Psi_{n}$ converges to $\mathring{\nabla}^{2}\Psi$ in
$L^{\infty}$ weak-* and $\Theta_{n}\to\Theta$ in $\dot{H}^{-1+\delta}$.
However, it is not clear whether this continuity property uniquely
characterizes the extension as we do not know whether $C_{c}^{\infty}$ is
sequentially weak-* dense in $\mathring{W}^{2,\infty}$.

The statement of Theorem 9 implicitly takes advantage of the following
fact222Without this theorem we can state that the nonlinearity extends to test
functions in $\mathring{W}^{2,\infty}$ that are tempered distributions, but
Theorem 10 assures that such distributions contain all of
$\mathring{W}^{2,\infty}$., which we were motivated by Theorem 9 to discover
and believe to be of independent interest: Every distribution whose Hessian
has bounded trace free part is a tempered distribution. More precisely, we
have the following Theorem

######  Theorem 10 ([24]).

Let $\Psi\in{\cal D}^{\prime}(\mathbb{R}^{d})$ be a distribution such that the
trace free part of the weak Hessian
$\mathring{\nabla}^{2}_{j\ell}\Psi=\nabla_{j}\nabla_{\ell}\Psi-\frac{1}{d}\Delta\Psi\delta_{j\ell}$
belongs to $L^{\infty}$. Then $\Psi$ is a tempered distribution. Moreover,
$\Psi$ has log-Lipschitz continuous first derivatives and satisfies the growth
estimates

$\displaystyle|\Psi(x)|=O(|x|^{2}\log|x|),\qquad|\nabla\Psi(x)|=O(|x|\log|x|)\qquad\mbox{
as }|x|\to\infty.$

These estimates are sharp in view of the example
$\Psi(x)=|x|^{2}\log|x|^{2}=2|x|^{2}\log|x|\in\mathring{W}^{2,\infty}$.

Two open questions related to this theorem are the following: First, is the
space of $C_{c}^{\infty}$ functions dense in $\mathring{W}^{2,\infty}$ in the
weak-* topology? (If so, the weak-* continuity and boundedness properties
would uniquely characterize the continuous extension of the nonlinearity.)
Likewise, is the space $\mathring{W}^{2,\infty}$ complete in the sense of a
non-Hausdorff space? That is, given $\Psi_{n}$ Cauchy in the semi-norm, does
there exist $\Psi\in\mathring{W}^{2,\infty}$ that is the (non-unique) limit of
$\Psi_{n}$? Completeness is true for the space $\dot{W}^{2,\infty}$ (see the
Appendix), but it is not obvious whether completeness holds in
$\mathring{W}^{2,\infty}$.

We conclude this introduction with the organization of the paper and some
basic definitions.

###  1.1 Organization of the paper

Sections 2, 3 and 4 are devoted to proving the conservation of the Hamiltonian
stated in Theorem 2. These sections rely on a divergence form expression that
will be a basic tool in proving all of the remaining Theorems of the paper.
This tool is developed in Section 5. In Section 6 we derive our new definition
of the nonlinearity. We prove Theorem 8 on the continuous extension of the
operator in Section 7. The special estimates and continuity properties of
Theorems 4 and 9 are obtained by establishing a certain divergence form in
Section 7.4. Section 7.4 also establishes the angular momentum conservation
stated in Theorems 1 and 7.

Theorem 5 on the sharpness of the mSQG bound is proven in Section 8. Theorem
6, which characterizes the mSQG nonlinearity, is proven in Section 9.

The paper concludes with an Appendix that lays out the important properties of
the homogeneous function spaces used in the paper.

Throughout the paper, the results are obtained in both the periodic setting
and the nonperiodic setting wherever it makes sense to do so. We have made
efforts to clarify which setting we consider at each instance where the
presentation switches settings.

#### Acknowledgments

The first author acknowledges the support of a Sloan fellowship and the NSF
grant DMS 2055019.

###  1.2 Definitions of basic function spaces

For $X$ a Banach space of distributions on $\mathbb{T}^{d}$ or
$\mathbb{R}^{d}$, we denote by $L_{t}^{p}X$ to be the space of distributions
$u$ such that $u$ is strongly measurable in time with values in $X$ and the
norm $\|\|u(t)\|_{X}\|_{L^{p}}$ is finite.

The Banach space $\dot{H}^{-1+\delta}$ is defined and its properties are
studied in detail in the Appendix.

We define Littlewood-Paley projections with the conventions that $P_{\leq 0}$
has a Fourier multiplier $\hat{\chi}_{\leq 0}(\xi)$ that equals $1$ in
$|\xi|\leq 1/2$ and has compact support in $|\xi|\leq 1$. With this convention
$P_{\leq q}$ has Fourier multiplier $\hat{\chi}_{\leq q}\equiv\hat{\chi}_{\leq
0}(2^{-q}\xi)$ and we define $P_{q}\equiv P_{\leq q+1}-P_{\leq q}$, which has
a Fourier multiplier $\hat{\chi}_{q}$ localized to $2^{q-1}\leq|\xi|\leq
2^{q+1}$.

We say that a distribution $u$ belongs to the homogeneous Besov space
$\dot{B}^{\alpha}_{p,c(\mathbb{N})}$ if there exists a sequence of
$C_{c}^{\infty}$ functions $u_{n}$ such that
$\|u-u_{n}\|_{\dot{B}^{\alpha}_{p,c(\mathbb{N})}}\to 0$, where

$\displaystyle\|u\|_{\dot{B}^{\alpha}_{p,c(\mathbb{N})}}$
$\displaystyle=\sup_{q}2^{\alpha q}\|P_{q}u\|_{L^{p}}.$

Distributions in this class inherit properties from test functions such as
$\|P_{q}u\|_{L^{p}}=o(2^{-\alpha q})$ as $q\to\infty$.

##  2 Hamiltonian conservation: The test function step

The first step in the proof of Hamiltonian conservation is to justify
multiplying the equation by a test function that is not necessarily smooth
with compact support. This step is less technical on the torus as opposed to
the full Euclidean space, so we concentrate on the $\mathbb{T}^{2}$ case
first. We explain the additional technicalities for the Euclidean case in
Section 3.

Let $\theta$ be an mSQG solution with $\delta\in[0,1/2]$ fixed and $\theta\in
L_{t}^{3}\dot{H}^{-1+\delta}$ with $\Lambda^{-1+\delta}\theta\in
L^{3}_{t}B^{\alpha}_{3,\infty}$ for $\alpha\geq(1+\delta)/3$. Note that on the
torus the assumption $\theta\in L_{t}^{3}\dot{H}^{-1+\delta}$ follows from our
other assumption on $\Lambda^{-1+\delta}\theta$. Indeed, if
$\Lambda^{-1+\delta}\theta\in L^{3}_{t}B^{\alpha}_{3,\infty}$ for $\alpha>0$
then $\theta\in L^{3}_{t}\dot{H}^{-1+\delta}_{x}$ due to the Littlewood-Paley
inequality

$\|\theta\|_{\dot{H}^{-1+\delta}(\mathbb{T}^{2})}^{2}\sim\sum_{q\geq
0}\|2^{(-1+\delta)q}P_{q}\theta\|^{2}_{L^{2}(\mathbb{T}^{2})}\lesssim\sum_{q\geq
0}\left(2^{-\alpha
q}\|\Lambda^{-1+\delta}\theta\|_{B^{\alpha}_{3,\infty}(\mathbb{T}^{2})}\right)^{2}\lesssim\|\Lambda^{-1+\delta}\theta\|_{B^{\alpha}_{3,\infty}(\mathbb{T}^{2})}^{2}$

We refer to Section 1.2 for the definitions of the operators $P_{q}$ and the
Besov space $B^{\alpha}_{3,\infty}$.

Define $\phi(t)\in C_{c}^{\infty}(\mathbb{R})$ to be a radial bump function of
unit integral and let $\phi_{\tau}(t)=\tau^{-1}\phi(\tau^{-1}t)$ be a
mollifying kernel in time. In addition, let $\eta(t)\in C_{c}^{\infty}(I)$ be
a test function in time. Define the test function

$\psi_{\hat{q},\tau}\coloneqq\left(\eta(t)\left(P_{\leq\hat{q}}^{2}\Lambda^{2(-1+\delta)}\theta\right)_{\tau}\right)_{\tau}\in
C^{\infty}_{c}(\mathbb{T}^{2}\times\mathbb{R})$

where $(\cdot)_{\tau}$ denotes mollification in time with $\phi_{\tau}$.

We may safely apply $\psi_{\hat{q},\tau}$ as a test function to mSQG in order
to obtain

$\displaystyle-\int_{\mathbb{R}}\int_{\mathbb{T}^{2}}\theta\partial_{t}\psi_{\hat{q},\tau}\,dxdt-\int
B[\psi_{\hat{q},\tau}(t),\theta(t),\theta(t)]dt=0$ (7)

where
$B[\psi_{\hat{q},\tau},\theta,\theta]\coloneqq\int_{\mathbb{T}^{2}}u^{l}\theta\nabla_{l}\psi_{\hat{q},\tau}\,\textrm{`}dx\textrm{'}$
is the nonlinear term defined in Definition 1.3. Using the self-adjointness of
$P_{\leq\hat{q}}^{2}$ and $(\cdot)_{\tau}$, we have that

$\displaystyle\frac{1}{2}\int_{\mathbb{R}\times\mathbb{T}^{2}}\eta^{\prime}(t)\cdot\left|\left(\Lambda^{-1+\delta}P_{\leq\hat{q}}\theta\right)_{\tau}\right|^{2}\,dxdt$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{T}^{2}}\theta\partial_{t}\psi_{\hat{q},\tau}\,dxdt$

By the definition of a weak solution, we get

$\displaystyle\frac{1}{2}\int_{\mathbb{R}\times\mathbb{T}^{2}}\eta^{\prime}(t)\cdot\left|\left(\Lambda^{-1+\delta}P_{\leq\hat{q}}\theta\right)_{\tau}\right|^{2}\,dxdt=\int_{\mathbb{R}\times\mathbb{T}^{2}}u^{l}\theta\nabla_{l}\left(\eta\left(P_{\leq\hat{q}}^{2}\Lambda^{2(-1+\delta)}\theta\right)_{\tau}\right)_{\tau}\,\textrm{`}dxdt\textrm{'}$
(8)

where the formal integral on the right hand side is interpreted in terms of
the linear-quadratic form for mSQG as

$\displaystyle\int
B\left[\left(\eta\left(P_{\leq\hat{q}}^{2}\Lambda^{2(-1+\delta)}\theta\right)_{\tau}\right)_{\tau},\theta(t),\theta(t)\right]dt$

We would like to take a limit of $\tau\rightarrow 0$ of both sides of (8). To
do this we use the next Lemma.

######  Lemma 2.1.

$\psi_{\hat{q},\tau}\longrightarrow\eta(t)\Lambda^{2(-1+\delta)}P_{\leq\hat{q}}^{2}\theta\eqqcolon\psi_{\hat{q}}\text{
in }L^{3}_{t}\dot{W}^{2,\infty}\text{ as }\tau\rightarrow 0$

###### Proof.

For each fixed $\hat{q}$ we note that
$\Lambda^{2(-1+\delta)}P^{2}_{\leq\hat{q}}:L^{3}_{t}\dot{H}^{-1+\delta}_{x}\longrightarrow
L^{3}_{t}\dot{W}^{2,\infty}$ is a bounded linear operator with an operator
norm that depends on $\hat{q}$. So

$\|\psi_{\hat{q},\tau}-\psi_{\hat{q}}\|_{L^{3}_{t}\dot{W}^{2,\infty}}=\left\|\Lambda^{2(-1+\delta)}P_{\leq\hat{q}}^{2}\left((\eta(t)\theta_{\tau})_{\tau}-\eta(t)\theta\right)\right\|_{L^{3}_{t}\dot{W}^{2,\infty}}\lesssim_{\hat{q}}\left\|(\eta(t)\theta_{\tau})_{\tau}-\eta(t)\theta\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}$

Hence, it is enough to show

$\lim_{\tau\rightarrow
0}\left\|(\eta(t)\theta_{\tau})_{\tau}-\eta(t)\theta\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}=0$

We proceed via an approximation argument. Define $\psi\coloneqq\eta(t)\theta$.
Banach valued simple functions are dense in $L^{3}_{t}\dot{H}^{-1+\delta}_{x}$
[21] and [38]. Using this face, we can approximate $\psi$ in the
$L^{3}_{t}\dot{H}^{-1+\delta}_{x}$-norm by a smooth in time approximation
$f\in C_{c}^{\infty}(\mathbb{R};\dot{H}^{-1+\delta}(\mathbb{T}^{2}))$ [38,
Chapter 4, Theorem 4.11]. Given an $\epsilon>0$, take $f\in
C_{c}^{\infty}(\mathbb{R};\dot{H}^{-1+\delta}(\mathbb{T}^{2}))$ such that
$\left\|\theta-f\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}<\epsilon$. Define
$\psi_{\tau,\tau}\coloneqq\left(\eta(t)\theta_{\tau}\right)_{\tau}$ and
$f_{\tau,\tau}\coloneqq\left(\eta(t)f_{\tau}\right)_{\tau}$ and consider the
inequality

$\displaystyle\left\|\psi_{\tau,\tau}-\psi\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}$
$\displaystyle\leq\left\|\psi_{\tau,\tau}-f_{\tau,\tau}\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}+\left\|f_{\tau,\tau}-\eta
f\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}+\left\|\eta
f-\psi\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}$ (9) $\displaystyle\coloneqq
I+II+III$

We will discuss how to estimate each term on the RHS of (9). Beginning with
$I$,
$\left\|\psi_{\tau,\tau}-f_{\tau,\tau}\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}<\epsilon$
due to a modified argument of the proof that convolutions are bounded linear
operators in $L^{p}$-norms.

For $II$, $\lim_{\tau\rightarrow 0}\left\|f_{\tau,\tau}-\eta
f\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}=0$ by a dominated convergence
argument and observing that for sufficiently small $\tau$, the family of
functions $\\{f_{\tau,\tau}\\}$ can yield a sequence of uniformly bounded
functions converging to $\eta f$ in $\dot{H}^{-1+\delta}$ pointwise uniformly
in time.

Finally for $III$, $\left\|\eta
f-\psi\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}\leq\|\eta\|_{L^{\infty}(\mathbb{R})}\left\|f-\theta\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}<\epsilon$
by Young’s inequality and our assumption
$\left\|f-\theta\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}<\epsilon$. By
combining the estimates of $I,II$, and $III$ and taking a limit of (9) we get
$\limsup_{\tau\rightarrow
0}\left\|\psi_{\tau,\tau}-\psi\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}\leq\limsup_{\tau\rightarrow
0}\left(2\epsilon+\left\|f_{\tau,\tau}-\eta
f\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}\right)\leq 2\epsilon$. Since this
inequality holds for any $\epsilon>0$ we conclude that $\lim_{\tau\rightarrow
0}\left\|\psi_{\tau,\tau}-\psi\right\|_{L^{3}_{t}\dot{H}^{-1+\delta}_{x}}=0$ ∎

Next, by the assumption $\theta\in L^{3}_{t}B^{\alpha}_{3,\infty}\subseteq
L^{3}_{t}\dot{H}^{-1+\delta}_{x}$, Lemma 2.1, and the bound on the nonlinear
term given in Theorem 3 we may safely take a limit of $\tau\rightarrow 0$ of
line (8). Formally, the identity we have established is

$\int_{\mathbb{R}}\int_{\mathbb{T}^{2}}\eta^{\prime}\left(P_{\leq\hat{q}}\Lambda^{-1+\delta}\theta\right)^{2}dxdt=-\int_{\mathbb{R}}\int_{\mathbb{T}^{2}}u^{l}\theta\nabla_{l}\left(\eta
P_{\leq\hat{q}}^{2}\Lambda^{2(-1+\delta)}\theta\right)\textrm{`dxdt'}$ (10)

where in the line above we define

$\displaystyle\psi_{\hat{q}}\coloneqq\eta(t)P_{\leq\hat{q}}^{2}\Lambda^{2(-1+\delta)}\theta.$
(11)

Rigorously the right hand side of (10) must be interpreted as

$\displaystyle-\int_{\mathbb{R}}\int_{\mathbb{T}^{2}}u^{l}\theta\nabla_{l}\left(\eta
P_{\leq\hat{q}}^{2}\Lambda^{2(-1+\delta)}\theta\right)\textrm{`dxdt'}\coloneqq\int
B(\psi_{\hat{q}}(t),\theta(t),\theta(t))dt$

where $B$ is the linear-quadratic form that is bounded on
$\dot{W}^{2,\infty}\times\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$.

We observe that the left hand side of (10) is an approximation of the weak
derivative of the Hamiltonian, $\frac{d}{dt}H(t)$, against the smooth bump
function in time $-\eta(t)$ and we observe that, by dominated convergence,

$\lim_{\hat{q}\rightarrow\infty}\int_{\mathbb{R}}\int_{\mathbb{T}^{2}}\eta^{\prime}\left(P_{\leq\hat{q}}\Lambda^{-1+\delta}\theta\right)^{2}dxdt=\int_{\mathbb{R}}\int_{\mathbb{T}^{2}}\eta^{\prime}\left(\Lambda^{-1+\delta}\theta\right)^{2}dxdt$

Our goal is thus to show
$\lim_{\hat{q}\rightarrow\infty}\int_{\mathbb{R}}\eta^{\prime}(t)\int_{\mathbb{T}^{2}}\left(P_{\leq\hat{q}}\Lambda^{-1+\delta}\theta\right)^{2}dxdt=0$.
In other words, we would like to prove that the RHS of (10) obeys the limit

$\displaystyle\lim_{\hat{q}\rightarrow\infty}\int
B(\psi_{\hat{q}}(t),\theta(t),\theta(t))dt=0$ (12)

To investigate this limit we will make use of an approximating sequence.

##  3 The test function step in Euclidean space

In this section we show how to generalize the analysis of Section 2 from the
torus to the case of $\mathbb{R}^{2}$. The reader interested mainly in the
periodic case may wish to skip to Section 4 upon first reading.

We assume $\theta\in L_{t}^{3}\dot{H}^{-1+\delta}$. We start by observing
that, as in the periodic case

$\displaystyle\int\eta^{\prime}(t)\int\left|\Lambda^{-1+\delta}\theta(t,x)\right|^{2}dxdt$
$\displaystyle=\lim_{\hat{q}\to\infty}\lim_{\tau\to
0}\int\eta^{\prime}(t)\int\left|P_{\leq\hat{q}}^{2}(\Lambda^{-1+\delta}\theta(t,x))_{\tau}\right|^{2}dxdt$
(13)

by a simple approximation argument using density of
$\dot{H}^{-1+\delta}$-valued simple functions of time.

We now deviate from the periodic case to write

$\displaystyle\eqref{eq:sameAsT2}$
$\displaystyle=\lim_{\hat{q}\to\infty}\lim_{\tau\to
0}\lim_{m\to\infty}\int\eta^{\prime}(t)\int\left|P_{(-m,\hat{q}]}(\Lambda^{-1+\delta}\theta(t,x))_{\tau}\right|^{2}dxdt$
(14)

where $P_{(-m,\hat{q}]}\coloneqq P_{\leq\hat{q}}^{2}-P_{\leq-m}$ is a
truncation of $P_{\leq\hat{q}}^{2}$. Again we have applied dominated
convergence using that $\theta\in L_{t}^{2}\dot{H}^{-1+\delta}$.

Our next step is to introduce a physical space cutoff $\chi_{R}(x)=\chi(x/R)$,
where $\chi(x)=1$ on the unit ball and $\chi\in C_{c}^{\infty}$. We claim that

$\displaystyle\eqref{eq:diffFromT2}$
$\displaystyle=\lim_{\hat{q}\to\infty}\lim_{\tau\to
0}\lim_{m\to\infty}\lim_{R\to\infty}\int\eta^{\prime}(t)\int\left|P_{(-m,\hat{q}]}\Lambda^{-1+\delta}[\chi_{R}\theta(t,x)_{\tau}]\right|^{2}dxdt$
(15)

To prove this claim, we decompose the function on the right hand side as

$\displaystyle P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}[\chi_{R}\theta_{\tau}]$
$\displaystyle=I+II+III+IV$ $\displaystyle I$
$\displaystyle=P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}[P_{\leq-m-2}\chi_{R}P_{\leq-m-2}\theta_{\tau}]$
$\displaystyle II$
$\displaystyle=P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}[P_{(-m-2,\hat{q}+2]}\chi_{R}P_{\leq-m-2}\theta_{\tau}]$
$\displaystyle III$
$\displaystyle=P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}[P_{\leq\hat{q}+2}\chi_{R}P_{(-m-2,\hat{q}+2]}\theta_{\tau}]$
$\displaystyle IV$
$\displaystyle=P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}\sum_{k\geq\hat{q}+1}[P_{k+1}\chi_{R}P_{k}\theta_{\tau}+P_{k}\chi_{R}P_{k+1}\theta_{\tau}+P_{k+1}\chi_{R}P_{k+1}\theta_{\tau}]$

Note that $IV$ contains all the terms involving high frequencies since, if
$k\geq\hat{q}+1$, we have that $P_{k+1}\chi_{R}P_{n}\theta_{\tau}$ and
$P_{k+1}\theta_{\tau}P_{n}\theta_{\tau}$ have frequency support disjoint from
that of $P_{(-m,\hat{q}]}^{2}$ for $n<k-1$.

By frequency support considerations $I=0$, so it suffices to take the limit in
$L^{2}$ of the other terms.

We start by estimating $II$. All estimates are taken at a fixed time

$\displaystyle\|II(t)\|_{L_{x}^{2}}$
$\displaystyle\lesssim_{m,\hat{q}}\|P_{(-m-2,\hat{q}+2]}\chi_{R}\|_{L^{\infty}}\|P_{\leq-m-2}\theta_{\tau}\|_{L^{2}}$
$\displaystyle\lesssim_{m,\hat{q}}\|P_{(-m-2,\hat{q}+2]}\chi_{R}\|_{L^{\infty}}\|\theta_{\tau}(t)\|_{\dot{H}^{-1+\delta}}$

Furthermore, as $R\to\infty$, we have

$\displaystyle\|P_{(-m-2,\hat{q}+2]}\chi_{R}\|_{L^{\infty}}$
$\displaystyle\lesssim_{m,\hat{q}}\|\nabla\chi_{R}\|_{L^{\infty}}\to
0\qquad\mbox{ as }R\to\infty$

and we conclude $II$ converges to $0$ in $L^{2}_{x}$ at any time $t$, with
$L^{2}$ norms bounded uniformly by a square-integrable function of $t$.

We now consider the limit of $III$, which is the only term that does not
vanish in the limit. As $\hat{q}$ is fixed, the functions

$\displaystyle P_{\leq\hat{q}+2}\chi_{R}(x)$
$\displaystyle=\int\chi_{R}(y)\eta_{\leq\hat{q}+2}(x-y)dy$

are uniformly bounded in $L^{\infty}$ and converge pointwise (but not
uniformly) in $x$ to $1=\int\eta_{\leq\hat{q}+2}dy$ as $R\to\infty$. Using
dominted convergence and the $L^{2}$ boundedness of
$P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}$, it follows that

$\displaystyle III(t)$ $\displaystyle\to
P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}P_{(-m-2,\hat{q}+2]}\theta_{\tau}=P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}\theta_{\tau}$

in $L^{2}_{x}$ for any time $t$, with $L^{2}$ norms bounded uniformly by a
square-integrable function of $t$.

Finally we estimate $IV$ as follows. Using the fact that
$P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}$ is bounded on $L^{2}$ we have

$\displaystyle\|IV\|_{L_{x}^{2}}$
$\displaystyle\lesssim_{m,\hat{q}}\sum_{k\geq\hat{q}}\|P_{\approx
k}\chi_{R}\|_{L^{\infty}}\|P_{\approx k}\theta_{\tau}\|_{L^{2}_{x}}$
$\displaystyle\lesssim_{m,\hat{q}}\sum_{k\geq\hat{q}}\|P_{\approx
k}\chi_{R}\|_{L^{\infty}}2^{(1-\delta)k}\|\Lambda^{-1+\delta}P_{\approx
k}\theta_{\tau}\|_{L^{2}_{x}}$
$\displaystyle\lesssim_{m,\hat{q}}\sum_{k\geq\hat{q}}2^{-3k}\|\nabla^{3}\chi_{R}\|_{L^{\infty}}2^{(1-\delta)k}\|\theta_{\tau}(t)\|_{\dot{H}^{-1+\delta}}$

Since $\nabla^{3}\chi_{R}\to 0$ uniformly, we have that $IV$ converges to $0$
in $L_{x}^{2}$ at any time $t$, with $L^{2}$ norms uniformly bounded by a
square-integrable function of $t$.

This analysis together with the dominated convergence theorem justifies the
claim (15). We now observe that, for each fixed value of $\hat{q},\tau,m$ and
$R$, we have

$\displaystyle-\frac{1}{2}\int\eta^{\prime}(t)\int$
$\displaystyle\left|P_{(-m,\hat{q}]}(\Lambda^{-1+\delta}\chi_{R}\theta(t,x))_{\tau}\right|^{2}dxdt=-\int\theta\partial_{t}\psi_{\hat{q},\tau,m,R}dxdt$
(16) $\displaystyle\psi_{\hat{q},\tau,m,R}$ $\displaystyle=\chi_{R}\left(\eta
P_{(-m,\hat{q}]}^{2}\Lambda^{2(-1+\delta)}(\chi_{R}\theta_{\tau})\right)_{\tau}.$
(17)

using the self-adjointness of both the mollification operators involved and of
multiplication by $\chi_{R}$. Note that $\psi_{\hat{q},\tau,m,R}\in
C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{2})$ is a smooth test function with
compact support, which enables us to apply the definition of a weak solution
to write the right hand side as

$\displaystyle\eqref{eq:phewLimit}$ $\displaystyle=\int
B(\psi_{\hat{q},\tau,m,R}(t),\theta(t),\theta(t))dt.$

We now aim to take the limits as $R\to\infty,m\to\infty$ and $\tau\to\infty$
in that order. We first claim that

$\displaystyle\psi_{\hat{q},\tau,m,R}\to\psi_{\hat{q},\tau,m}\coloneqq\left(\eta
P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}(\theta_{\tau})\right)_{\tau}\qquad\mbox{
in }L_{t}^{3}\dot{W}^{2,\infty},$ (18)

which is sufficient given that $\theta(t)\in L_{t}^{3}\dot{H}^{-1+\delta}$ and
the $\dot{W}^{2,\infty}\times\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$
boundedness of $B$.

To take this limit, first observe that the convergence

$\displaystyle\eta
P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}(\chi_{R}\theta_{\tau})\to\eta
P_{(-m,\hat{q}]}\Lambda^{2(-1+\delta)}(\theta_{\tau})$ (19)

holds in $L_{t}^{3}L_{x}^{2}$ by the same argument used to take the
$R\to\infty$ limit in (15). The convergence also holds in the
$L_{t}^{3}C_{x}^{2}$ norm by the boundedness of the frequency localization
operator $P_{(-m,\hat{q}]}=P_{(-m-2,\hat{q}+2]}P_{(-m,\hat{q}]}$.

Recalling that the $\dot{W}^{2,\infty}$ semi-norm is
$\|\nabla^{2}f\|_{L^{\infty}}$, it is not hard to check that the second
mollification in time preserves the $L_{t}^{3}\dot{W}^{2,\infty}$ convergence,
as does multiplying by $\chi_{R}$ and letting $R\to\infty$. This observation
suffices to take the $R\to\infty$ limit in (18).

We now study the $m\to\infty$ limit where we re-insert the low frequencies.
Observe that, by the Berenstein inequality,

$\displaystyle\|\psi_{\hat{q},\tau,m}(t)-\psi_{\hat{q},\tau}(t)\|_{\dot{W}^{2,\infty}}$
$\displaystyle=\|\nabla^{2}P_{\leq-m}\Lambda^{-1+\delta}\Lambda^{-1+\delta}\theta_{\tau}\|_{L^{\infty}}$
$\displaystyle\lesssim
2^{-2m/2}\|\nabla^{2}P_{\leq-m}\Lambda^{-1+\delta}\Lambda^{-1+\delta}\theta_{\tau}\|_{L^{2}}$
$\displaystyle\lesssim
2^{-m}\|\nabla^{2}P_{\leq-m}\Lambda^{-1+\delta}\|_{L^{2}\to
L^{2}}\|\Lambda^{-1+\delta}\theta_{\tau}\|_{L^{2}}$ $\displaystyle\lesssim
2^{-m}2^{-(1+\delta)m}\|\Lambda^{-1+\delta}\theta_{\tau}(t)\|_{L^{2}}$

The bound for $\|\nabla^{2}P_{\leq-m}\Lambda^{-1+\delta}\|_{L^{2}\to L^{2}}$
used in the last line can be shown by a dyadic decomposition and summing a
geometric series. This estimate and an application of the dominated
convergence theorem suffice for the $L_{t}^{3}\dot{W}^{2,\infty}$ convergence.
(In the case $\delta=0$ of $2D$ Euler, we employ the bound
$\|\Lambda^{-1}\theta_{\tau}(t)\|_{L^{2}}\lesssim\|v^{i}\|_{L^{2}}\approx\|\theta_{\tau}(t)\|_{\dot{H}^{-1}}$,
where $v$ is the Helmholtz solution to $\nabla_{i}v^{i}=\theta_{\tau}$, and we
use that $\Lambda^{-1}\nabla_{i}$ is $L^{2}$ bounded. See Appendix Section
A.2.)

Next we claim that the convergence

$\displaystyle\psi_{\hat{q},\tau}=\left(\eta
P_{\leq\hat{q}}^{2}\Lambda^{2(-1+\delta)}(\theta_{\tau})\right)_{\tau}\to\psi_{\hat{q}}=\eta
P_{\leq\hat{q}}^{2}\Lambda^{2(-1+\delta)}\theta\quad\mbox{in
}L_{t}^{3}\dot{W}^{2,\infty}\mbox{ as }\tau\to 0$

for $\theta\in L_{t}^{3}\dot{H}^{-1+\delta}$. The proof follows a similar line
of reasoning as in Section 2, but we now use the fact that
$\nabla^{2}P_{\leq\hat{q}}^{2}\Lambda^{-1+\delta}$ is bounded from
$L_{t}^{3}L^{2}_{x}(I\times\mathbb{R}^{2})$ to
$L_{t}^{3}L^{\infty}_{x}(I\times\mathbb{R}^{2})$, which again can be proved by
a dyadic decomposition.

We now proceed with the proof of Hamiltonian conservation, which is primarily
a high frequency analysis.

##  4 Conservation of the Hamiltonian

We now return to the setting of $\mathbb{T}^{d}$ periodic solutions and
explain the modifications needed in the nonperiodic setting in Section 4.1.

By the convention laid out in Section 1.2, $P_{\leq q}$ has a multiplier that
equals $1$ for $|\xi|\leq 2^{q-1}$ and is supported in $|\xi|<2^{q}$.
Consequently, $P_{q}\equiv P_{\leq q+1}-P_{\leq q}$ has a multiplier supported
in $2^{q-1}\leq|\xi|\leq 2^{q+1}$.

Let $\theta_{N}=P_{\leq N}\theta$ and
$\psi_{\hat{q}}\equiv\Delta^{-1+\delta}P_{\leq\hat{q}}^{2}\theta$ (recall that
$\Delta=\Lambda^{2}$). By the boundedness of $B$ on
$\dot{W}^{2,\infty}\times\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$, and by
dominated convergence, we have that

$\displaystyle\int
B(\psi_{\hat{q}}(t),\theta(t),\theta(t))dt=\lim_{N\to\infty}\int
B(\psi_{\hat{q}}(t),\theta_{N}(t),\theta_{N}(t))dt$ (20)

However, the right hand side (at least in the case of the torus) is a
classical integral. Specifically, we are looking at

$\displaystyle B_{\hat{q}N}=B(\psi_{\hat{q}};\theta_{N},\theta_{N})$
$\displaystyle\equiv\int\eta(t)\int
T^{\ell}(\theta_{N})\theta_{N}\nabla_{\ell}\psi_{\hat{q}}dxdt$ (21)
$\displaystyle=\int\eta(t)\int\nabla_{\ell}(-\Delta)^{-1+\delta}P_{\leq\hat{q}}^{2}\theta
P_{\leq N}\theta\epsilon^{\ell a}\nabla_{a}(-\Delta)^{-1+\delta}P_{\leq
N}\theta dxdt$ (22)

We can assume that the mean value $\Pi_{0}\theta$ of $\theta_{N}$ is zero, for
in general we may write $\theta=\Pi_{0}\theta+\Pi_{\neq 0}\theta$, where
$T^{\ell}(\Pi_{0}\theta)=0$ (since the multiplier is odd), and

$\displaystyle B(\psi_{\hat{q}};\theta_{N},\theta_{N})$
$\displaystyle=\int\eta(t)\int T^{\ell}(\Pi_{\neq
0}\theta_{N})\Pi_{0}\theta_{N}\nabla_{\ell}\psi_{\hat{q}}dxdt+\int\eta(t)\int
T^{\ell}(\Pi_{\neq 0}\theta_{N})\Pi_{\neq
0}\theta_{N}\nabla_{\ell}\psi_{\hat{q}}dxdt.$ (23)

The first integral vanishes as $\Pi_{0}\theta_{N}$ is constant,
$T^{\ell}(\Pi_{\neq 0}\theta_{N})$ is divergence free, and there is no
boundary.

The part of (22) from $\epsilon^{\ell
a}\nabla_{\ell}(-\Delta)^{-1+\delta}P_{\leq\hat{q}}^{2}\theta\nabla_{a}(-\Delta)^{-1+\delta}P_{\leq\hat{q}}^{2}\theta$
also vanishes, leaving us with

$\displaystyle B(\psi_{\hat{q}};\theta_{N},\theta_{N})$
$\displaystyle=\int\eta(t)\int\nabla_{\ell}(-\Delta)^{-1+\delta}P_{\leq\hat{q}}^{2}\theta
P_{\leq N}\theta T_{>\hat{q}}^{\ell}(P_{\leq N}\theta)dxdt$ (24)
$\displaystyle T_{>\hat{q}}^{\ell}(f)$
$\displaystyle\equiv(1-P_{\leq\hat{q}}^{2})T^{\ell}f=(1-P_{\leq\hat{q}}^{2})\epsilon^{\ell
a}\nabla_{a}(-\Delta)^{-1+\delta}f$

We remark here that there is an additional complication related to low
frequencies in the case of $\mathbb{R}^{2}$. We address this complication in
Section 4.1.

Since $T_{>\hat{q}}$ restricts to frequencies above $2^{\hat{q}-1}$, we have

$\displaystyle
B_{\hat{q}N}=\sum_{q=\hat{q}-2}^{N}\delta_{q}B_{\hat{q}N},\quad\delta_{q}B_{\hat{q}N}=\int\eta(t)\int\nabla_{\ell}\psi_{\hat{q}}(P_{\leq
q+1}\theta T_{>\hat{q}}^{\ell}P_{\leq q+1}\theta-P_{\leq q}\theta
T_{>\hat{q}}^{\ell}P_{\leq q}\theta)dxdt.$

Indeed, $P_{\leq q}\theta T_{>\hat{q}}^{\ell}P_{\leq q}\theta=0$ for
$q<\hat{q}-2$. Now decompose
$\delta_{q}B_{\hat{q}N}=\delta_{qH1}B_{\hat{q}N}+\delta_{qH2}B_{\hat{q}N}+\delta_{qL}B_{\hat{q}N}$,

$\displaystyle\delta_{qH1}B_{\hat{q}N}$
$\displaystyle=\int\eta(t)\int\nabla_{\ell}\psi_{\hat{q}}\left(P_{q+1}\theta
T_{>\hat{q}}^{\ell}P_{q+1}\theta\right)dxdt$ (25)
$\displaystyle\delta_{qH2}B_{\hat{q}N}$
$\displaystyle=\int\eta(t)\int\nabla_{\ell}\psi_{\hat{q}}\left(P_{q+1}\theta
T_{>\hat{q}}^{\ell}P_{q}\theta+P_{q}\theta
T_{>\hat{q}}^{\ell}P_{q+1}\theta\right)dxdt$ (26)
$\displaystyle\delta_{qL}B_{\hat{q}N}$
$\displaystyle=\int\eta(t)\int\nabla_{\ell}\psi_{\hat{q}}\underbrace{\left(P_{q+1}\theta
T_{>\hat{q}}^{\ell}P_{\leq q-1}\theta+P_{\leq q-1}\theta
T_{>\hat{q}}^{\ell}P_{q+1}\theta\right)}_{Q_{qL}(\theta)}dxdt$ (27)

Observe that, since
${\mathcal{F}}\mbox{supp}\,P_{q+1}\theta\subseteq\\{2^{q}\leq|\xi|\leq
2^{q+2}\\},{\mathcal{F}}\mbox{supp}\,P_{\leq
q-1}\theta\subseteq\\{|\xi|<2^{q-1}\\}$, we have

$\displaystyle{\mathcal{F}}\mbox{supp}\,\,Q_{qL}(\theta)$
$\displaystyle\subseteq\\{2^{q-1}\leq|\xi|\leq 2^{q+3}\\}$ (28)

In particular, $\delta_{qL}B_{\hat{q}N}=0$ for $q>\hat{q}+3$ since
${\mathcal{F}}\mbox{supp}\,\psi_{\hat{q}}\subseteq\\{|\xi|<2^{\hat{q}+1}\\}$.

To estimate $\delta_{qL}B_{\hat{q}N}$, write
$Q_{qL}(\theta)=\nabla_{j}\nabla^{j}\Delta^{-1}Q_{qL}(\theta)$ and integrate
by parts

$\displaystyle\delta_{qL}B_{\hat{q}N}$
$\displaystyle=-\int\eta(t)\int\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}\Delta^{-1}\nabla^{j}Q_{qL}(\theta)dxdt.$
(29)

Let $t$ be a time for which $\Lambda^{-1+\delta}\theta$ belongs to
$B_{3,c(\mathbb{N})}^{\alpha}$. Then

$\displaystyle\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L^{3}_{x}}$
$\displaystyle=\|\nabla_{j}\nabla_{\ell}(-\Delta)^{-1+\delta}P_{\leq\hat{q}}^{2}\theta(t)\|_{L_{x}^{3}}\lesssim\sum_{q=-\infty}^{\hat{q}+1}\|\nabla_{j}\nabla_{\ell}\Lambda^{-1+\delta}P_{q}P_{\leq\hat{q}}^{2}\Lambda^{-1+\delta}\theta\|_{L_{x}^{3}}$
$\displaystyle\lesssim\sum_{q=-\infty}^{\hat{q}+1}\|\nabla_{j}\nabla_{\ell}\Lambda^{-1+\delta}P_{\approx
q}\|\cdot\|P_{q}\Lambda^{-1+\delta}\theta\|_{L_{x}^{3}}$

Here we recall that we write $\|C\ast\|\equiv\|C\|_{L^{1}(\mathbb{R}^{d})}$
for a convolution operator $C\ast$, and that $P_{\approx q}$ has a Fourier
multiplier that is a smooth bump function adapted to $|\xi|\sim 2^{q}$. Thus

$\displaystyle\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L^{3}_{x}}$
$\displaystyle\lesssim\sum_{q=-\infty}^{\hat{q}+1}2^{(2-1+\delta)q}2^{-\alpha
q}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}\lesssim
2^{(1+\delta-\alpha)\hat{q}}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}.$
(30)

since $1+\delta-\alpha>0$ for $\alpha\leq(1+\delta)/3$. Furthermore, for any
$t$ such that $\Lambda^{-1+\delta}\theta\in B_{3,c(\mathbb{N})}^{\alpha}$, we
have

$\displaystyle\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L^{3}_{x}}$
$\displaystyle=\|\nabla_{j}\nabla_{\ell}(-\Delta)^{-1+\delta}P_{\leq\hat{q}}^{2}\theta(t)\|_{L_{x}^{3}}=o(2^{(1+\delta-\alpha)\hat{q}})\qquad\mbox{
as }\hat{q}\to\infty$ (31)

by a simple approximation argument. The key points are the uniform in
$\hat{q}$ boundedness of the operators
$2^{-(1+\delta-\alpha)\hat{q}}\nabla_{j}\nabla_{\ell}\Lambda^{-1+\delta}$ on
$B_{3,\infty}^{\alpha}$, and the fact that $\Lambda^{-1+\delta}\theta(t)$ is
the strong limit in $B_{3,\infty}^{\alpha}$ of $\Lambda^{-1+\delta}f_{(k)}$,
where the $f_{(k)}$ are smooth and satisfy an $O(1)$ estimate as
$\hat{q}\to\infty$.

Meanwhile, since we can take $\theta$ to have zero mean, we have $P_{\leq
q}\theta=\sum_{r<q}\Lambda^{1-\delta}P_{r}\Lambda^{-1+\delta}\theta$. We
obtain using $1-\delta-\alpha\geq 2/3-4\delta/3>0$ for $\delta<1/2$

$\displaystyle\|P_{\leq q}\theta(t)\|_{L^{3}}$
$\displaystyle\lesssim\sum_{r<q}\|\Lambda^{1-\delta}P_{\approx r}\|2^{-\alpha
r}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}\lesssim
2^{(1-\delta-\alpha)q}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}.$

In the case of SQG where $\delta=1/2$, $\alpha=(1+\delta)/3=1/2$ at the
critical exponent, we have $1-\delta-\alpha=0$ and the above estimate breaks
down. Instead, in this case, we require $\theta(t)\in L_{x}^{3}$ to bound
$P_{\leq\hat{q}}\theta(t)$ uniformly in $L^{3}$ by
$\lesssim\|\theta(t)\|_{L^{3}_{x}}$.

We now use the Fourier support of $Q_{qL}$ in (28), the fact that
$\delta_{qL}B_{\hat{q}N}=0$ unless $\hat{q}-2\leq q\leq\hat{q}+3$, and Hölder
to obtain

$\displaystyle\|(-\Delta)^{-1}\nabla^{j}Q_{qL}(\theta)\|_{L^{3/2}_{x}}$
$\displaystyle\lesssim 2^{-q}\|Q_{qL}\|_{L^{3/2}_{x}}\lesssim
2^{-q}\|P_{q}\theta\|_{L_{x}^{3}}\|P_{\approx q}T_{>\hat{q}}\|\|P_{\leq
q-1}\theta(t)\|_{L_{x}^{3}}$ $\displaystyle\lesssim 2^{-q}\|P_{\approx
q}\Lambda^{1-\delta}\|\,\|P_{q}\Lambda^{-1+\delta}\theta(t)\|\,\|P_{\approx
q}\nabla(-\Delta)^{-1+\delta}\|\,\|P_{\leq q-1}\theta(t)\|_{L_{x}^{3}}$
$\displaystyle\lesssim
2^{(-1+(1-\delta)-\alpha+(2\delta-1)+(1-\delta-\alpha))q}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}^{2}$
$\displaystyle\|(-\Delta)^{-1}\nabla^{j}Q_{qL}(\theta)\|_{L^{3/2}_{x}}$
$\displaystyle\lesssim 2^{-2\alpha
q}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}^{2}.$ (32)

Substituting into (29) leaves us with

$\displaystyle\sup_{N}\sum_{q=\hat{q}-2}^{N}|\delta_{qL}B_{\hat{q}N}|$
$\displaystyle\lesssim\int\eta(t)\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L_{x}^{3}}2^{-2\alpha\hat{q}}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}^{2}dt.$
(33)

(In the SQG case, we simply bound $\|P_{q}\theta(t)\|_{L_{x}^{3}},\|P_{\leq
q-1}\theta(t)\|_{L_{x}^{3}}\lesssim\|\theta(t)\|_{L_{x}^{3}}$ in deriving (32)
to obtain $\|\theta(t)\|_{L_{x}^{3}}^{2}$ in place of
$\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}^{2}$ on the
right hand side of (33).)

We now proceed to establish the same bound for (25) and (26). In fact we will
only consider (26) since (25) can be treated identically. Consider the
bilinear form appearing in (26)

$\displaystyle Q_{q}^{\ell}[\theta,\theta]=P_{q+1}\theta
T_{>\hat{q}}^{\ell}P_{q}\theta+P_{q}\theta
T_{>\hat{q}}^{\ell}P_{q+1}\theta=Q_{q}^{\ell}[P_{\approx q}\theta,P_{\approx
q}\theta]$

We suppress the dependence of $Q_{q}$ on $\hat{q}$ to ease notation. In
Section 5 below we derive the following divergence form for $Q_{q}$

$\displaystyle Q_{q}^{\ell}[\theta,\theta]$
$\displaystyle=\nabla_{j}B_{q}^{j\ell}[P_{\approx q}\theta,P_{\approx
q}\theta]$ (34) $\displaystyle B_{q}^{j\ell}[f,f]$
$\displaystyle=\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}f(t,x-h_{1})f(t,x-h_{2})\widetilde{K}_{q}^{j\ell}(h_{1},h_{2})dh_{1}dh_{2}$
$\displaystyle\|\widetilde{K}_{q}\|_{L^{1}(\mathbb{R}^{2}\times\mathbb{R}^{2})}$
$\displaystyle\lesssim
2^{2(-1+\delta)q},\quad\|B_{q}^{j\ell}[f,f]\|_{L_{x}^{3/2}}\leq\|\widetilde{K}_{q}\|_{L^{1}(\mathbb{R}^{2}\times\mathbb{R}^{2})}\|f\|_{L_{x}^{3}}^{2},$
(35)

where again we suppress the dependence of $B_{q}$ and $K_{q}$ on $\hat{q}$.
The last estimate follows by Hölder and Fubini using the duality between
$L^{3/2}$ and $L^{3}$ together with the translation invariance of the $L^{3}$
norm.

Substituting (34) into (26), integrating by parts and using (35), we have

$\displaystyle\delta_{qH2}B_{\hat{q}N}$
$\displaystyle=-\int\eta(t)\int\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}B_{q}^{j\ell}[P_{\approx
q}\theta,P_{\approx q}\theta]dxdt$ $\displaystyle|\delta_{qH2}B_{\hat{q}N}|$
$\displaystyle\leq\int\eta(t)\,\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L_{x}^{3}}\|B_{q}^{j\ell}[P_{\approx
q}\theta,P_{\approx q}\theta]\|_{L_{x}^{3/2}}$
$\displaystyle\lesssim\int\eta(t)\,\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L_{x}^{3}}2^{2(-1+\delta)q}\|P_{\approx
q}\theta(t)\|_{L_{x}^{3}}^{2}dt$ (36)
$\displaystyle\lesssim\int\eta(t)\,\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L_{x}^{3}}\|P_{\approx
q}\Lambda^{-1+\delta}\theta(t)\|_{L_{x}^{3}}^{2}dt$
$\displaystyle\lesssim\int\eta(t)\,\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L_{x}^{3}}2^{-2\alpha
q}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}^{2}dt$
$\displaystyle\sup_{N}\sum_{q=\hat{q}-2}^{N}|\delta_{qH2}B_{\hat{q}N}|$
$\displaystyle\lesssim\int\eta(t)\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L_{x}^{3}}2^{-2\alpha\hat{q}}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}^{2}dt$
(37)

In the SQG case ($\delta=\alpha=1/2$), we replace
$\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}$ with
$\|\theta(t)\|_{L_{x}^{3}}$, in which case the bound follows from (36).

We similarly obtain $\sup_{N}\sum_{q=\hat{q}-2}^{N}|\delta_{qH1}B_{\hat{q}N}|$
bounded by the right hand side of (37), which, in view of (33), means that our
bound for (22) is

$\displaystyle|B(\psi_{\hat{q}};\theta,\theta)|=\lim_{N}|B(\psi_{\hat{q}};\theta_{N},\theta_{N})|\leq\int\eta(t)\|\nabla_{j}\nabla_{\ell}\psi_{\hat{q}}(t)\|_{L_{x}^{3}}2^{-2\alpha\hat{q}}\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}^{2}dt.$

Using (30), the $dt$ integrand on the right hand side is bounded by
$\eta(t)\|\Lambda^{-1+\delta}\theta(t)\|_{\dot{B}_{3,\infty}^{\alpha}}^{3}$
(or bounded by $\eta(t)\|\theta(t)\|_{L_{x}^{3}}^{3}$ in the SQG case), which
is $L^{1}$ in time since $\Lambda^{-1+\delta}\theta(t)\in
L_{t}^{3}\dot{B}_{3,c(\mathbb{N})}^{\alpha}$. Moreover, the little-Oh bound
(31) shows that the $dt$ integrand goes to $0$ for almost every $t$ as
$\hat{q}\to\infty$. By the dominated convergence theorem, we conclude that
$|B(\psi_{\hat{q}};\theta,\theta)|\to 0$ as $\hat{q}\to\infty$.

###  4.1 Hamiltonian conservation in the Euclidean case

In the Euclidean case we replace $\theta_{N}=P_{\leq N}\theta$ by
$\theta_{N}=P_{[-N,N]}\theta\equiv(P_{\leq N}-P_{\leq-N})\theta$. Then
equation (20) still holds, but it is not quite a classical integral. Rather,
for $P_{[-N,\hat{q}]}\theta\equiv(P_{\leq\hat{q}}^{2}-P_{\leq-N})\theta$ we
have

$\displaystyle B_{\hat{q}N}=B(\psi_{\hat{q}};\theta_{N},\theta_{N})$
$\displaystyle\equiv\int\eta(t)\int
T^{\ell}(P_{[-N,N]}\theta)P_{[-N,N]}\theta\nabla_{\ell}\Delta^{-1+\delta}P_{[-N,\hat{q}]}\theta
dxdt$ (38)
$\displaystyle+\int\eta(t)B(P_{\leq-N}\Delta^{-1+\delta}\theta(t),P_{[-N,N]}\theta(t))dt$
(39)

The main point is that the term (39) is an error term, for $\theta\in
L_{t}^{3}\dot{H}^{-1+\delta}$. In fact,

$\displaystyle|B(P_{\leq-N}\Delta^{-1+\delta}\theta,P_{[-N,N]}\theta)|$
$\displaystyle\lesssim\|\nabla^{2}P_{\leq-N}\Delta^{-1+\delta}\theta\|_{L^{\infty}}\|P_{[-N,N]}\theta\|_{\dot{H}^{-1+\delta}}^{2}$
$\displaystyle\lesssim\|P_{\leq-N}\nabla\Lambda^{\delta}\|_{L^{2}\to
L^{\infty}}\|\nabla\Lambda^{-2+\delta}\theta\|_{L^{2}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$
$\displaystyle\lesssim o(1)\cdot\|\theta(t)\|_{\dot{H}^{-1+\delta}}^{3}$

The remainder of the proof in the Euclidean case, which involves estimating
(38), follows along the lines of the torus case but with $-N$ as the lower
bound for the frequency sums. Specifically, after observing the key
cancellation, the quantity that must be estimated is

$\displaystyle\int\eta(t)\int\nabla_{\ell}(-\Delta)^{-1+\delta}P_{[-N,\hat{q}]}\theta
P_{[-N,N]}\theta T_{>\hat{q}}^{\ell}(P_{[-N,N]}\theta)dxdt,$

and one takes $P_{[-N,q]}\theta\equiv(P_{\leq q}-P_{\leq-N})\theta$ in place
of $P_{\leq q}\theta$ during the analysis.

##  5 Deriving the bilinear form

In this section, we include the derivation of the bilinear form (34) and the
bound (35). We follow [26], which in turn generalizes a computation of [3]. We
only consider the case where the domain is the torus since the nonperiodic
case is simpler.

Consider the bilinear form appearing in (26)

$\displaystyle Q_{q}^{\ell}[\theta,\theta]$ $\displaystyle=P_{q+1}\theta
T_{>\hat{q}}^{\ell}P_{q}\theta+P_{q}\theta
T_{>\hat{q}}^{\ell}P_{q+1}\theta=Q_{q}^{\ell}[P_{\approx q}\theta,P_{\approx
q}\theta]$ (40) $\displaystyle\widehat{T_{>\hat{q}}^{\ell}f}(\xi)$
$\displaystyle=m_{\hat{q}}^{\ell}(\xi)\hat{f}(\xi)=(1-\hat{\eta}_{\leq\hat{q}}^{2}(\xi))\epsilon^{\ell
a}(i\xi_{a})|\xi|^{2(-1+\delta)}\hat{f}(\xi).$

The most crucial point is the anti-symmetry of the operator $T$, reflected in
the fact that $m_{\hat{q}}^{\ell}$ is odd. Without this feature, (40) may not
have integral $0$ and an anti-divergence would not exist. The idea to obtain
the divergence form will be to Taylor expand in frequency space. Another
example of Taylor expansion in frequency space to derive an anti-divergence
operator can be found in [28].

We start by approximating the periodic function $\theta$ by Schwartz
functions. By mollifying in frequency space, we may assume that the Schwartz
approximations to $P_{q}\theta$ and $P_{q+1}\theta$ have frequency support in
$\\{2^{q-2}\leq|\xi|<2^{q+2}\\}$, and that they converge pointwise in physical
space while also remaining uniformly bounded there. By abuse of notation,
assume now that $P_{q}\theta$ and $P_{q+1}\theta$ are the Schwartz
approximations. For these approximations we construct a bilinear anti-
divergence as follows.

$\displaystyle\hat{Q}_{q}^{\ell}(\xi)$
$\displaystyle=\int_{\hat{\mathbb{R}}^{2}}m_{\hat{q}}^{\ell}(\xi-\eta)\widehat{P_{q+1}\theta}(\xi-\eta)\widehat{P_{q}\theta}(\eta)+\widehat{P_{q+1}\theta}(\xi-\eta)m_{\hat{q}}^{\ell}(\eta)\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}$
$\displaystyle=\int_{\hat{\mathbb{R}}^{2}}m_{\hat{q}}^{\ell}(\xi-\eta)\hat{\chi}_{q}\widehat{P_{q+1}\theta}(\xi-\eta)\widehat{P_{q}\theta}(\eta)+\widehat{P_{q+1}\theta}(\xi-\eta)m_{\hat{q}}^{\ell}(\eta)\hat{\chi}_{q}\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}$
$\displaystyle\text{Define }m_{q}(\cdot)\coloneqq
m_{\hat{q}}(\cdot)\hat{\chi}_{q}(\cdot)$
$\displaystyle=\int_{\hat{\mathbb{R}}^{2}}\left[m^{\ell}_{q}(\xi-\eta)+m^{\ell}_{q}(\eta)\right]\widehat{P_{q+1}\theta}(\xi-\eta)\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}$
$\displaystyle\text{By the oddness of the multiplier }m^{\ell}_{q}$
$\displaystyle=\int_{\hat{\mathbb{R}}^{2}}\left[m^{\ell}_{q}(\xi-\eta)-m^{\ell}_{q}(-\eta)\right]\widehat{P_{q+1}\theta}(\xi-\eta)\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}$
$\displaystyle=i\xi_{j}\int_{\hat{\mathbb{R}}^{2}}\int_{0}^{1}-i\nabla^{j}m^{\ell}_{q}(u_{\sigma})d\sigma\widehat{P_{q+1}\theta}(\zeta)\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}$
$\displaystyle\hat{Q}_{q}^{\ell}(\xi)$ $\displaystyle\eqqcolon
i\xi_{j}\int_{\hat{\mathbb{R}}^{2}}\hat{K}^{j\ell}_{q}(\zeta,\eta)\widehat{P_{q+1}\theta}(\zeta)\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}},$
(41)

where we have set

$\zeta\coloneqq\xi-\eta,\quad
u_{\sigma}\coloneqq\sigma(\xi-\eta)-(1-\sigma)\eta=\sigma\zeta-(1-\sigma)\eta,$

and we have used Taylor’s Remainder Theorem in the last equality. In the last
line we defined

$\hat{K}^{j\ell}_{q}(\zeta,\eta)\coloneqq\int_{0}^{1}-i\nabla^{j}m^{\ell}_{q}(u_{\sigma})\hat{\chi}_{q}(\zeta)\hat{\chi}_{q}(\eta)d\sigma.$
(42)

Here we suppress the dependence of $K_{q}^{j\ell}$ on $\hat{q}$. For
$q>\hat{q}+1$, the cutoff $(1-\hat{\eta}_{\leq\hat{q}}^{2})$ in $m_{q}$ is $1$
on the support of $\hat{\chi}_{q}$, and the homogeneity of $m^{\ell}$ leads to
the scaling
$\widehat{K}_{q}^{j\ell}(\zeta,\eta)=2^{2(-1+\delta)q}f^{j\ell}(2^{-q}\zeta,2^{-q}\eta)$,
where $f^{j\ell}$ is Schwartz, which in turn implies the bound

$\displaystyle\|K_{q}\|_{L^{1}(\mathbb{R}^{2}\times\mathbb{R}^{2})}$
$\displaystyle\lesssim 2^{2(-1+\delta)q}$ (43)

for $q>\hat{q}+1$. The same bound also holds for $\hat{q}-1\leq
q\leq\hat{q}+1$ by scaling; for example
$K_{\hat{q}-1}^{j\ell}(\zeta,\eta)=2^{2(-1+\delta)\hat{q}}f_{0}^{j\ell}(2^{-\hat{q}}\zeta,2^{-\hat{q}}\eta)$,
for some Schwartz function $f_{0}$. Finally, $K_{q}=0$ for $q<\hat{q}-1$.

Take the inverse Fourier Transform of (41) to obtain

$\displaystyle Q_{q}^{\ell}(x)$
$\displaystyle=\nabla_{j}\int_{\hat{\mathbb{R}}^{2}}e^{i\xi\cdot
x}\int_{\hat{\mathbb{R}}^{2}}\hat{K}^{j\ell}_{q}(\zeta,\eta)\widehat{P_{q+1}\theta}(\zeta)\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}\frac{d\xi}{(2\pi)^{2}}$
$\displaystyle=\nabla_{j}\int_{\hat{\mathbb{R}}^{2}\times\hat{\mathbb{R}}^{2}}e^{i(\zeta+\eta)\cdot
x}\hat{K}^{j\ell}_{q}(\zeta,\eta)\widehat{P_{q+1}\theta}(\zeta)\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}\frac{d\zeta}{(2\pi)^{2}}.$

Observe that $\hat{K}_{q}^{j\ell}$ is compactly supported in frequency and
smooth due to the factors of $\hat{\chi}_{\approx\lambda}$, making
$\hat{K}_{q}^{j\ell}$ a Schwartz function. Thus we can define the
corresponding physical space kernel $K^{j\ell}_{q}(h_{1},h_{2})$ as the
inverse Fourier Transform of $\hat{K}^{j\ell}_{q}(\zeta,\eta)$ so that

$\hat{K}^{j\ell}_{q}(\zeta,\eta)=\int_{\hat{\mathbb{R}}^{2}\times\hat{\mathbb{R}}^{2}}e^{-i(\zeta,\eta)\cdot(h_{1},h_{2})}K^{j\ell}_{q}(h_{1},h_{2})dh_{1}dh_{2}.$

Then

$\displaystyle Q_{q}^{\ell}(x)$
$\displaystyle=\nabla_{j}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}e^{i(\zeta+\eta)\cdot
x}\int_{\hat{\mathbb{R}}^{2}\times\hat{\mathbb{R}}^{2}}e^{-i(\zeta,\eta)\cdot(h_{1},h_{2})}K^{j\ell}_{q}(h_{1},h_{2})dh_{1}dh_{2}\widehat{P_{q+1}\theta}(\zeta)\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}\frac{d\zeta}{(2\pi)^{2}}$
$\displaystyle=\nabla_{j}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}K^{j\ell}_{q}(h_{1},h_{2})\int_{\hat{\mathbb{R}}^{2}}e^{i\zeta\cdot(x-h_{1})}\widehat{P_{q+1}\theta}(\zeta)\frac{d\zeta}{(2\pi)^{2}}\int_{\hat{\mathbb{R}}^{2}}e^{i\eta\cdot(x-h_{2})}\widehat{P_{q}\theta}(\eta)\frac{d\eta}{(2\pi)^{2}}dh_{1}dh_{2}$
$\displaystyle=\nabla_{j}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}K^{j\ell}_{q}(h_{1},h_{2})P_{q+1}\theta(x-h_{1})P_{q}\theta(x-h_{2})dh_{1}dh_{2}$
$\displaystyle\eqqcolon\nabla_{j}B^{j\ell}_{q}[\theta,\theta](x)$

Here we define the bilinear anti-divergence operator as

$\displaystyle B^{j\ell}_{q}[F_{1},F_{2}](x)$
$\displaystyle\coloneqq\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}K^{j\ell}_{q}(h_{1},h_{2})P_{q+1}F_{1}(x-h_{1})P_{q}F_{2}(x-h_{2})dh_{1}dh_{2},$
$\displaystyle B^{j\ell}_{q}[F_{1},F_{2}](x)$
$\displaystyle=\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}\widetilde{K}^{j\ell}_{q}(h_{1},h_{2})F_{1}(x-h_{1})F_{2}(x-h_{2})dh_{1}dh_{2}$
(44)

for scalar fields $F_{1},F_{2}\in C^{\infty}(\mathbb{T}^{2})$. Here we obtain
$\widetilde{K}_{q}=(P_{q+1}\otimes P_{q})K_{q}$ from $K_{q}$ by writing
$P_{q+1}F_{1}$ (and $P_{q}F_{2}$) as a convolution in the variable $h_{1}$
($h_{2}$ in the latter case) and we pass $P_{q}$ and $P_{q+1}$ onto the kernel
$K_{q}$ by a change of variables. This bi-convolution operator satisfies the
equality

$\displaystyle
T^{\ell}[P_{q+1}\theta]\nabla_{\ell}P_{q}\theta+T^{\ell}[P_{q}\theta]\nabla_{\ell}P_{q+1}\theta=\nabla_{j}B^{j\ell}_{q}[P_{\approx
q}\theta,P_{\approx q}\theta](x)$ (45)

for the Schwartz approximations to $P_{q+1}\theta,P_{q}\theta$ whose Fourier
supports lie in $\\{|\xi|\sim 2^{q}\\}$. Since
$\|\widetilde{K}_{q}\|_{L^{1}(\mathbb{R}^{2})}$ is finite and the Schwartz
approximations to $P_{\approx q}\theta$ are bounded and converge pointwise, we
may apply dominated convergence to pass to the limit in (44) with
$F_{1}=F_{2}\to P_{\approx q}\theta$ to conclude that (45) holds (both in
${\cal D}^{\prime}$ and classically) for the periodic functions
$P_{q+1}\theta,P_{q}\theta$. Thus we have shown (34). Finally, note that the
bound (43) for $K_{q}$ holds as well for $\widetilde{K}_{q}$, which
establishes (35).

Remark: For future reference, we note here the important property that $K_{q}$
is trace-free in the special case of the mSQG multiplier. To check that
$K_{q}^{j\ell}$ is trace-free, we use the fact that the frequency cutoff
$\chi_{q}(\xi)=\tilde{\chi}_{q}(|\xi|^{2})$ is radial and it suffices check
that $\nabla^{j}m_{q}^{\ell}(\xi)$ is trace-free:

$\displaystyle m_{q}^{\ell}(\xi)$ $\displaystyle=\epsilon^{\ell
a}i\xi_{a}|\xi|^{2(-1+\delta)}\chi_{q}(\xi)=\epsilon^{\ell
a}i\xi_{a}|\xi|^{2(-1+\delta)}\tilde{\chi}_{q}(|\xi|^{2})$
$\displaystyle\nabla^{j}m_{q}^{\ell}(\xi)$ $\displaystyle=i\epsilon^{\ell
j}|\xi|^{2(-1+\delta)}\chi_{q}(\xi)+\epsilon^{\ell
a}i\xi_{a}\xi^{j}F_{q,\delta}(|\xi|^{2}).$ (46)

Here $F_{q,\delta}$ is some radial smooth function adapted to
$|\xi|^{2}\sim(2^{q})^{2}$. As the above expression is trace-free
($\epsilon^{\ell j}$ is antisymmetric and $\epsilon^{\ell a}\xi_{a}$ is
perpendicular to $\xi$), this calculation confirms that $K_{q}^{j\ell}$ is
trace-free.

##  6 Defining the nonlinearity

For scalar functions $\psi$, $\theta\in C^{\infty}(\mathbb{T}^{2})$, we define
the form

$\displaystyle B[\psi,\theta]$
$\displaystyle=-\int\psi(x)T^{\ell}\theta(x)\nabla_{\ell}\theta(x)dx=\int\nabla_{\ell}\psi\theta
T^{\ell}\theta dx$ (47)

The main result in this section is the estimate

$\displaystyle|B[\psi,\theta]|$
$\displaystyle\lesssim_{\delta}\|\nabla^{2}\psi\|_{C^{0}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2},\qquad
0<\delta<\infty,$ (48)

where the constant depends on $\delta>0$. The same bound also holds for the
case of Euler ($\delta=0$), in which case the nonlinearity is given by
$B[\psi,\theta]=-\int\psi\cdot\epsilon_{a_{\ell}}\nabla^{a}\nabla_{j}(T^{j}\theta
T^{\ell}\theta)dx$, $\epsilon$ the Levi-Civita symbol and
$T^{\ell}\theta=-\epsilon^{\ell b}\nabla_{b}\Delta^{-1}\theta$.

The associated trilinear form
$T[\psi,\theta,\phi]\coloneqq\int\nabla_{\ell}\psi(\theta T^{\ell}\phi+\phi
T^{\ell}\theta)dx$, for which
$B[\psi,\theta]=\frac{1}{2}T[\psi,\theta,\theta]$ and
$T[\psi,\theta,\phi]=\frac{1}{4}(B[\psi,\theta+\phi]-B[\psi,\theta-\phi])$,
will consequently satisfy a bound

$\displaystyle|T[\psi,\theta,\phi]|\lesssim\|\nabla^{2}\psi\|_{C^{0}}\|\theta\|_{\dot{H}^{-1+\delta}}\|\phi\|_{\dot{H}^{-1+\delta}}.$
(49)

This bound on $T$ guarantees a unique continuous extension to $\theta\in
L_{t}^{2}\dot{H}^{-1+\delta}$ for the nonlinearity defining the SQG equation
by a Cauchy sequence argument. We now begin the proof of (48).

Note that when $\psi$ is a constant, $B[\psi,\theta]=0$. Thus we obtain the
following decomposition of paraproduct type.

$\displaystyle B[\psi,\theta]$ $\displaystyle=\sum_{k}\int
P_{k}\nabla_{\ell}\psi\theta\,T^{\ell}\theta dx=LL+HH+HL+LH$ (50)
$\displaystyle LL$ $\displaystyle=\sum_{k}\int P_{k}\nabla_{\ell}\psi P_{\leq
k-3}\theta\,T^{\ell}P_{\leq k-3}\theta dx=0$ (51) $\displaystyle HL$
$\displaystyle=\sum_{k}\sum_{q\geq k-3}\int P_{k}\nabla_{\ell}\psi
P_{q+1}\theta\,T^{\ell}P_{\leq q-1}\theta dx$ (52) $\displaystyle LH$
$\displaystyle=\sum_{k}\sum_{q\geq k-3}\int P_{k}\nabla_{\ell}\psi P_{\leq
q-1}\theta\,T^{\ell}P_{q+1}\theta dx$ (53) $\displaystyle HH$
$\displaystyle=\sum_{k}\sum_{q\geq k-3}\int
P_{k}\nabla_{\ell}\psi\underbrace{\left(P_{q}\theta\,T^{\ell}P_{q+1}\theta+P_{q+1}\theta\,T^{\ell}P_{q}\theta+P_{q+1}\theta\,T^{\ell}P_{q+1}\theta\right)}_{Q_{q}^{\ell}(\theta)}dx$
(54)

That $LL=0$ follows from Plancharel and the bound on the frequency support of
$P_{\leq k-3}\theta T^{\ell}P_{\leq k-3}$.

We now estimate $HH$. From Section 5, we have a divergence form for
$Q_{q}^{\ell}(\theta)=\nabla_{j}B_{q}^{j\ell}[P_{\approx q}\theta]$, where the
multiplier for $P_{\approx q}$ is a bump function adapted to
$\\{\xi\in\widehat{\mathbb{R}}^{2}~{}:~{}2^{q-4}\leq|\xi|\leq 2^{q+4}\\}$,

$\displaystyle B_{q}^{j\ell}[P_{\approx q}\theta](x)$
$\displaystyle=\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}K_{q}^{j\ell}(h_{1},h_{2})P_{\approx
q}\theta(x-h_{1})P_{\approx q}\theta(x-h_{2})dh_{1}dh_{2},$ (55)

and $\|K_{q}\|_{L^{1}(\mathbb{R}^{2}\times\mathbb{R}^{2})}\lesssim
2^{2(-1+\delta)q}$. (The $K_{q}$ is the sum of the kernels coming from the
interaction of $P_{q+1}\theta$ with itself, and the other coming from the
symmetric interaction of $P_{q+1}\theta$ with $P_{q}\theta$.) Plugging in the
anti-divergence, interchanging the order of summation (which is easily
justified for smooth $\psi$ and $\theta$), and integrating by parts yields

$\displaystyle-HH$ $\displaystyle=\sum_{q}\int\left(\sum_{k\leq
q-3}P_{k}\nabla_{j}\nabla_{\ell}\psi\right)B_{q}^{j\ell}[P_{\approx
q}\theta]dx$ (56) $\displaystyle|HH|$ $\displaystyle\leq\sum_{q}\|P_{\leq
q-3}\nabla^{2}\psi\|_{C^{0}}\|B_{q}[P_{\approx q}\theta]\|_{L^{1}}dx$
$\displaystyle|HH|$
$\displaystyle\leq\sup_{\ell}\|P_{\leq\ell}\nabla^{2}\psi\|_{C^{0}}\sum_{q}\|K_{q}\|_{L^{1}(\mathbb{R}^{2}\times\mathbb{R}^{2})}\|P_{\approx
q}\theta\|_{L^{2}(\mathbb{T}^{2})}^{2}$
$\displaystyle\lesssim\|\nabla^{2}\psi\|_{C^{0}}\sum_{q}\left(2^{(-1+\delta)q}\|P_{\approx
q}\theta\|_{L^{2}(\mathbb{T}^{2})}\right)^{2}$ $\displaystyle|HH|$
$\displaystyle\lesssim\|\nabla^{2}\psi\|_{C^{0}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$
(57)

Now we consider $HL$ and $LH$. Notice that
${\mathcal{F}}\mbox{supp}\,(P_{q+1}\theta\,T^{\ell}P_{\leq
q-1}\theta)\subseteq\\{|\xi|\geq 2^{q-1}\\}$ and similarly for the quadratic
term in $LH$. Thus the sum over $q$ extends only from $k-3$ to $k+2$. To
simplify the presentation, we treat the sum as though there is only one term
with $q=k$.

$\displaystyle HL$ $\displaystyle\sim\sum_{k}\int P_{k}\nabla_{\ell}\psi
P_{k+1}\theta\,T^{\ell}P_{\leq k-1}\theta dx,\qquad LH\sim\sum_{k}\int
P_{k}\nabla_{\ell}\psi P_{\leq k-1}\theta\,T^{\ell}P_{k+1}\theta dx$

We first treat $HL$. Recall from the remarks surrounding (23) that we may
assume that the zero Fourier mode of $\theta$ vanishes. Thus we decompose

$\displaystyle HL$ $\displaystyle\sim\sum_{k}\sum_{j\leq k-1}\int
P_{k}\nabla_{\ell}\psi P_{k+1}\theta T^{\ell}P_{j}\theta dx$ (58)

Using
$\|T^{\ell}P_{j}\theta\|_{L^{2}}=\|\nabla^{\ell}(-\Delta)^{-1+\delta}P_{j}\theta\|_{L^{2}}\lesssim
2^{(-1+2\delta)j}\|P_{j}\theta\|_{L^{2}}$, we obtain

$\displaystyle|HL|$ $\displaystyle\lesssim\sum_{k}\sum_{j\leq
k-1}\|P_{k}\nabla\psi\|_{L^{\infty}}\|P_{k+1}\theta\|_{L^{2}}2^{\delta
j}2^{(-1+\delta)j}\|P_{j}\theta\|_{L^{2}}$
$\displaystyle\lesssim\sum_{j}\sum_{k\geq
j+1}\|P_{k}\nabla\psi\|_{L^{\infty}}2^{(1-\delta)k}\|\Lambda^{-1+\delta}P_{k+1}\theta\|_{L^{2}}2^{\delta
j}\|\Lambda^{-1+\delta}P_{j}\theta\|_{L^{2}}$
$\displaystyle\lesssim\sum_{j}\sum_{k}\|P_{k}\nabla^{2}\psi\|_{L^{\infty}}2^{-\delta|j-k|}\|\Lambda^{-1+\delta}P_{k+1}\theta\|_{L^{2}}\|\Lambda^{-1+\delta}P_{j}\theta\|_{L^{2}}$
$\displaystyle\lesssim\sup_{\ell}\|P_{\ell}\nabla^{2}\psi\|_{L^{\infty}}\sum_{j,k}2^{-\delta|j-k|}\|\Lambda^{-1+\delta}P_{k+1}\theta\|_{L^{2}}\|\Lambda^{-1+\delta}P_{j}\theta\|_{L^{2}}$
$\displaystyle|HL|$
$\displaystyle\lesssim\sup_{\ell}\|P_{\ell}\nabla^{2}\psi\|_{L^{\infty}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$
(59)

upon applying the Schur test to the last term. Here we have used the standard
bound

$\displaystyle 2^{k}\|P_{k}\nabla_{\ell}\psi\|_{L^{\infty}}$
$\displaystyle=2^{k}\|\nabla^{j}\Delta^{-1}P_{\approx
k}P_{k}\nabla_{j}\nabla_{\ell}\psi\|_{L^{\infty}}\lesssim\|P_{k}\nabla^{2}\psi\|_{L^{\infty}}.$

Finally, the bound for $LH$ follows without interchanging the order of
summation

$\displaystyle LH$ $\displaystyle\sim\sum_{k}\sum_{j\leq k-1}\int
P_{k}\nabla_{\ell}\psi\,P_{j}\theta\,T^{\ell}P_{k+1}\theta dx$ (60)
$\displaystyle|LH|$ $\displaystyle\lesssim\sum_{k}\sum_{j\leq
k-1}\|P_{k}\nabla_{\ell}\psi\|_{L^{\infty}}\|P_{j}\theta\|_{L^{2}}2^{(-1+2\delta)k}\|P_{k+1}\theta\|_{L^{2}}$
$\displaystyle\lesssim\sum_{k}\sum_{j\leq
k-1}\|P_{k}\nabla_{\ell}\psi\|_{L^{\infty}}2^{(1-\delta)j}\|\Lambda^{-1+\delta}P_{j}\theta\|_{L^{2}}2^{\delta
k}\|\Lambda^{-1+\delta}P_{k+1}\theta\|_{L^{2}}$
$\displaystyle\leq\sum_{k}\sum_{j\leq
k-1}2^{k}\|P_{k}\nabla_{\ell}\psi\|_{L^{\infty}}2^{-\delta|k-j|}\|\Lambda^{-1+\delta}P_{j}\theta\|_{L^{2}}\|\Lambda^{-1+\delta}P_{k+1}\theta\|_{L^{2}}$
$\displaystyle|LH|$
$\displaystyle\lesssim\sup_{\ell}\|P_{\ell}\nabla^{2}\psi\|_{L^{\infty}}\sum_{j,k}2^{-\delta|k-j|}\|\Lambda^{-1+\delta}P_{j}\theta\|_{L^{2}}\|\Lambda^{-1+\delta}P_{k+1}\theta\|_{L^{2}}$
$\displaystyle|LH|$
$\displaystyle\lesssim\sup_{\ell}\|P_{\ell}\nabla^{2}\psi\|_{L^{\infty}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}.$
(61)

Again we have applied the Schur test. Combining (57), (59) and (61), we have
proven (48).

###  6.1 The nonperiodic case

We now explain how to extend the estimate (48) where $\mathbb{T}^{2}$ is
replaced by $\mathbb{R}^{2}$. In this case, we define

$\displaystyle B[\psi,\theta]$
$\displaystyle=-\int\psi(x)\nabla_{\ell}[T^{\ell}\theta(x)\theta(x)]dx=\int\nabla_{\ell}\psi\theta
T^{\ell}\theta dx$

for $\psi$ in the Schwartz class and for $\theta\in\SS_{0}$ in the class of
Schwartz functions with compact frequency support away from $0$. The
additional details involve handling the low frequencies.

For any integer $m\geq 0$, we have
$\psi(x)=P_{\leq-m}\psi+\sum_{k=-m+1}^{\infty}P_{k}\psi$, and the series
converges uniformly. Furthermore,
$\|P_{\leq-m}\nabla\psi\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim
2^{-2m}\|\nabla\psi\|_{L^{1}(\mathbb{R}^{2})}$,
$\|P_{k}\nabla\psi\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim
2^{2k}\|\nabla\psi\|_{L^{1}(\mathbb{R}^{2})}$, and
$\|P_{k}\nabla\psi\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim
2^{-k}\|\nabla^{2}\psi\|_{L^{\infty}(\mathbb{R}^{2})}$ together with $\theta
T^{\ell}\theta\in L^{1}$ for $\theta\in\SS_{0}$ allow us to obtain the
expansion (50) by letting $m\to\infty$ and applying dominated convergence and
Fubini-Tonelli to interchange the sum and the integral. Equations (51) through
(54) then follow routinely from $\int P_{k}\nabla_{\ell}\psi\theta
T^{\ell}\theta dx=\lim_{N\to\infty}\int P_{k}\nabla_{\ell}\psi P_{\leq
N}\theta T^{\ell}P_{\leq N}\theta dx$ once the absolute convergence of the
series has been obtained.

Bounding each term in (51) through (54) requires the decompositions (58) and
(60). In the case of $\mathbb{T}^{2}$, the summation over $j$ includes only
finitely terms, but it extends to all $j>-\infty$ on $\mathbb{R}^{2}$. Since
$P_{\leq k-1}=P_{\leq-m}+\sum_{j=-m+1}^{k-1}P_{j}$ for any $m$, the key point
is to verify the limit

$\displaystyle\lim_{m\to\infty}\int
P_{k}\nabla_{\ell}\psi(P_{k+1}\theta\,T^{\ell}P_{\leq-m}\theta
dx+P_{\leq-m}\theta\,T^{\ell}P_{k+1}\theta)dx=0$ (62)

for each fixed $k$, every Schwartz function $\psi$ and every
$\theta\in\SS_{0}$. This limit is clear by the compact frequency support of
$\theta$ away from the origin, since the left hand side is $0$ for
sufficiently large $m$.

###  6.2 Remark on generalized estimates of this type

Our technique for estimating the nonlinearity can be applied to a more general
class of nonlinearities that have the form $\int\psi\theta T[\theta]dx$. The
only properties of the bilinear form
$B(\psi,\theta)=\int\nabla_{\ell}\psi\theta T^{\ell}\theta dx$ that we have
used in the proof are the scaling and anti-symmetry of $T^{\ell}$, reflected
in the oddness and the homogeneity of the multiplier. For example, the bound

$\displaystyle\left|\int\psi_{\ell}\theta\nabla^{\ell}\Delta\theta dx\right|$
$\displaystyle\lesssim\|\nabla\psi\|_{L^{\infty}}\|\nabla\theta\|_{L^{2}}^{2},\qquad\psi_{\ell},\theta\in
C_{c}^{\infty}$ (63)

can be proven by a similar (nonlocal) argument based on the fact that the
symbol of $\nabla\Delta$ is odd and degree $3$ homogeneous.

It is natural to wonder whether the estimate (63) can be proven by an
integration by parts argument, but it appears that a nonlocal proof is
necessary. To see that an integration by parts argument would not apply, note
that the bound is false if we replace the $L^{2}(\mathbb{R}^{d})$ norms with
local $L^{2}(\mathrm{supp}\,\psi)$ norms. To see this failure of the localized
version of the estimate, consider a $\theta_{0}$ with $\nabla\Delta\theta_{0}$
nonzero, and choose $\psi_{\ell}=\chi\nabla_{\ell}\Delta\theta_{0}$ where
$\chi$ is a bump function supported where $\nabla\Delta\theta_{0}$ is nonzero.
Now set $\theta_{R}=\theta_{0}+R\tilde{\chi}$, where $R$ is large and
$\tilde{\chi}=1$ on the support of $\chi$. Then
$\|\nabla\theta_{R}\|_{L^{2}(\mathrm{supp}\,\psi)}=\|\nabla\theta_{0}\|_{L^{2}(\mathrm{supp}\,\psi)}$
does not depend on $R$, but the left hand side of (63) becomes unbounded as
$R$ gets large.

These considerations also motivate why our definition of the space
$\dot{H}^{-1+\delta}$ requires that we exclude polynomials. After all the
estimate $|\int\psi_{\ell}\theta\nabla^{\ell}\Delta\theta
dx|\lesssim\|\nabla\psi\|_{L^{\infty}}\|\theta\|_{\dot{H}^{1}}^{2}$ would be
false for high degree polynomials for which the right hand side would vanish.

##  7 Continuous extension of the operator

We now have the homogeneous estimate

$\displaystyle|B[\psi,\theta]|\lesssim_{\delta}\|\nabla^{2}\psi\|_{L^{\infty}}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$
(64)

for $\psi,\theta$ Schwartz, and this bound extends by continuity for any
Schwartz $\psi$ and any $\theta\in\dot{H}^{-1+\delta}$ using the corresponding
bound (49) for the trilinear form and a density and Cauchy sequence argument.
The completion of the Schwartz space under the semi-norm
$\|\nabla^{2}\psi\|_{L^{\infty}}$ is the space $\dot{C}_{0}^{2}$ of $C^{2}$
functions with bounded second derivatives that satisfy
$|\nabla^{2}\psi(x)|=o(1)$ as $|x|\to\infty$ (see Appendix Section A). The
more delicate matter is proving the continuous extension with respect to
weak-* convergence, which is the subject of the next section.

###  7.1 Extension to $\dot{W}^{2,\infty}$: Definition, boundedness and
uniqueness

Now let $\dot{W}^{2,\infty}(\mathbb{R}^{2})$ and
$\dot{W}^{2,\infty}(\mathbb{T}^{2})$ denote the space of distributions $\Psi$
such that $\nabla^{2}\Psi\in L^{\infty}$. This space includes functions (such
as quadratic polynomials) for which there is no sequence
$\psi_{n}\in\dot{C}_{0}^{2}$ satisfying $\nabla^{2}\psi_{n}\to\nabla^{2}\Psi$
in $L^{\infty}$, as the latter convergence would imply
$\Psi\in\dot{C}_{0}^{2}$. Consequently, our estimate (64) on
$|B[\psi,\theta]|$ does not on its own provide a unique continuous extension
to allow for $\Psi\in\dot{W}^{2,\infty}$.

Nonetheless we can uniquely extend the bilinear form $B(\psi,\theta)$ to the
space of $(\Psi,\theta)\in\dot{W}^{2,\infty}\times\dot{H}^{-1+\delta}$ such
that the extension $B(\Psi,\theta)$ also satisfies the homogeneous estimate
(64), and such that whenever $(\Psi_{n},\theta_{n})$ is a sequence in
$\dot{W}^{2,\infty}\times\dot{H}^{-1+\delta}$ satisfying
$\nabla^{2}\Psi_{n}\rightharpoonup\nabla^{2}\Psi$ in the $L^{\infty}$ weak-*
topology, and $\theta_{n}\to\theta$ in the $\dot{H}^{-1+\delta}$ norm, we have
that $B(\psi_{n},\theta_{n})\to B(\Psi,\theta)$.

To define this extension, first let $\theta\in\dot{H}^{-1+\delta}$ be fixed
and $\Psi\in\dot{W}^{2,\infty}$ be given. Choose a sequence $\psi_{n}\in
C_{c}^{\infty}$ such that $\nabla^{2}\psi_{n}\hookrightarrow\nabla^{2}\Psi$ in
$L^{\infty}$ weak-*. (A construction can be found in Section A.) In
particular, there is a uniform bound on $\|\nabla^{2}\psi_{n}\|_{L^{\infty}}$.
For the sequence $\psi_{n}$, we use the formulas

$\displaystyle B(\psi_{n},\theta)$
$\displaystyle=HL(\psi_{n},\theta)+LH(\psi_{n},\theta)+HH(\psi_{n},\theta)$
(65) $\displaystyle HL(\psi_{n},\theta)+LH(\psi_{n},\theta)$
$\displaystyle\sim\sum_{k}\sum_{j\leq k-1}\left\\{\int
P_{k}\nabla_{\ell}\psi_{n}(P_{k+1}\theta
T^{\ell}P_{j}\theta+P_{j}\theta\,T^{\ell}P_{k+1}\theta)dx\right\\}$ (66)
$\displaystyle HH(\psi_{n},\theta)$ $\displaystyle=\sum_{q}\left\\{\int
P_{\leq q-3}\nabla_{j}\nabla_{\ell}\psi_{n}B_{q}^{j\ell}[P_{\approx
q}\theta]dx\right\\}$ (67)

For $\Psi\in\dot{W}^{2,\infty}$, $\theta\in\dot{H}^{-1+\delta}$, we define
$B(\Psi,\theta)$ to be the right hand side with $\psi_{n}$ replaced by $\Psi$.
For each fixed $k,q\in\mathbb{Z}$, the functions $P_{\leq
q-3}\nabla_{j}\nabla_{\ell}\psi_{n}$ and
$2^{k}P_{k}\nabla_{\ell}\psi_{n}=[2^{k}\Delta^{-1}\nabla^{i}P_{\approx
k}]P_{k}\nabla_{i}\nabla_{\ell}\psi_{n}$ (along with their derivatives) are
uniformly bounded and they converge uniformly to $P_{\leq
q-3}\nabla_{j}\nabla_{\ell}\Psi$ and $2^{k}P_{k}\nabla_{\ell}\Psi$ by a
compactness and contradiction argument. The terms in the braces then converge
as $n\to\infty$ for any fixed $k,j\in\mathbb{Z}^{2}$ or $q\in\mathbb{Z}$, and
they are bounded by an absolutely summable series in $(k,j)$ or $q$ by
repeating the proof of the bounds for $LH$, $HL$ and $HH$ in Section 6. By
dominated convergence, we obtain $B(\psi_{n},\theta)\to B(\Psi,\theta)$ for
any fixed $\theta\in\dot{H}^{-1+\delta}$.

If we desire $B(\Psi,\theta)$ to be sequentially continuous in $\Psi$ with
respect to weak-* convergence, this argument proves that $B(\Psi,\theta)$ is
uniquely determined and has the desired continuity property. The continuous
extension maintains the homogeneous bound (64), though the optimal constant
may in principle be different.

We also desire the property that $B(\Psi_{n},\theta_{n})\to B(\Psi,\theta)$
whenever $\theta_{n}\to\theta$ in the $\dot{H}^{-1+\delta}$ seminorm, and
$\nabla^{2}\Psi_{n}\rightharpoonup\nabla^{2}\Psi$ in $L^{\infty}$ weak-*. This
last statement is a corollary of the continuity proven in Section 7.2 below,
but it is desirable to have a direct proof. If $\theta_{n}$ were constant in
$n$, the convergence follows as in the previous argument after noting that
$\sup_{n}\|\nabla^{2}\Psi_{n}\|_{L^{\infty}}<\infty$ by the uniform
boundedness principle. Now let $T[\Psi,\theta,\phi]$ be the trilinear form,
symmetric in $\theta,\phi$ for which $B(\Psi,\theta)=T[\Psi,\theta,\theta]/2$.
Then $T[\cdot,\cdot,\cdot]$ is bounded on
$\dot{W}^{2,\infty}\times\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$ as in
(49). We then conclude that

$\displaystyle
B(\Psi_{n},\theta_{n})-B(\Psi,\theta)=T[\Psi_{n},\theta_{n}-\theta,\theta_{n}]/2+T[\Psi_{n},\theta,\theta_{n}-\theta]/2+B(\Psi_{n}-\Psi,\theta)$

converges to $0$ as $n\to\infty$.

###  7.2 Continuity property of the extended operator

In this section, we prove another continuity property of the extended operator
as the proof will be useful to us later on.

Let $X\subseteq\dot{W}^{2,\infty}$ be bounded in the semi-norm. We want to
show that $B$ is continuous on $X\times\dot{H}^{-1+\delta}$ with the
appropriate (weak-*$\times$strong) topologies. We obtain this continuity by
showing that the trilinear form $T[\cdot,\cdot,\cdot]$ is continuous as a map
on the product $X\times\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$ into
$\mathbb{R}$ when we endow $\dot{H}^{-1+\delta}$ with the strong topology
induced by the homogeneous norm, and $X$ with the (non-Hausdorff) weak-*
topology obtained as the pullback of the map $\Psi\mapsto\nabla^{2}\Psi$ into
$L^{\infty}$ weak-*. The continuity of $B(\cdot,\cdot)$ follows then by
restricting to the diagonal in $\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$.
(One can approach this problem by passing to quotients to make
$\dot{W}^{2,\infty}$ a normed vector space and cite the metrizability of the
weak-* topology on norm bounded subsets of $L^{\infty}(\mathbb{R}^{2})$
together with the sequential continuity just established; however, we believe
the following direct approach is more insightful.)

The key point is to obtain a representation of the form

$\displaystyle B(\Psi,\theta)=T[\Psi,\theta,\theta]/2$
$\displaystyle=\int\nabla_{j}\nabla_{\ell}\Psi\,\widetilde{T}^{j\ell}[\theta,\theta]dx$
(68)

for some bilinear forms $\widetilde{T}^{j\ell}$ that are symmetric and bounded
mapping $\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$ into $L^{1}$.
Inspecting (66)-(67), we can write
$P_{k}\nabla_{\ell}\Psi=\Delta^{-1}\nabla^{j}P_{k}\nabla_{j}\nabla_{\ell}\Psi$,
take appropriate adjoint operators, and define
$\widetilde{T}^{j\ell}=\widetilde{T}^{j\ell}_{L}+\widetilde{T}^{j\ell}_{H}$,
where each form is obtained by polarizing the quadratic forms

$\displaystyle\widetilde{T}^{j\ell}_{L}[\theta,\theta]$
$\displaystyle\sim-\sum_{k}\sum_{i\leq
k-1}\Delta^{-1}\nabla^{j}P_{k}\left\\{P_{k+1}\theta
T^{\ell}P_{i}\theta+P_{i}\theta\,T^{\ell}P_{k+1}\theta\right\\}$ (69)
$\displaystyle\widetilde{T}^{j\ell}_{H}[\theta,\theta]$
$\displaystyle=\sum_{q}P_{\leq q-3}\left\\{B_{q}^{j\ell}[P_{\approx
q}\theta]\right\\}$ (70)

The bound
$\|\widetilde{T}^{j\ell}[\theta,\theta]\|_{L^{1}}\leq\widetilde{C}\|\theta\|_{\dot{H}^{-1+\delta}}^{2}$
follows as in the bounds on $LH,HL$, and $HH$ established before, where we use
the uniform bounds on the operator norms $\sup_{q}\|P_{\leq
q}\|+\sup_{k}\|2^{k}\Delta^{-1}\nabla P_{k}\|<\infty$.

Once we have the representation (68), we can obtain continuity of
$T[\cdot,\cdot,\cdot]$ as a mapping from $X\times(\dot{H}^{-1+\delta})^{2}$
into $\mathbb{R}$ with the appropriate product topology (weak-* on $X$ and
normed on $\dot{H}^{-1+\delta}$) as follows. Fix a trio
$(\Psi_{0},\theta_{0},\phi_{0})\in X\times(\dot{H}^{-1+\delta})^{2}$ and let
$\epsilon>0$ be given. Letting $\widetilde{C}$ be the constant above and
setting $M_{X}=\sup_{\Psi\in X}\|\nabla^{2}\Psi\|_{L^{\infty}}$, we define the
following subset of $X\times\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$

$\displaystyle
U_{\delta_{1},\delta_{2}}\coloneqq\Big{\\{}(\Psi,\theta,\phi):\|\theta_{0}{-}\theta_{0}\|_{\dot{H}^{-1+\delta}}{+}\|\phi{-}\phi_{0}\|_{\dot{H}^{-1+\delta}}<\delta_{1},\left|\int\nabla_{j}\nabla_{\ell}(\Psi{-}\Psi_{0})\widetilde{T}^{j\ell}[\theta_{0},\phi_{0}]dx\right|<\delta_{2}\Big{\\}}.$

Then $U_{\delta_{1},\delta_{2}}$ is an open neighborhood of
$(\Psi_{0},\theta_{0},\phi_{0})$ because the $\widetilde{T}^{j\ell}$ take
values in $L^{1}$ and it is the (finite) product of open neighborhoods in $X$,
and $(\dot{H}^{-1+\delta})^{2}$. If we now choose $\delta_{1}$ and
$\delta_{2}$ small enough depending on $\epsilon$, $\widetilde{C}$, $M_{X}$,
$\|\theta_{0}\|_{\dot{H}^{-1+\delta}}$ and
$\|\phi_{0}\|_{\dot{H}^{-1+\delta}}$, and use the polarization of (68) we have
the inequality
$|T[\Psi,\theta,\phi]-T[\Psi_{0},\theta_{0},\phi_{0}]|<\epsilon$ for
$(\Psi,\theta,\phi)\in U_{\delta_{1},\delta_{2}}$, which establishes the
continuity of $T[\cdot,\cdot,\cdot]$.

###  7.3 On the optimal constant of the extension

For $\delta>0$, define the “optimal constants” for the form $B(\cdot,\cdot)$
on the Euclidean space and its extensions

$\displaystyle\|B\|_{c}$
$\displaystyle=\sup\\{|B(\psi,\theta)|~{}:~{}\|\nabla^{2}\psi\|_{C^{0}}\leq
1,\|\theta\|_{\dot{H}^{-1+\delta}}^{2}\leq 1,\psi,\theta\in
C_{c}^{\infty}(\mathbb{R}^{2})\\}$ (71) $\displaystyle\|B\|_{0}$
$\displaystyle=\sup\\{|B(\Psi,\theta)|~{}:~{}\|\nabla^{2}\Psi\|_{L^{\infty}}\leq
1,\|\theta\|_{\dot{H}^{-1+\delta}}^{2}\leq
1,\Psi\in\dot{W}_{0}^{2,\infty},\theta\in{\dot{H}}^{-1+\delta}(\mathbb{R}^{2})\\}.$
(72) $\displaystyle\|B\|_{\infty}$
$\displaystyle=\sup\\{|B(\Psi,\theta)|~{}:~{}\|\nabla^{2}\Psi\|_{L^{\infty}}\leq
1,\|\theta\|_{\dot{H}^{-1+\delta}}^{2}\leq
1,\Psi\in\dot{W}^{2,\infty},\theta\in{\dot{H}}^{-1+\delta}(\mathbb{R}^{2})\\}..$
(73)

Here $\dot{W}_{0}^{2,\infty}$ is the space of distributions with
$\nabla^{2}\Psi(x)=o(1)$ as $|x|\to\infty$, $\nabla^{2}\Psi\in L^{\infty}$.
Since $C_{c}^{\infty}$ is norm dense in $\dot{H}^{-1+\delta}$ (see Appendix
Section A.2) and the form $B(\cdot,\cdot)$ is bounded, we can restrict
attention to $\theta\in C_{c}^{\infty}$ in (72)-(73).

It is clear that $\|B\|_{c}\leq\|B\|_{0}\leq\|B\|_{\infty}$ and the reverse
inequalities would also be clear from the boundedness of $B$ if
$C_{c}^{\infty}$ were also norm dense in $\dot{W}^{2,\infty}_{0}$ or
$\dot{W}^{2,\infty}$. While this strong density is not the case, we do still
have $\|B\|_{0}=\|B\|_{c}$. To see this fact, given
$\Psi\in\dot{W}^{2,\infty}_{0}$ and $\theta\in C_{c}^{\infty}$, we can choose
a sequence $\psi_{n}\in C_{c}^{\infty}$ such that
$\nabla^{2}\psi_{n}\rightharpoonup\nabla^{2}\Psi$ in $L^{\infty}$ weak-* and
$\|\nabla^{2}\psi_{n}\|_{L^{\infty}}\to\|\nabla^{2}\Psi\|_{L^{\infty}}$ as
$n\to\infty$. (See Appendix Section A.) Then for each such
$\Psi\in\dot{W}^{2,\infty}_{0}$ we have

$\displaystyle|B(\Psi,\theta)|$
$\displaystyle=|\lim_{n\to\infty}B(\psi_{n},\theta)|\leq\liminf_{n\to\infty}\|B\|_{c}\|\nabla^{2}\psi_{n}\|_{C^{0}}\|\theta\|_{\dot{H}^{-1+\delta}}=\|B\|_{c}\|\nabla^{2}\Psi\|_{L^{\infty}}\|\theta\|_{\dot{H}^{-1+\delta}}.$

This inequality shows that $\|B\|_{0}\leq\|B\|_{c}$, so the two constants are
equal.

For $\Psi\in\dot{W}^{2,\infty}$, it is not clear whether such a sequence
exists, and we do not know whether $\|B\|_{\infty}=\|B\|_{c}$. (The
approximating sequence $\psi_{n}$ in Section 7.1 does satisfy
$\|\nabla^{2}\psi_{n}\|_{L^{\infty}}\lesssim\|\nabla^{2}\Psi\|_{L^{\infty}}$,
but may not satisfy
$\|\nabla^{2}\psi_{n}\|_{L^{\infty}}\to\|\nabla^{2}\Psi\|_{L^{\infty}}$ as
$n\to\infty$.) However, the above argument does show the equality of constants
$\|B\|_{\infty}=\|B\|_{c}$ in the periodic setting, where we restrict to mean-
zero $\theta\in C^{\infty}(\mathbb{T}^{2})$.

###  7.4 Special properties and conservation laws of mSQG

In this Section, we now prove the estimates that are unique to SQG, 2D Euler
and the mSQG family, which stem from the following Theorem:

######  Theorem 11.

Let $m^{\ell}=\epsilon^{\ell a}\nabla_{a}(-\Delta)^{-1+\delta}$ be the mSQG
multiplier, $\delta\geq 0$, and $T^{\ell}$ denote the corresponding operator.
Then there is a bounded bilinear operator $\widetilde{T}^{j\ell}$ mapping
$\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$ to $L^{1}$ that takes values in
the space of symmetric trace-free tensor fields such that
$\nabla_{\ell}[\theta
T^{\ell}\theta]=\nabla_{\ell}\nabla_{j}\widetilde{T}^{j\ell}[\theta,\theta]$
in the sense of distributions.

This theorem implies the bound in (3) by integration by parts, and thus
concludes the proof of Theorem 3. We note that in the case of the Euler
equations ($\delta=0$), the theorem follows easily from the derivation of the
Euler equations in vorticity form. Indeed, in this case the nonlinearity is
the curl of the divergence of $T\theta\otimes T\theta$, which is the second
order divergence of a trace-free tensor field
($\nabla_{j}\nabla_{\ell}[T^{j}\theta\epsilon^{\ell a}T_{a}\theta]$, where the
Biot-Savart law $T^{\ell}\theta=\epsilon^{\ell a}\nabla_{a}\Delta^{-1}\theta$
is bounded from $\dot{H}^{-1}$ to $L^{2}$). The condition of symmetry can then
be enforced by replacing this tensor with its symmetric part using the formula
$\nabla_{j}\nabla_{\ell}A^{j\ell}=\nabla_{j}\nabla_{\ell}A_{S}^{j\ell}$,
$A_{S}^{j\ell}=\frac{1}{2}(A^{j\ell}+A^{\ell j})$. For SQG and more generally
mSQG with $\delta>0$, Theorem 11 is apparently much less immediate.

In our analysis so far, the key properties we have used to bound the
nonlinearity are the homogeneity and oddness of the multiplier. These
properties are only enough to construct a bilinear form
$\widetilde{T}=\widetilde{T}_{L}+\widetilde{T}_{H}$ defined in (69)-(70) that
has the desired boundedness but does not necessarily take on trace-free
values.

We derive the desired trace-free bilinear form $\widetilde{T}$ by examining
the derivations of equations (69)-(70) in the specific case of mSQG333We
remark that one can alternatively obtain the double-divergence form by
decomposing $\theta T^{\ell}\theta$ without decomposing $\psi$, as is used in
[25].. These equations give a representation $\nabla_{\ell}[\theta
T^{\ell}\theta]=\nabla_{j}\nabla_{\ell}[\widetilde{T}_{L}^{j\ell}+\widetilde{T}_{H}^{j\ell}]$.
Let us first consider the term $\widetilde{T}_{L}^{j\ell}$ that arises from
interactions involving low frequencies. This term may not be trace free, but
there exists a trace-free tensor field $\widetilde{T}_{L2}^{j\ell}$ with the
same second order divergence and the same boundedness properties.

To obtain this tensor field, consider the following operator ${\cal R}$, which
acts on scalar fields and inverts the second order divergence operator while
taking values in the space of symmetric, trace-free tensor fields: ${\cal
R}^{j\ell}[f]=A\delta^{j\ell}\Delta^{-1}f+B\Delta^{-2}\nabla^{j}\nabla^{\ell}f$.
The constants $A$ and $B$ solve $A+B=1$, $dA+B=0$ so that ${\cal R}f$ is trace
free and solves $\nabla_{j}\nabla_{\ell}{\cal R}^{j\ell}f=f$ for $f$ with
Fourier support in an annulus excluding $0$.

We now wish to define $\widetilde{T}_{L2}^{j\ell}\equiv{\cal
R}^{j\ell}\nabla_{a}\nabla_{b}\widetilde{T}_{L}^{ab}$ in the hope that we will
have
$\nabla_{j}\nabla_{\ell}\widetilde{T}_{L2}^{j\ell}=\nabla_{j}\nabla_{\ell}\widetilde{T}_{L}^{j\ell}$.
However, the fact that the zeroth order operator ${\cal
R}^{j\ell}\nabla_{a}\nabla_{b}$ is unbounded on $L^{1}$ makes it unclear this
bilinear form is bounded from $\dot{H}^{-1+\delta}\times\dot{H}^{-1+\delta}$
into $L^{1}$, or even that it maps into the space of distributions.
Nonetheless, the boundedness of $\widetilde{T}_{L2}$ is true: ${\cal
R}^{j\ell}\nabla_{a}\nabla_{b}\Delta^{-1}\nabla^{a}P_{k}={\cal
R}^{j\ell}\nabla_{b}P_{k}$ satisfies estimates similar to
$\Delta^{-1}\nabla^{j}P_{k}$ (since ${\cal R}$ has a symbol that is degree
$-2$ homogeneous and smooth away from the origin), and

$\displaystyle\widetilde{T}^{j\ell}_{L2}[\theta,\theta]$
$\displaystyle\sim-\sum_{k}\sum_{i\leq k-1}{\cal
R}^{j\ell}\nabla_{a}\nabla_{b}\Delta^{-1}\nabla^{a}P_{k}\left\\{P_{k+1}\theta
T^{b}P_{i}\theta+P_{i}\theta\,T^{b}P_{k+1}\theta\right\\}$ (74)

has the same structure as $\widetilde{T}_{L}$, so that the same boundedness
argument based on Schur’s Test applies (modifying the proofs of lines (59) and
(61)), and the equality of the second order divergences is also guaranteed.

This last argument does not apply to $\widetilde{T}_{H}$ and does not use the
full structure of the mSQG multiplier. The unique feature of the mSQG
multiplier is that the $\widetilde{T}_{H}^{j\ell}$ defined in (70) is in fact
already trace-free. Namely, thanks to formulas (55) and (42), this trace-free
property follows from the calculation of line (46), which shows that the
Jacobian of the frequency-truncated multiplier is trace-free. From the
identity $\nabla_{\ell}[\theta
T^{\ell}\theta]=\nabla_{j}\nabla_{\ell}[\widetilde{T}_{L2}^{j\ell}+\widetilde{T}_{HS}^{j\ell}]$,
where $\widetilde{T}_{HS}^{j\ell}$ is the symmetric part of
$\widetilde{T}_{H}$, we conclude Theorem 11.

We now explain how Theorem 11 quickly implies the conservation of impulse and
angular momentum for general weak solutions to mSQG, whose Hamiltonian is
locally integrable. The proof is based on the following Lemma:

######  Lemma 7.1.

Let $Q(x)$ be a polynomial on $\mathbb{R}^{2}$ of the form
$A+Bx_{1}+Cx_{2}+D(x_{1}^{2}+x_{2}^{2})$, so that the trace free part of the
Hessian of $Q$ vanishes. Let $\mathring{F}^{j\ell}$ be a trace-free tensor
field in $L^{1}(\mathbb{R}^{2})$ and let $\chi\in
C_{c}^{\infty}(\mathbb{R}^{2})$ be equal to $1$ in a neighborhood of the
origin. Then

$\displaystyle\lim_{R\to\infty}\int_{\mathbb{R}^{2}}\mathring{F}^{j\ell}\nabla_{j}\nabla_{\ell}(\chi(x/R)Q(x))dx=0$

Indeed, the term $\mathring{F}^{j\ell}\nabla_{j}\nabla_{\ell}Q=0$ vanishes
pointwise, since it is equal to
$\mathring{F}^{j\ell}\mathring{\nabla}^{2}_{j\ell}Q$, and the trace-free part
of the Hessian
$\mathring{\nabla}^{2}_{j\ell}Q=\nabla_{j}\nabla_{\ell}Q-(1/d)\Delta
Q\delta_{j\ell}$ vanishes. When the derivatives hit the cutoff, we can bound
$|\mathring{F}||\nabla[\chi(\cdot/R)]||\nabla Q|$ and
$|\mathring{F}||\nabla^{2}[\chi(\cdot/R)]||Q|$ pointwise by
$\lesssim|\mathring{F}|\in L^{1}(\mathbb{R}^{2})$. We then apply the dominated
convergence theorem using the fact that the derivatives of the cutoff are
supported in $|x|\sim R$, to conclude that these latter terms tend to zero as
$R\to\infty$.

Now let $\theta\in L_{t}^{2}\dot{H}^{-1+\delta}$ be a weak solution to mSQG on
the time interval $[0,T]$ and fix $t_{0}\in(0,T]$. We define the restriction
to time $t_{0}$, $\theta(t_{0},\cdot)$, as the unique distribution on
$\mathbb{R}^{2}$ that satisfies

$\displaystyle\theta(t_{0},x)$
$\displaystyle=\theta(t,x)-\int_{t_{0}}^{t}\nabla_{j}\nabla_{\ell}\widetilde{T}^{j\ell}[\theta(s),\theta(s)]ds,$
(75)

where $\widetilde{T}$ is the bilinear form of Theorem 11. The equation
$\partial_{t}\theta+\nabla_{\ell}(\theta T^{\ell}\theta)=0$ implies that the
element of ${\cal D}^{\prime}((0,T)\times\mathbb{R}^{2})$ defined by the right
hand side satisfies $\partial_{t}\theta(t_{0},x)=0$ in ${\cal
D}^{\prime}((0,T)\times\mathbb{R}^{2})$, so we can identify it with a unique
element of ${\cal D}^{\prime}(\mathbb{R}^{2})$.

In particular, taking the difference $\theta(t,\cdot)-\theta(0,\cdot)$, we
have

$\displaystyle\theta(t,x)$
$\displaystyle=\theta(0,x)+\nabla_{j}\nabla_{\ell}\mathring{F}^{j\ell}_{t}(x),\qquad\mathring{F}_{t}^{j\ell}=-\int_{0}^{t}\widetilde{T}^{j\ell}[\theta(s),\theta(s)]ds.$
(76)

From $\theta\in L_{t}^{2}\dot{H}^{-1+\delta}$ and Theorem 11, we have
$\mathring{F}_{t}\in L^{1}(\mathbb{R}^{2})$ for any $t$, so that Lemma 7.1
applies to give

$\displaystyle\lim_{R\to\infty}\int_{\mathbb{R}^{2}}\theta(t,x)Q(x)\chi(x/R)dx$
$\displaystyle=\lim_{R\to\infty}\int_{\mathbb{R}^{2}}\theta(0,x)Q(x)\chi(x/R)dx,$
(77)

for all polynomials whose Hessian has vanishing trace-free part, which
includes the polynomials $x_{1}^{2}+x_{2}^{2}$ and $x_{i}$ that (up to
constants) define angular momentum and the components of impulse. In
particular the left hand limit is defined if and only if the right hand limit
is defined, where both integrals are interpreted in the sense of
distributions, viewing $Q(x)\chi(x/R)$ as a test function.

Note that even if the limit (77) does exist, it may a priori depend on the
cutoff function $\chi$. A sufficient condition for the limit to exist and to
be independent of $\chi$ is that the initial condition has the form
$\theta(0,x)=f(x)+\nabla_{j}g^{j}(x)+\nabla_{j}\nabla_{\ell}h^{j\ell}(x)$ for
tensors $f,g$ and $h$ such that $|x|^{2}|f|,|x||g|$, and $|h|$ belong to
$L^{1}$. The proof is similar to that of Lemma 7.1.

##  8 Sharpness of the estimate on the nonlinearity

Let $B(\psi,\theta)$ be the multilinear form associated to mSQG and let
$\|\,\|_{X}$ be a semi-norm that satisfies

$\displaystyle|B(\psi,\theta)|\lesssim\|\psi\|_{X}\|\theta\|_{H^{-1+\delta}}^{2}$
(78)

We have shown that the $\mathring{W}^{2,\infty}$ semi-norm
$\|\psi\|_{X}=\|\mathring{\nabla}^{2}\psi\|_{L^{\infty}}$ satisfies the above
estimate. We now prove Theorem 5, which states that every such semi-norm $X$
that satisfies (78) must control the $\mathring{W}^{2,\infty}$ semi-norm,
meaning that the estimate we have proven is sharp (when we assume only
$H^{-1+\delta}$ control of the scalar).

The proof proceeds by a geometric optics technique that is related to the
convex integration construction in [3, 26], but with a different normalization
that extracts an approximate delta function from the bilinear form. In
particular we employ different techniques to estimate the error.

Let $\psi(x_{1},x_{2})$ be an arbitrary $C_{c}^{\infty}$ test function.
Consider the sequence

$\displaystyle\begin{split}\Theta_{\lambda}=P_{\lambda}F_{\lambda}&\equiv\lambda^{1-\delta}P_{\lambda}(e^{i\lambda
x_{1}}\lambda^{1/2}\varphi(\lambda^{1/2}x)),\\\
\bar{\Theta}_{\lambda}=P_{-\lambda}\bar{F}_{\lambda}&\equiv\lambda^{1-\delta}P_{-\lambda}(e^{-i\lambda
x_{1}}\lambda^{1/2}\varphi(\lambda^{1/2}x)),\end{split}$ (79)

where $P_{\lambda}$ is a convolution operator whose symbol
$\hat{u}_{\approx\lambda}$ is a bump function of size $1$ adapted to the
frequency region $|\xi-\lambda e_{1}|\leq\lambda/3$. The family
$\theta_{\lambda}=\Theta_{\lambda}+\bar{\Theta}_{\lambda}$ is then real-
valued. Let us normalize $\varphi$ to be a bump function supported in the unit
ball in physical space with $\int\varphi^{2}dx=1$. Observe that the family
$\theta_{\lambda}$ is then normalized to have size $\sim 1$ in
$\dot{H}^{-1+\delta}$, so by our assumption on the $X$ seminorm we have
$\sup_{\lambda}|B(\psi,\theta_{\lambda})|\lesssim\|\psi\|_{X}$.

Our claim is that

$\displaystyle\lim_{\lambda\to\infty}B(\psi,\theta_{\lambda})=2(-1+\delta)\partial_{1}\partial_{2}\psi(0,0),$
(80)

hence $|\partial_{1}\partial_{2}\psi(0,0)|\lesssim\|\psi\|_{X}$ according to
(78). By performing a rotation we can modify the construction to show
$|(\partial_{1}-\partial_{2})(\partial_{1}+\partial_{2})\psi(0,0)|\lesssim\|\psi\|_{X}$,
which implies $|\mathring{\nabla}^{2}\psi(0,0)|\lesssim\|\psi\|_{X}$ when
combined with (80). By translation invariance, we conclude that
$\|\mathring{\nabla}^{2}\psi\|_{L^{\infty}}\lesssim\|\psi\|_{X}$ as desired.

We have thus proven Theorem 5 under the assumption of claim (80), whose proof
occupies the remainder of this section. We start by expanding

$\displaystyle
B(\psi,\theta_{\lambda})=B(\psi,\Theta_{\lambda}+\bar{\Theta}_{\lambda})$
$\displaystyle=B(\psi,\Theta_{\lambda})+B(\psi,\bar{\Theta}_{\lambda})$ (81)
$\displaystyle+T(\psi,\Theta_{\lambda},\bar{\Theta}_{\lambda})+T(\psi,\bar{\Theta}_{\lambda},\Theta_{\lambda}),$
(82)

with $T$ is the usual trilinear form associated to $B$.

The terms of line (81) converge weakly to zero. For example, by Plancharel we
have

$\displaystyle B(\psi,\Theta_{\lambda})$
$\displaystyle=\int_{\hat{\mathbb{R}}^{2}}\hat{\psi}\left(\widehat{\Theta_{\lambda}}\ast\widehat{T^{\ell}}[\Theta_{\lambda}]\right)d\xi$

Due to the projection $P_{\approx\lambda}$ in (79), the convolution is
supported in frequencies of size $\sim\lambda$. It is also uniformly bounded
by a polynomial in frequency. Since $\hat{\psi}$ is Schwartz, we conclude that
this term, and the similar term $B(\psi,\bar{\Theta}_{\lambda})$, tend to $0$
by dominated convergence.

We now isolate the main term of line (82) and show that it converges to (80).
We express (82) more fully as

$\displaystyle\eqref{eq:interactionLowFreqTerms}$
$\displaystyle=\int\psi(T^{\ell}\Theta_{\lambda}\nabla_{\ell}\bar{\Theta}_{\lambda}+T^{\ell}\bar{\Theta}_{\lambda}\nabla_{\ell}\Theta_{\lambda})dx$
(83)

We now exploit the divergence form of the symmetric interaction from Section
5.

$\displaystyle T^{\ell}\Theta_{\lambda}\nabla_{\ell}\bar{\Theta}_{\lambda}$
$\displaystyle+T^{\ell}\bar{\Theta}_{\lambda}\nabla_{\ell}\Theta_{\lambda}=\nabla_{j}\nabla_{\ell}B_{\lambda}^{j\ell}[\Theta_{\lambda},\bar{\Theta}_{\lambda}]$
$\displaystyle B_{\lambda}^{j\ell}[\Theta_{\lambda},\Theta_{\lambda}]$
$\displaystyle=\int
F_{\lambda}(x-h_{1})\bar{F}_{\lambda}(x-h_{2})\tilde{K}_{\lambda}^{j\ell}(h_{1},h_{2})dh_{1}dh_{2}$
(84)

In making this decomposition, we have absorbed the operator
$P_{\lambda}\otimes P_{-\lambda}$ in the definition (79) into the kernel
$\tilde{K}_{\lambda}$, which is Schwartz, similar to line (44). Doing so
results in appearances of the bump function $\hat{u}$ in the following
formulas:

$\displaystyle\hat{K}_{\lambda}^{j\ell}(\zeta,\eta)$
$\displaystyle=\int_{0}^{1}\nabla^{j}m_{\lambda}^{\ell}(\sigma\zeta-(1-\sigma)\eta)\hat{u}(\zeta)\hat{u}(-\eta)d\sigma$
(85) $\displaystyle m_{\lambda}^{\ell}(\xi)$ $\displaystyle=i\epsilon^{\ell
a}\xi_{a}|\xi|^{2(-1+\delta)}\chi_{\approx\lambda}(\xi)$
$\displaystyle\nabla^{j}m_{\lambda}^{\ell}(\xi)$
$\displaystyle=2(-1+\delta)i\epsilon^{\ell
a}\xi_{a}\xi^{j}|\xi|^{2(-2+\delta)}\chi_{\approx\lambda}+...+O(\epsilon^{\ell
j})$ (86)

The anti-symmetric part, which is proportional to $\epsilon^{\ell j}$, will
not be important in our calculations, and we have also neglected the term
where the derivative hits the cutoff function $\chi_{\approx\lambda}$ since it
will also vanish in the forthcoming computation of the main term.

Using coordinates $h_{1}=(h_{1}^{1},h_{1}^{2})$ and
$h_{2}=(h_{2}^{1},h_{2}^{2})$ on $\mathbb{R}^{2}\times\mathbb{R}^{2}$, we now
obtain a leading order term expansion of (84)

$\displaystyle\eqref{eq:bilinearFormConvolve1}$
$\displaystyle=\lambda^{2-2\delta}\int e^{i\lambda h_{1}^{1}}e^{-i\lambda
h_{2}^{1}}\lambda\varphi^{2}(\lambda^{1/2}x)\tilde{K}_{\lambda}^{j\ell}(h_{1},h_{2})dh_{1}dh_{2}$
(87) $\displaystyle+\underbrace{\lambda^{2-2\delta}\int e^{i(\lambda
h_{1}^{1}-\lambda
h_{2}^{1})}\int_{0}^{1}\frac{d}{d\sigma}\lambda\varphi(\lambda^{1/2}(x-\sigma
h_{1}))\varphi(\lambda(x-\sigma
h_{2}))\tilde{K}_{\lambda}^{j\ell}(h_{1},h_{2})dh_{1}dh_{2}d\sigma}_{\equiv
E_{\lambda}}$ (88)

Let us first isolate the leading term, (87), and show how it leads to (80).
Here we recognize that the terms involving $\chi$ do not depend on
$(h_{1},h_{2})$ and what appears is the Fourier transform of $\tilde{K}$. We
have

$\displaystyle\eqref{eq:leadingTerm}^{j\ell}$
$\displaystyle=\lambda^{2-2\delta}\lambda\varphi^{2}(\lambda^{1/2}x)\widehat{\tilde{K}^{j\ell}}(\lambda
e_{1},-\lambda e_{1})$ (89)
$\displaystyle=2(-1+\delta)\lambda^{2-2\delta}\lambda\varphi^{2}(\lambda^{1/2}x)\delta^{j}_{1}\delta^{\ell}_{2}\lambda^{2(-1+\delta)}+O(\epsilon^{\ell
j})$ (90)

Here we have used that $\chi_{\approx\lambda}(\xi)$ and the $\hat{u}$ factors
are equal to $1$ in an $O(\lambda)$ neighborhood of $\lambda e_{1}$; in
particular, the factor $\nabla^{j}\chi_{\approx\lambda}$ vanishes at $\lambda
e_{1}$.

Returning to (83), we integrate the main term against the test function $\psi$
to obtain

$\displaystyle\int\psi\nabla_{j}\nabla_{\ell}\eqref{eq:leadingTerm}dx$
$\displaystyle=2(-1+\delta)\int\nabla_{j}\nabla_{\ell}\psi\lambda\varphi^{2}(\lambda^{1/2}x)\delta^{j}_{1}\delta^{\ell}_{2}dx$
(91)
$\displaystyle=2(-1+\delta)\partial_{1}\partial_{2}\psi(0,0)+o(1)\quad\mbox{
as $\lambda\to\infty$}.$ (92)

Hence the leading term establishes the key claim (80). It remains to treat the
error term $E_{\lambda}$ from (88).

We now treat the error term $E_{\lambda}$ from (88). We claim that
$E_{\lambda}$ converges to $0$ in $L^{1}$. To see this, we apply Fubini-
Tonelli and apply Cauchy-Schwartz, using the translation invariance of the
$L^{2}$ norm, to obtain

$\displaystyle\int|E_{\lambda}(x)|dx\lesssim\lambda^{2-2\delta+1+\frac{1}{2}}\int_{0}^{1}\left[\int|\nabla\varphi(\lambda^{1/2}(x-\sigma
h_{1}))||\varphi(\lambda^{1/2}(x-\sigma
h_{2}))|dx\right]|h||\tilde{K}_{\lambda}(h_{1},h_{2})|dh_{1}dh_{2}d\sigma$
$\displaystyle\lesssim(\lambda^{1/2}\|\nabla\varphi(\lambda^{1/2}\cdot)\|_{L^{2}})\cdot(\lambda^{1/2}\|\varphi(\lambda^{1/2}\cdot)\|_{L^{2}})\cdot\lambda^{2-2\delta+1/2}\int|h||\tilde{K}_{\lambda}(h_{1},h_{2})|dh_{1}dh_{2}\lesssim(1)\cdot(1)\cdot\lambda^{-1/2}.$

The last bound follows due to the scaling of the kernel
$\tilde{K}_{\lambda}(h_{1},h_{2})=\lambda^{-2+2\delta}\lambda^{2d}\tilde{K}_{1}(\lambda
h_{1},\lambda h_{2})$, which is Schwartz. Note that the estimate relies
crucially on the gain of $\lambda^{-1}$ coming from the small factor of $|h|$
and scaling. Thus $\|E_{\lambda}\|_{L^{1}}=O(\lambda^{-1/2})$ goes to $0$ as
$\lambda\to\infty$.

##  9 Uniqueness of the main estimate for the mSQG multiplier

In this section we show that the special estimate for the mSQG nonlinearity in
Theorem 3 is not satisfied by any other active scalar nonlinearities with the
same translation and scaling symmetries other than the constant multiples of
the mSQG nonlinearity.

Consider an active scalar nonlinearity on $\mathbb{R}^{2}$, which is a
nonlinearity of the form

$\displaystyle B(\psi,\theta)\equiv\int_{\mathbb{R}^{2}}\psi
T^{\ell}\theta\nabla_{\ell}\theta dx$

initially defined for $\theta$ in the space $\SS_{0}$ of Schwartz functions
with compact Fourier support away from the origin and for $\psi\in
C_{c}^{\infty}$. We restate here the theorem we are aiming to prove for the
convenience of the reader.

######  Theorem 12.

Assume that the operator $T^{\ell}$ has a Fourier multiplier $m^{\ell}$ that
is homogeneous of degree $-1+2\delta$, $\delta\geq 0$, $\delta\neq 1$. If the
nonlinearity $B(\psi,\theta)$ satisfies the bound

$\displaystyle|B(\psi,\theta)|$
$\displaystyle\lesssim\|\mathring{\nabla}^{2}\psi\|_{L^{\infty}}\|\theta\|_{H^{-1+\delta}}^{2}$
(S)

for $\psi\in C_{c}^{\infty}(\mathbb{R}^{2})$ and
$\theta\in\SS_{0}(\mathbb{R}^{2})$, then $T^{\ell}$ is a constant multiple of
the mSQG Biot Savart law.

###  9.1 Reducing to the case of a smooth multiplier

Let us reduce to the case where the multiplier is smooth. We can argue by
contradiction. Suppose $m^{\ell}$ is a homogeneous multiplier that is not a
constant multiple of the mSQG multiplier, but for which the corresponding
nonlinearity nonetheless satisfies (S). We first reduce to the case where
$m^{\ell}$ is smooth away from the origin by exploiting a natural action of
the group of rotations.

For $\alpha\in\mathbb{R}$, let $R_{\alpha}$ denote the one parameter group of
linear operators that act on $\mathbb{R}^{2}$ by rotation by the angle
$\alpha$, and let $[R_{\alpha}]_{k}^{\ell}$ denote the corresponding matrix
entries. The multiplier $m^{\ell}(\xi)$ can be regarded as a vector field on
$\widehat{\mathbb{R}}^{2}$, and the group $R_{\alpha}$ acts on this space of
vector fields by pushforward:
$(R_{\alpha}m)^{\ell}(\xi)\equiv[R_{\alpha}]_{k}^{\ell}m^{k}(R_{-\alpha}\xi)$.
The mSQG multiplier, which we denote by $M^{\ell}(\xi)$, is up to a constant
the unique vector field invariant under this action with the $-1+2\delta$
homogeneity.

Letting $f\circ R_{\alpha}(x)=f(R_{\alpha}x)$, since $R_{\alpha}$ is
orthogonal, the formula $\widehat{f\circ
R_{\alpha}}(\xi)=\hat{f}(R_{\alpha}\xi)$ yields that the rotation group
correspondingly acts on the operator $T^{\ell}$ associated with $m^{\ell}$ by
$(R_{\alpha}T)^{\ell}[\theta](x)\equiv[R_{\alpha}]^{\ell}_{k}T^{k}[\theta\circ
R_{\alpha}](R_{-\alpha}x)$. From this identity, we can check that the action
of the rotation group respects the estimate (S), for if (S) holds for
$T^{\ell}$ then for the rotated form $B_{\alpha}$, we have

$\displaystyle B_{\alpha}[\psi,\theta]$
$\displaystyle\equiv\int\psi(x)(R_{\alpha}T)^{\ell}[\theta]\nabla_{\ell}\theta(x)dx=\int\psi(x)[R_{\alpha}]_{k}^{\ell}T^{k}[\theta\circ
R_{\alpha}](R_{-\alpha}x)\nabla_{\ell}\theta(x)dx$
$\displaystyle=\int\psi(R_{\alpha}x)T^{k}[\theta\circ
R_{\alpha}](x)[R_{\alpha}]_{k}^{\ell}\nabla_{\ell}\theta(R_{\alpha}x)=\int\psi(R_{\alpha}x)T^{k}[\theta\circ
R_{\alpha}](x)\nabla_{k}[\theta\circ R_{\alpha}](x)$ $\displaystyle
B_{\alpha}[\psi,\theta]$ $\displaystyle=B_{0}[\psi\circ R_{\alpha},\theta\circ
R_{\alpha}]$

Thus the bilinear form $B_{\alpha}$ associated to $R_{\alpha}T$ maintains the
bound (S), since both $\|\mathring{\nabla}^{2}\psi\|_{L^{\infty}}$ and
$\|\theta\|_{\dot{H}^{s}}$ are invariant under rotations.

Now consider again the operator $T$ that we have assumed is not the mSQG
operator but still satisfies (S). Define the twisted mollification of $T$ to
be

$\displaystyle(T_{\epsilon})^{\ell}=\int_{\mathbb{R}}(R_{\alpha}T)^{\ell}\eta_{\epsilon}(\alpha)d\alpha$
(93)

where $\eta_{\epsilon}$ is a standard mollifier supported in
$|\alpha|\leq\epsilon$. Then $T_{\epsilon}$ inherits the estimate (S) since we
are simply averaging over rotations that preserve the estimate. Moreover, we
claim the multiplier of $T_{\epsilon}$ is smooth and distinct from the mSQG
multiplier for $\epsilon$ small enough.

Indeed, since $m^{\ell}$ is a homogeneous distribution on
$\widehat{\mathbb{R}^{2}}$, away from the origin $m$ has the form
$m^{\ell}(\xi)=h^{\ell}(\omega)|\xi|^{-1+2\delta}$, where
$\omega\in\mathbb{R}/\mathbb{Z}$ is the polar angle on
$\widehat{\mathbb{R}^{2}}$ and $h^{\ell}$ is an $\mathbb{R}^{2}$-valued
distribution on the $1D$ torus $\mathbb{R}/\mathbb{Z}$ (see e.g. [23]). The
operation of twisted mollification forming the operator $T_{\epsilon}$ simply
replaces the distribution $h$ by the distribution
$h_{\epsilon}^{\ell}(\omega)=\int_{\mathbb{R}}h^{k}(\omega-\alpha)\eta_{\epsilon}(\alpha)[R_{\alpha}]_{k}^{\ell}d\alpha$,
where the integral and the translation of $h$ are interpreted in the sense of
distributions using duality.

In the same way that a usual convolution produces a smooth function from a
distribution, the distribution $h_{\epsilon}$ can be shown to be smooth.
Furthermore, since $R_{\alpha}$ is smooth and $R_{0}=\mbox{Id}$, one can check
that $h_{\epsilon}$ converges to $h$ in ${\cal
D}^{\prime}(\mathbb{R}/\mathbb{Z})$ as $\epsilon\to 0$ by calculating the dual
operator that acts on test functions, and finally showing that the dual
twisted mollification converges in $C^{k}(\mathbb{R}/\mathbb{Z})$ for any test
function and any $k\geq 0$. We omit the details, which are similar to the
usual process of convolution for distributions (see e.g. [23]).

As a result, $T_{\epsilon}$ must be distinct from the mSQG operator for
$\epsilon$ sufficiently small, for we have assumed that the limiting
multiplier $m^{\ell}(\xi)=h^{\ell}(\omega)|\xi|^{-1+2\delta}$ is distinct from
the mSQG multiplier. Also, as pointed out $T_{\epsilon}$ inherits the estimate
(S). Thus, if we can show $T_{\epsilon}$ is the mSQG Biot-Savart Law for every
$\epsilon>0$, the limiting operator $T$ is also.

We now assume $T$ has a smooth multiplier satisfying (S) and $-1+2\delta$
homogeneity, and proceed to show $T$ is a multiple of the Biot-Savart law for
mSQG.

###  9.2 The case of an odd multiplier

By the analysis of the previous section, if the bound (S) holds for an
operator $T^{\ell}$ with multiplier $m^{\ell}$, the same bound holds for the
odd part of the operator $T_{o}^{\ell}=(T^{\ell}+(R_{\pi}T)^{\ell})/2$ whose
multiplier is $m_{o}^{\ell}(\xi)=(m^{\ell}(\xi)-m^{\ell}(-\xi))/2$, and also
for the even part of the operator
$T_{e}^{\ell}=(T^{\ell}-(R_{\pi}T)^{\ell})/2$. We will show that
$T_{o}^{\ell}$ is a multiple of the mSQG Biot-Savart law and that
$T_{e}^{\ell}$ is $0$ (except possibly in the case where the multiplier is
degree $1$ homogeneous).

Let us now consider the case where the multiplier $m^{\ell}=m_{o}^{\ell}$ is
odd. We first establish that, assuming oddness of the multiplier, the velocity
field $T^{\ell}$ must be incompressible in the sense that $m(\xi)\cdot\xi=0$.

######  Proposition 9.1.

If the velocity field $T^{\ell}$ is not incompressible, then there is a test
function $\psi$ and a sequence $\theta_{\lambda}$ with Fourier support in an
annulus $\\{|\xi|\sim\lambda\\}$ such that
$\|\theta_{\lambda}\|_{\dot{H}^{\delta}}=O(1)$ and such that
$B(\psi,\theta_{\lambda})$ has a nontrivial limit. As a consequence, there
cannot exist any estimate of the form

$\displaystyle|B(\psi,\theta)|\lesssim\|\psi\|_{X}\|\theta\|_{H^{s}}^{2},\qquad\psi\in{\cal
C}_{c}^{\infty},\theta\in H^{s}$ (94)

for any $s<\delta$ and any norm $X$ on ${\cal C}_{c}^{\infty}$.

Indeed, if an estimate of the type (94) were to exist, it would force
$B(\psi,\theta_{\lambda})\to 0$ for the example we now construct in view of
the estimate
$\|\theta_{\lambda}\|_{H^{s}}\lesssim\lambda^{s-\delta}\|\theta_{\lambda}\|_{\dot{H}^{\delta}}$
for functions with Fourier support in a annulus of radius $\sim\lambda$.

###### Proof.

As $T$ is not incompressible-valued, there exists a frequency $\bar{\xi}$,
$|\bar{\xi}|=1$ such that $m(\bar{\xi})\cdot\bar{\xi}\neq 0$. Let
$\eta_{\lambda}$ be a Schwartz function that is a bump function in frequency
space adapted to $|\lambda\bar{\xi}-\xi|\leq\lambda/2$. Define the operator
$P_{\lambda}=\eta_{\lambda}\ast$ and define a bump function
$\chi_{\lambda}(x)=\lambda^{-\delta}\lambda^{d/4}\chi(\lambda^{1/2}x)$ (with
$d=2$) and define

$\displaystyle\theta_{\lambda}=(\Theta_{\lambda}+\overline{\Theta}_{\lambda})=(\Theta_{\lambda}+\Theta_{-\lambda}),\qquad\Theta_{\lambda}=P_{\lambda}[e^{i\lambda\bar{\xi}\cdot
x}\chi_{\lambda}(x)].\qquad$

We need the following geometric optics Lemma:

######  Lemma 9.1.

There is a leading order expansion of the form

$\displaystyle T^{\ell}\Theta_{\lambda}$
$\displaystyle=e^{i\lambda\bar{\xi}\cdot
x}\lambda^{-1+2\delta}\chi_{\lambda}m^{\ell}(\bar{\xi})+\mathrm{ErrV}^{\ell}$
(95) $\displaystyle\|\mathrm{ErrV}\|_{L^{2}}$
$\displaystyle=O(\lambda^{-1/2}\lambda^{-1+2\delta}\lambda^{-\delta})=O(\lambda^{-3/2+\delta})$
(96) $\displaystyle\nabla_{\ell}\Theta_{\lambda}$
$\displaystyle=(i\lambda)\bar{\xi}_{\ell}e^{i\lambda\bar{\xi}\cdot
x}\chi_{\lambda}+\mathrm{ErrS}_{\ell}$ (97)
$\displaystyle\|\mathrm{ErrS}\|_{L^{2}}$
$\displaystyle=O(\lambda^{1-\delta-1/2})$ (98)

We prove the expansion for the velocity field. The proof for the scalar
gradient is similar. Let $K_{\lambda}^{\ell}$ denote the convolution kernel
representing $T^{\ell}P_{\lambda}$. Observe that

$\displaystyle T^{\ell}\Theta_{\lambda}$ $\displaystyle=\int
e^{i\lambda\bar{\xi}\cdot(x-h)}\chi_{\lambda}(x-h)K_{\lambda}^{\ell}(h)dh$
(99) $\displaystyle=e^{i\lambda\bar{\xi}\cdot x}\int
e^{-i\lambda\bar{\xi}\cdot h}\chi_{\lambda}(x-h)K_{\lambda}^{\ell}(h)dh$ (100)
$\displaystyle=e^{i\lambda\bar{\xi}\cdot x}\int e^{-i\lambda\bar{\xi}\cdot
h}\chi_{\lambda}(x)K_{\lambda}^{\ell}(h)dh+\mathrm{ErrV}$ (101)

Notice that the leading order term is equal to the one in (95). We estimate
the error term by writing

$\displaystyle\mathrm{ErrV}$ $\displaystyle=e^{i\lambda\bar{\xi}\cdot
x}\int\int_{0}^{1}e^{-i\lambda\bar{\xi}\cdot
h}\nabla_{a}\chi_{\lambda}(x-\sigma h)h^{a}K_{\lambda}^{\ell}(h)d\sigma dh$
$\displaystyle\|\mathrm{ErrV}\|_{L^{2}}$
$\displaystyle\lesssim\sup_{\sigma,h}\|\nabla\chi_{\lambda}(\cdot-\sigma
h)\|_{L^{2}}\cdot\|h|K_{\lambda}|\|_{L^{1}}$
$\displaystyle\lesssim\lambda^{1/2-\delta}\cdot\lambda^{-1-1+2\delta}=O(\lambda^{-3/2+\delta})$

Note that the reason we gain by a factor of $\lambda^{-1/2}$ compared to the
main term is that differentiating $\chi$ costs only $\lambda^{1/2}$, but gains
a factor of $\lambda^{-1}$ coming from the factor of $h$. The same
observations lead to the other equations (97) and (98).

We now let $\psi$ be any test function with $\psi(0)=1$. We claim that

$\displaystyle B(\psi,\theta_{\lambda})$ $\displaystyle=\int\psi
T^{\ell}\theta_{\lambda}\nabla_{\ell}\theta_{\lambda}dx=\int\psi(T^{\ell}\Theta_{\lambda}\nabla_{\ell}\Theta_{-\lambda}+T^{\ell}\Theta_{\lambda}\nabla_{\ell}\Theta_{-\lambda})dx+o(1)$

To see that the other terms are $o(1)$, consider $\int\psi
T^{\ell}\Theta_{\lambda}\nabla_{\ell}\Theta_{\lambda}$. We can write this as
an integral in frequency space by Plancharel, and note that the Fourier
transform of $T^{\ell}\Theta_{\lambda}\nabla_{\ell}\Theta_{\lambda}$, which is
localized to frequencies of the order $\lambda$, has $L^{2}$ norm bounded by a
polynomial in $\lambda$. By applying dominated convergence in frequency space,
the integral tends to zero as claimed.

We now consider the main term, which we expand as

$\displaystyle B(\psi,\theta_{\lambda})$
$\displaystyle=(-i\lambda)\lambda^{-1+2\delta}\int\psi\chi_{\lambda}^{2}(m^{\ell}(\bar{\xi})\bar{\xi}_{\ell}-m^{\ell}(-\bar{\xi})\bar{\xi}_{\ell})dx$
$\displaystyle+\lambda^{-1+2\delta}\int\psi\chi_{\lambda}m^{\ell}(\bar{\xi})\mathrm{ErrS}_{\ell}-(i\lambda)\int\psi\chi_{\lambda}\bar{\xi}_{\ell}\mathrm{ErrV}^{\ell}$
$\displaystyle+\int\psi\mathrm{ErrS}_{\ell}\mathrm{ErrV}^{\ell}dx$

By bounding $\|\psi\|$ in $L^{\infty}$ and taking the other factors in $L^{2}$
using $\|\chi_{\lambda}\|_{L^{2}}\lesssim\lambda^{-\delta}$ and (96),(98), we
see that the terms on the second line are $O(\lambda^{-1/2})$ while the error
term in the last line is $O(\lambda^{-1})$.

The term on the first line is therefore the main term. The cutoff
$\chi_{\lambda}$ is normalized so that
$\lambda^{2\delta}\chi_{\lambda}^{2}~{}\rightharpoonup~{}A\delta_{0}$ for some
$A\neq 0$. Since we have assumed a choice of $\bar{\xi}$ such that
$m^{\ell}(\bar{\xi})\bar{\xi}_{\ell}-m^{\ell}(-\bar{\xi})\bar{\xi}_{\ell}=2m^{\ell}(\bar{\xi})\bar{\xi}_{\ell}$
is not zero (recall that $m$ is odd) and a choice of $\psi$ that does not
vanish at the origin, we see as desired that $B(\psi,\theta_{\lambda})$ has a
nontrivial limit as $\lambda\to\infty$. ∎

We have now established that the velocity field is incompressible; that is,
the multiplier satisfies $m(\xi)\cdot\xi=0$. Our next task is to show that the
mSQG nonlinearity is the only one consistent with the special bound (S). This
fact follows from the following proposition.

######  Proposition 9.2.

Let $T^{\ell}$ be incompressible with odd multiplier $m$ that is homogeneous
of degree $-1+2\delta$. If $m$ is not a scalar multiple of the mSQG
multiplier, then there cannot exist a bound of the form (S) for the
corresponding nonlinearity.

###### Proof.

Let $M^{\ell}$ be the mSQG multiplier that is homogeneous of degree
$-1+2\delta$, which is a vector field tangent to the unit circle of constant