# Small scale creation of the Lagrangian flow in 2d perfect fluids

Ayman Rimah Said

( )

###### Abstract

In this paper we prove that for all solutions of the 2d Euler equations with
initial vorticity with finite Sobolev smoothness an initial data dependent
norm of the associated Lagrangian flow blows up in infinite time at least like
$t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact
$L^{2}$ decay of the Fourier transform of the solution. This adapted norm
turns out to be the exact quantity that controls a low to high frequency
cascade which we then show to be the quantitative phenomenon behind the
Lyapunov construction by Shnirelman in [24].

##  1 Introduction

We study 2d inviscid flows

$\partial_{t}\omega+u\cdot\nabla\omega=0,$ (1.1) $u=\nabla^{\perp}\psi\text{
and }\Delta\psi=\omega.$ (1.2)

Here, the scalar vorticity
$\omega:\mathbb{R}^{2}\times\mathbb{R}\rightarrow\mathbb{R}$ is transported by
the velocity field $u:\mathbb{R}^{2}\times\mathbb{R}\rightarrow\mathbb{R}^{2}$
which is uniquely determined at each time $t\in\mathbb{R}$ from $\omega$ using
the Newtonian potential:

$u(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\frac{(x-y)^{\perp}}{|x-y|^{2}}\omega(y)dy,\text{
and reciprocally }\omega=\nabla\times u=\partial_{1}u_{2}-\partial_{2}u_{1}.$
(1.3)

We adopt the standard notation $v^{\perp}=(-v_{2},v_{1})$ for
$v=(v_{1},v_{2})\in\mathbb{R}^{2}.$ It is well known that smooth enough
solutions to the 2d Euler equations (1.1)-(1.2) retain their smoothness for
all finite times. Much less is known in the infinite-time limit. The long time
behavior seems to consistently show some type of small scale creation for
smooth solutions[8, 26], which can be summed up in the following conjecture by
Yudovich.

######  Conjecture 1.1 (Yudovich (1974), [28, 29], quote from [21]).

There is a “substantial set” of inviscid incompressible flows whose vorticity
gradients grow without bound. At least this set is dense enough to provide the
loss of smoothness for some arbitrarily small disturbance of every steady
flow.

The literature towards this conjecture is rich. Of note is the result of Koch
[16] in which strong growth of Hölder and Sobolev norms of the vorticity is
established near any background solution (stationary or time-dependent) for
which the gradient of the flow map is unbounded in time. Yudovich also
established (boundary induced) growth results under some mild assumption on
the data near the boundary of the domain [29] (see also [21] for an extension
of [29]). The conjecture was established within m-fold symmetry for $m\geq 3$
by Elgindi, Murray and the author in [11]. There are also numerous important
results on growth of solutions in the neighborhood of stable steady states [8,
10, 22, 15, 30]. In the case of open neighborhoods of shearing stable steady
states a finer version of the conjecture including generic fluid aging has
been recently established by Drivas, Elgindi and Jeong [9].

The main Theorem of this paper can be stated informally as follows.

######  Theorem 1.2.

Consider the 2d Euler equation on $\mathbb{R}^{2}.$ Then for $\omega_{0}\in
H^{s}(\mathbb{R}^{2})\setminus H^{s+\epsilon}(\mathbb{R}^{2})$ for some $s>1$
and all $\epsilon>0$, then roughly the s+1 derivative of the Lagrangian flow
blows up in infinite time at least like $t^{\frac{1}{3}}$.

All the results in this paper can be naturally generalised to the periodic 2d
Euler equation on $\mathbb{T}^{2}$ and an analogue of Theorem 1.2 holds on
$\mathbb{T}^{2}$.

###  1.1 Propagation of exact smoothness in the 2d Euler equation

We recall the standard global well-posedness theory of the 2d Euler equation
in sub-critical Sobolev spaces, see for example the following excellent
monographs as an introduction to the study of the Euler equations [5, 6, 13,
19]. We define the Lagrangian flow

$\frac{d}{dt}\Phi_{t}=u\circ\Phi_{t}\text{ with }\Phi_{0}(\cdot)=Id.$

######  Theorem 1.3.

Consider $s>1$ and $\omega_{0}\in H^{s}\left(\mathbb{R}^{2}\right)$, then
there exists a unique solution $\omega\in
C\left(\mathbb{R},H^{s}\left(\mathbb{R}^{2}\right)\right)$ of (1.1)-(1.2) with
initial data $\omega_{0}$. Moreover there exists a universal constant $C_{s}$
such that we have following estimate

$\left\|\omega(t)\right\|_{H^{s}}+\left\|\Phi_{t}-Id\right\|_{H^{s+1}}\leq
C_{s}\left\|\omega_{0}\right\|_{H^{s}}\exp\left(\exp\left(C_{s}\left\|\omega_{0}\right\|_{L^{2}\cap
L^{\infty}}t\right)\right).$

One of the first results of this paper is to describe a maximal propagation of
smoothness result for the 2d Euler equation. First we set, for $\omega_{0}\in
L^{2}$, $\epsilon\geq 0$

$dr_{\omega_{0}}(\epsilon)=\left(\int_{\mathbb{R}^{2}\setminus
B\left(0,\frac{1}{\epsilon}\right)}\left|\mathscr{F}(\omega_{0})(\xi)\right|^{2}d\xi\right)^{\frac{1}{2}}$

We note that $dr_{\omega_{0}}$ is an increasing function of $\epsilon$ with
$\displaystyle\lim_{\epsilon\to 0}dr_{\omega_{0}}(\epsilon)=0$ and
$\displaystyle\lim_{\epsilon\to+\infty}dr_{\omega_{0}}(\epsilon)=\left\|\omega_{0}\right\|_{L^{2}(\mathbb{R}^{2})}$.
For $f\in\mathscr{S}(\mathbb{R}^{2})$ such that $dr_{f}$ vanishes at $0$ at
least like $dr_{\omega_{0}}$, we define

$\left\|f\right\|_{\omega_{0}}=\sup_{\epsilon\geq
0}\frac{dr_{f}(\epsilon)}{dr_{\omega_{0}}(\epsilon)}.$

Note that by construction $\left\|\omega_{0}\right\|_{\omega_{0}}=1$.

######  Theorem 1.4.

Consider $s>1$ and $\omega_{0}\in H^{s}\left(\mathbb{R}^{2}\right)$ and
$\omega\in C\left(\mathbb{R},H^{s}\left(\mathbb{R}^{2}\right)\right)$ the
unique solution of (1.1)-(1.2) with initial data $\omega_{0}$. Suppose that
for all $\lambda<1$

$C_{\omega_{0}}(\lambda)dr_{\omega_{0}}(\epsilon)\leq
dr_{\omega_{0}}(\lambda\epsilon),$ (1.4)

for some function $C_{\omega_{0}}$ independent of $\epsilon$. Then there
exists a constant $C_{\omega_{0}}^{\prime}$ such that

$\left\|\omega(t)\right\|_{\omega_{0}}+\left\|D\Phi_{t}-Id\right\|_{\omega_{0}}\leq\exp\left(\exp\left(C_{\omega_{0}}^{\prime}t\right)\right)$

From the proof of the of the previous theorem we show that the condition (1.4)
is equivalent to the fact that $\omega_{0}$’s Fourier transform decays at most
algebraically fast at infinity, in particular the previous theorem applies for
all $\omega_{0}\in H^{s_{1}}(\mathbb{R}^{2})\setminus
H^{s_{2}}(\mathbb{R}^{2})$ for a pair $1<s_{1}<s_{2}$. We believe that this
type of exact smoothness propagation holds more generally for hyperbolic
evolution PDEs. For example the proof here works in verbatim to give an
analogous result (locally in time) for the SQG equation.

In some sense this is an optimal sub-critical smoothness propagation statement
and answers the $L^{2}$ based version of Problem 23 of [26] for the
persistence of “kinks” in fluid flows. It is not hard to see from the proof of
Theorem 1.4, specifically from Proposition 2.3 that an analogous definition of
an $\omega_{0}$ dependent norm that captures sub-critical $L^{p}$ based
“kinks” are also being propagated by the flow of the 2d Euler equations for
$p\in[1,+\infty]$.

###  1.2 The forward frequency cascade

Towards understanding long time behavior of a dynamical system one of the main
tools is the construction of Lyapunov functions, in the case of the 2d Euler
equations very few such examples are known [11, 17, 21, 24, 29]. The proof of
Theorem 1.2 relies on the careful study of Shnirelman’s pioneering Lyapunov
construction [24] and supplementing it with optimal quantitative estimates
which turn out to be the construction’s most natural setting and in particular
for the so called “microlocal scalar product”. This allows us to significantly
simplify the construction.

######  Remark 1.5.

We record here part of S. Alinhac’s MathSciNet review of A. Shnirelman’s
paper: “ This may all seem very complicated, but it is not, the technical
“complications” arising naturally in the course of a basically simple
argument. Moreover, the new tool (the microlocal scalar product) introduced by
the author is certainly likely to have many other applications, just as
similar tools (microlocal defect measures, Young measures, etc.) already have
in the theory of weak solutions of nonlinear PDE, in homogenization, in
control theory, etc. Finally, we would like to emphasise what we believe can
be learned from the approach of the author: considering such an old problem as
the fluid flow, it is likely that not many new mathematical results are going
to be obtained by nineteenth-century PDE methods. Even researchers oriented
towards applications will have to incorporate Shnirelman’s results into their
research, just because these results, far from being some (irrelevant)
refinement of basically well understood things, are the first rigorous ones in
the subject.”

To state the theorem on the Lyapunov construction we need to introduce the
notion of paraproducts. For the reader unfamiliar with paraproducts we give a
heuristic construction below.

####  $\bullet$ Paraproducts

For the sake of this discussion let us pretend that $\partial_{x}$ is left-
invertible with a choice of $\partial_{x}^{-1}$ that acts continuously from
$H^{s}$ to $H^{s+1}$. We follow here analogous ideas to the ones presented by
Shnirelman in [25]. One way to define the paraproduct of two functions $f,g\in
H^{s}$ with $s$ sufficiently large is: we differentiate $fg$ $k$ times, using
the Leibniz formula, and then restore the function $fg$ by the $k$-th power of
$\partial_{x}^{-1}$:

$\displaystyle fg$ $\displaystyle=\partial_{x}^{-k}\partial_{x}^{k}(fg)$
$\displaystyle=\partial_{x}^{-k}\big{(}g\partial_{x}^{k}f+k\partial_{x}g\partial_{x}^{k-1}f+\dots+k\partial_{x}f\partial_{x}^{k-1}g+g\partial_{x}^{k}f\big{)}$
$\displaystyle=T_{g}f+T_{f}g+R,$

where,

$T_{g}f=\partial_{x}^{-k}\big{(}g\partial_{x}^{k}f\big{)},\ \
T_{f}g=\partial_{x}^{-k}\big{(}f\partial_{x}^{k}g\big{)},$

and $R$ is the sum of all remaining terms. The key observation is that if
$s>\frac{1}{2}+k$, then $g\mapsto T_{f}g$ is a continuous operator in $H^{s}$
for $f\in H^{s-k}$. The remainder $R$ is a continuous bilinear operator from
$H^{s}$ to $H^{s+1}$. The operator $T_{f}g$ is called the paraproduct of $g$
and $f$ and can be interpreted as follows. The term $T_{f}g$ takes into play
high frequencies of $g$ compared to those of $f$ and demands more regularity
in $g\in H^{s}$ than $f\in H^{s-k}$ thus the term $T_{f}g$ bears the
“singularities” brought on by $g$ in the product $fg$. Symmetrically $T_{g}f$
bears the ”singularities” brought on by $f$ in the product $fg$ and the
remainder $R$ is a smoother function ($H^{s+1}$) and does not contribute to
the main singularities of the product.

####  $\bullet$ Paradifferential operators

To get a good intuition of a paradifferential operator $T_{p(x,\xi)}$ with
symbol $p(x,\xi)\in\Gamma^{\beta}_{\rho}(\mathbb{R}^{2})$, as a first gross
approximation, one can think of $p(x,\xi)\approx f(x)m(\xi)$ and
$T_{p(x,\xi)}$ as the composition of a paraproduct $T_{f}$ with a Fourier
multiplier $m(D)$, that is:

$T_{p(x,\xi)}\approx T_{f}m(D),\text{ with }f\in W^{\rho,\infty}\text{ and
}m\text{ is of order }\beta.$

Indeed following Coifman and Meyer’s symbol reduction given in Proposition $5$
of [7], one can show that linear combinations of composition of a paraproduct
with a Fourier multiplier are dense in the space of paradifferential
operators. Definition A.7 gives the rigorous formula for $T_{p(x,\xi)}$.

We are now in position to state the theorem summarising the Lyapunov
construction.

######  Theorem 1.6.

Consider $\chi(\xi)\in C^{\infty}_{0}(\mathbb{R}^{2}\setminus B(0,1))$ and
$\omega_{0}\in H^{s}$ with $s>1$ verifying (1.4) then there exists a universal
constant $C$ and a constant $C_{\omega_{0}}$ such that for $\epsilon\geq 0$

$\displaystyle\frac{d}{dt}\left(\nabla\times
T_{[D\Phi_{t}]^{-1}}\Phi_{t},\chi(\epsilon
D)\omega_{0}\right)_{L^{2}}=\left\|T_{\frac{\left|\xi\right|}{\left|[D\Phi_{t}]^{-1}\xi\right|}}\chi(\epsilon
D)\omega_{0}\right\|^{2}_{L^{2}}+O\left(C_{\delta}e^{Ce^{C_{\omega_{0}}t}}\epsilon^{\min(s-1-\delta,1)}dr_{\omega_{0}}(\epsilon)^{2}\right),$

for all $0<\delta<s-1$ and $C_{\delta}>0$ is constant depending only on
$\delta$.

######  Remark 1.7.

In [24] Shnirelman proved the previous theorem in the case
$dr_{\omega_{0}}(\epsilon)=O(\epsilon^{s}),s>2$. Theorem 1.6 generalises this
result to the exact regularity of $\omega_{0}$ whatever it is and gives the
optimal control in $\epsilon$.

In particular the leading order decay in $\epsilon$ of $\left(\nabla\times
T_{[D\Phi_{t}]^{-1}}\Phi_{t},\chi(\epsilon D)\omega_{0}\right)$ is

$\int_{0}^{t}\left\|T_{\frac{\left|\xi\right|}{\left|[D\Phi_{r}]^{-1}\xi\right|}}\chi(\epsilon
D)\omega_{0}\right\|^{2}_{L^{2}}dr\underset{\epsilon\to
0}{\sim}c(t)dr_{\omega_{0}}(\epsilon)^{2}$

where $c(t)$ is an increasing function of time. The explicit estimate on the
residual term allows for the following interpretation of the previous result.
Fixing an outer frequency region $\\{\left|\xi\right|\geq R\\}$ then there
exists $T_{R}>0$ increasing in $R$ such that for $\left|t\right|\leq T_{R}$
there is an averaged forward frequency cascade of $\Phi_{t}$ in the signed
measure $\mathscr{F}(\omega_{0})(\xi)d\xi$ into the region
$\\{\left|\xi\right|\geq R\\}$. Thus there is always a positive flux of
frequency at “infinity” ($R\to\infty$) and the growth of that rate gives the
desired Lyapunov function. Using $dr_{\omega_{0}}$, it is given explicitly by
a re-normalised version of the semi-classical measure first introduced in [12]
and independently as the Wigner measure in [18] which is the so called
microlocal scalar product introduced by Shnirelman in [24]

$L_{\chi,\omega_{0}}(\Phi_{t})=\limsup_{\epsilon\to
0}\frac{\left(\chi(\epsilon D)\nabla\times
T_{[D\Phi_{t}]^{-1}}\Phi_{t},\chi(\epsilon
D)\omega_{0}\right)_{L^{2}}}{dr_{\omega_{0}}(\epsilon)^{2}}.$

To give a geometrical interpretation of what is measured by this Lyapunov
function we need to heuristically introduce the notion of paracomposition
which is defined rigorously in Theorem A.14.

####  $\bullet$ Paracomposition

We again work with $f\in H^{s}$ and $g\in C^{s}$ with $s$ large and consider
the composition of two functions $f\circ g$ which bears the singularities of
both $f$ and $g$, and our goal is to separate them. We proceed as before by
differentiating $f\circ g$ $k$ times, using the Faá di Bruno’s formula, and
then restore the function $fg$ by the $k$-th power of $\partial_{x}^{-1}$:

$\displaystyle f\circ g$
$\displaystyle=\partial_{x}^{-k}\partial_{x}^{k}(f\circ g)$
$\displaystyle=\partial_{x}^{-k}\big{(}(\partial_{x}^{k}f\circ
g)\cdot(\partial_{x}g)^{k}+\dots+(\partial_{x}f\circ
g)\cdot\partial_{x}^{k}g\big{)}$ $\displaystyle=g^{*}f+T_{\partial_{x}f\circ
g}g+R,$

where,

$g^{*}f=\partial_{x}^{-k}\big{(}(\partial_{x}^{k}f\circ
g)\cdot(\partial_{x}g)^{k}\big{)}\text{ is the paracomposition of $f$ by $g$}$

and $R$ is the sum of all remaining terms. Again the key observation is that
if $s>\frac{1}{2}+k$, then $f\mapsto g^{*}f$ is a continuous operator in
$H^{s}$ for $g\in C^{s-k}$. Thus this term bears essentially the singularities
of $f$ in $f\circ g$. As before $T_{\partial_{x}f\circ g}g$ bears essentially
the singularities of $g$ in $f\circ g$. The remainder $R$ is a continuous
bilinear operator from $H^{s}$ to $H^{s+1}$. Thus we have separated the
singularities of the composition $f\circ g$.

The geometric interpretation of the left hand side in Theorem 1.6 is computed
explicitly in the proof as

$\frac{d}{dt}T_{[D\Phi_{t}]^{-1}}\left(\Phi_{t}-Id\right)\approx
T_{[D\Phi_{t}]^{-1}}\Phi_{t}^{*}u,$

now recall that the pull back of $u$ by $\Phi_{t}$ is given by
$[D\Phi_{t}]^{-1}u\circ\Phi_{t}$, then the right hand side in the previous
identity can be interpreted as a paradifferential version of this pull-back
which “selects” the high frequencies of $u$ compared to $\Phi_{t}$. Thus

$\frac{d}{dt}\left(\nabla\times
T_{[D\Phi_{t}]^{-1}}\left(\Phi_{t}-Id\right),\chi(\epsilon
D)\omega_{0}\right)_{L^{2}}\approx\left(\nabla\times
T_{[D\Phi_{t}]^{-1}}\Phi_{t}^{*}u,\chi(\epsilon D)\omega_{0}\right)_{L^{2}},$

and the key observation is that the projection of the curl of the “para”pull-
back of the velocity to the high frequencies of the initial vorticity is
positive to leading order.

Another interpretation of the dynamical phenomena captured in the previous
theorem which is essentially the idea behind the proof and the key observation
in [24] is the following. We write

$\frac{d}{dt}\Phi_{t}=[\nabla^{\perp}\Delta^{-1}\left(\omega_{0}\circ\Phi_{t}^{-1}\right)]\circ\Phi_{t}=\left(\nabla^{\perp}\Delta^{-1}\right)^{*,\Phi_{t}}\omega_{0},$

where $\left(\nabla^{\perp}\Delta^{-1}\right)^{*,\Phi_{t}}$ is the pulled back
operator by $\Phi_{t}$. Thus morally the main term of the left hand is
$\nabla\times
T_{[D\Phi_{t}]^{-1}}\left(\nabla^{\perp}\Delta^{-1}\right)^{*,\Phi_{t}}$. Now
note that initially

$\nabla\times
T_{[D\Phi_{t}]^{-1}}\left(\nabla^{\perp}\Delta^{-1}\right)^{*,\Phi_{t}}_{|t=0}=\nabla\times
T_{Id}\nabla^{\perp}\Delta^{-1}\approx Id,$

thus the identity given in Theorem 1.6 can be interpreted as a measure of how
the flow “twists” the identity. More precisely the use of the semi-classical
cut-off in frequency, $\chi(\epsilon D)$, gives that the left hand-side of the
identity given in Theorem 1.6 is the study of the principal symbol of the
operator $\nabla\times
T_{[D\Phi_{t}]^{-1}}\left(\nabla^{\perp}\Delta^{-1}\right)^{*,\Phi_{t}}$. The
right hand side then gives that in the small scales ($\epsilon\to 0$) the
Lagrangian flow $\Phi_{t}$ is consistently getting farther and farther from
the identity, that is small scales are continuously being created.

The use of $T_{[D\Phi_{t}]^{-1}}$ in the previous operator is not only the
natural factor appearing in the pull-back of a velocity field $u$ by a flow
$\Phi_{t}$, it also concretly brings into play a key cancellation in the
principal symbol that can be seen as follows:

$\frac{d}{dt}\left([D\Phi_{t}]^{-1}\Phi_{t}\right)=[D\Phi_{t}]^{-1}\left(u\circ\Phi_{t}-Du\circ\Phi_{t}\Phi_{t}\right).$

The right hand side is of the form $G(x)=F(x)-F^{\prime}(x)x$ and we note the
cancellation $G^{\prime}(x)=-F^{\prime\prime}(x)x$. To get useful quantitative
bounds from this cancellation in the high frequencies it is natural to use
paraproducts and paracomposition. Then the formula obtained on the right hand
side in Theorem 1.6 can be seen as a direct consequence of this cancellation
combined with the algebra property of pseudodifferential and paradifferential
operators combined with the fact that restricted to principal symbols this
algebra becomes commutative.

######  Remark 1.8.

The proof of the previous theorem applies more generally for $\chi(\epsilon
D)$ replaced with $a(x,\epsilon D)$ with $a$ in the (standard) Hörmander
symbol class $S^{0}_{1,0}(\mathbb{R}^{2})$ and $T_{a}$ with $a$ in the
paradifferential symbol class $\Gamma_{0}^{1}(\mathbb{R}^{2})$. The control on
$a$ needed in the residual term is it’s symbolic semi-norm $M_{1}^{0}(a;2)$
given in Definition A.3. Note that a large number of such Lyapunov functions
can be constructed by changing the choice of $a$.

An immediate corollary of Theorem 1.6 is the following blow up result.

######  Corollary 1.9.

Consider $\omega_{0}\in H^{s}$ with $s>1$ verifying (1.4) then
$\left\|D\Phi_{t}-Id\right\|_{\omega_{0}}$ blows up at least like
$t^{\frac{1}{3}}$ in infinite time.

We note that by Proposition 2.3 that if
$dr_{\omega_{0}}(\epsilon)\underset{\epsilon\to 0}{\sim}\epsilon^{s}$ then
$\left\|\cdot\right\|_{B^{s}_{2,\infty}}$ and
$\left\|\cdot\right\|_{\omega_{0}}$ are equivalent. In particular the previous
corollary gives the growth of
$\left\|\Phi_{t}-Id\right\|_{B^{s+1}_{2,\infty}}$ for $\omega_{0}\in
B^{s}_{2,\infty}$ with $dr_{\omega_{0}}(\epsilon)\underset{\epsilon\to
0}{\sim}\epsilon^{s}$.

###  1.3 Active scalar equations

The construction given in Theorem 1.6 is not special to the 2d Euler equations
but can be adapted to a large class of active scalar equations as for example
the generalised SQG equations given by

$\partial_{t}\Theta+u\cdot\nabla\Theta=0,$ (1.5)
$u=\nabla^{\perp}(-\Delta)^{-\frac{\alpha}{2}}\Theta.$ (1.6)

Note that for $\alpha=2$ we get the 2d Euler equations. The results in this
paper generalise in verbatim to give the following.

######  Theorem 1.10.

Consider $\alpha>1$, $s>3-\alpha$, $\Theta_{0}\in
H^{s}\left(\mathbb{R}^{2}\right)$ and $\Theta\in
C\left([0,T],H^{s}\left(\mathbb{R}^{2}\right)\right)$ the unique solution of
(1.5)-(1.6) with initial data $\Theta_{0}$ for some
$T\geq\frac{C_{\delta}}{\left\|\Theta_{0}\right\|_{H^{3-\alpha-\delta}}}$ with
$0<\delta<s+\alpha-3$ and $C_{\delta}>0$ is a constant depending only on
$\delta$. Suppose moreover that $\Theta_{0}$ verifies (1.4), then there exists
a constant $C_{\Theta_{0}}$ such that for $0\leq t\leq T$

$\left\|\Theta(t)\right\|_{\Theta_{0}}+\left\|D\Phi_{t}-Id\right\|_{\Theta_{0}}\leq\exp\left(C_{\Theta_{0}}\int^{t}_{0}\left\|\nabla
u(s)\right\|_{L^{\infty}}ds\right).$

Moreover fix $\chi(\xi)\in C^{\infty}_{0}(\mathbb{R}^{2}\setminus B(0,1))$
then there exists a constant $C_{\Theta_{0}}$ such that for $\epsilon\geq 0$

$\frac{d}{dt}\left((-\Delta)^{\frac{\alpha-2}{2}}\nabla\times
T_{[D\Phi_{t}]^{-1}}\Phi_{t},\chi(\epsilon
D)\Theta_{0}\right)_{L^{2}}=-\left\|T_{\frac{\left|\xi\right|^{\frac{\alpha}{2}}}{\left|[D\Phi_{t}]^{-1}\xi\right|^{\frac{\alpha}{2}}}}\chi(\epsilon
D)\Theta_{0}\right\|^{2}_{L^{2}}\\\
+O\left(C_{\delta}\exp\left(C_{\Theta_{0}}\int^{t}_{0}\left\|\nabla
u(s)\right\|_{L^{\infty}}ds\right)\epsilon^{\min(s+\alpha-3-\delta,1,\alpha-1)}dr_{\Theta_{0}}(\epsilon)^{2}\right),$

for all $0<\delta<s+\alpha-3$ and $C_{\delta}>0$ is constant depending only on
$\delta$.

###  1.4 Organisation of the paper

In Section 2 we give the proof of Theorem 1.4. In Section 3 we discuss the
rates of convergence of semi-classical measures and show that they are exactly
given by $dr_{f}$. In Section 4 we give the proof of Theorem 1.6 and Corollary
1.9. Finally in Appendix A we give a review of the microlocal analysis notions
needed in this article.

## Acknowledgements

The author acknowledges funding from the UKRI grant SWAT. I would like to
thank T. Alazard for suggesting to go beyond the Besov setting of the previous
construction and his generosity in sharing techniques and knowledge, J.
Bedrossian for having suggested to look for singularity formation using
Shnirelman’s construction, T. M. Elgindi for helpful discussions and remarks
that helped the work take shape and N. Tzvetkov for an insightful discussion
that helped better my understanding of the new norm used here. The starting
idea of this work emanated from T. Drivas’s suggestion and encouragement to
study A. Shnirelman’s construction, he also contributed valuable input along
the realisation of this project for which I am deeply indebted.

##  2 Exact smoothness propagation in 2d Euler

In this section we give the proof of Theorem 1.4. First for
$\left\|\omega\right\|_{\omega_{0}}$, we paralinearise the Euler equation to
get

$\partial_{t}\omega+T_{u}\cdot\nabla\omega=-T_{\nabla\omega}\cdot
u-R(u,\nabla\omega),$

thus commuting with $\chi(\epsilon D)$ we get

$\partial_{t}\chi(\epsilon D)\omega+T_{u}\cdot\nabla\chi(\epsilon
D)\omega=-\left[\chi(\epsilon D),T_{u}\cdot\nabla\right]\omega-\chi(\epsilon
D)T_{\nabla\omega}\cdot u-\chi(\epsilon D)R(u,\nabla\omega).$

Now by the continuity of paradifferential operators given in Theorem A.10,
combined with the symbolic calculus given in Theorem A.11, the spectral
localisation property of paradifferential operators A.9 we see that

$\left\|\left[\chi(\epsilon
D),T_{u}\cdot\nabla\right]\omega\right\|_{L^{2}}\leq
C\left\|Du\right\|_{L^{\infty}}dr_{\omega_{0}}(\lambda\epsilon)\leq
C\left\|Du\right\|_{L^{\infty}}\left\|\omega\right\|_{\omega_{0}}dr_{\omega_{0}}(\lambda\epsilon),$

thus

$\left\|\left[\chi(\epsilon
D),T_{u}\cdot\nabla\right]\omega\right\|_{L^{2}}\leq
C_{\omega_{0}}\left\|Du\right\|_{L^{\infty}}\left\|\omega\right\|_{\omega_{0}}dr_{\omega_{0}}(\epsilon).$

The previous estimate holds analogously for $\chi(\epsilon
D)T_{\nabla\omega}\cdot u$ and $\chi(\epsilon D)R(u,\nabla\omega)$ and the
desired result follows from a standard energy estimate. To get the estimate on
$\left\|D\Phi_{t}-x\right\|_{\omega_{0}}$ we need a more refined
characterisation of $\left\|\cdot\right\|_{\omega_{0}}$ and the action of
multiplication and composition on this norm.

###  2.1 Analysis of the norm $\left\|\cdot\right\|_{\omega_{0}}$

First we show that (1.4) is equivalent to having finite smoothness.

######  Proposition 2.1.

Consider $f\in L^{2}(\mathbb{R}^{2})$ verifying (1.4) then there exits $s>0$
such that $f\notin H^{s}(\mathbb{R}^{2})$. Conversely if $f\in
L^{2}(\mathbb{R}^{2})$ and there exists $s>0$ such that $f\notin
H^{s}(\mathbb{R}^{2})$ then $f$ verifies (1.4) moreover $C_{f}$ can be chosen
in the form

$C_{f}(\lambda)\leq
C_{\delta}\left\|f\right\|_{H^{s_{f}}}\lambda^{s_{f}+\delta},$

where $s_{f}\geq 0$ is the largest index such that $f\in
H^{s_{f}}(\mathbb{R}^{2})\setminus H^{s_{f}+\delta}(\mathbb{R}^{2})$ for all
$\delta>0$.

###### Proof of Proposition 2.1.

The proof follows from the following lemma.

######  Lemma 2.2.

Consider a continuous decreasing function $F:\mathbb{R}_{+}\to\mathbb{R}_{+}$
such that $\displaystyle\lim_{x\to+\infty}F(x)=0$. Suppose that for all
$\lambda\geq 1$, $\displaystyle
C(\lambda)=\inf_{x\in\mathbb{R}_{+}}\frac{F(\lambda x)}{F(x)}>0$. Then there
exists $\alpha$ and $C_{\alpha}\geq 0$ such that $\displaystyle
C(\lambda)\geq\frac{C_{\alpha}}{1+\lambda^{\alpha}}$. Conversely if there
exits $\alpha>0$ such that $F(x)\geq\frac{C_{\alpha}}{1+x^{\alpha}}$. Then for
all $\lambda\geq 1$, $\displaystyle
C(\lambda)=\inf_{x\in\mathbb{R}_{+}}\frac{F(\lambda x)}{F(x)}>0$.

###### Poof of Lemma 2.2.

We note that $C$ is a decreasing function that goes from $1$ to $0$ which
moreover verifies a Cauchy functional inequality of the form

$C(\lambda_{1})C(\lambda_{2})\leq C(\lambda_{1}\lambda_{2}).$

For $x\geq 0$ we define $f(x)=-\ln(C(e^{x}))$, which is well defined as
$C(\lambda)>0$. Then we get that $f$ is a positive increasing subadditive
function

$f(x+y)\leq f(x)+f(y),$

thus $\lim_{x\to+\infty}\frac{f(x)}{x}=\inf_{x\geq 0}\frac{f(x)}{x}=\alpha\geq
0$ which gives the desired result.

For the second part of the lemma we proceed by contradiction and suppose that
there exists $\lambda\geq 1$ and a sequence
$x_{n}\underset{n\to+\infty}{\to}+\infty$ such that

$\frac{F(\lambda x_{n})}{F(x_{n})}\underset{n\to+\infty}{\to}0.$

By hypothesis there exists $\epsilon_{n}$ such that $F(x_{n})\leq
x_{n}^{-\alpha+\epsilon_{n}}$ thus

$\frac{F(\lambda
x_{n})}{F(x_{n})}\geq\lambda^{\alpha}\underbrace{x_{n}^{\epsilon_{n}}}_{\geq
1}\geq\lambda^{\alpha},$

which is a contradiction. ∎

∎

Next we give a characterisation of $\left\|\cdot\right\|_{\omega_{0}}$.

######  Proposition 2.3.

Consider $\omega_{0}\in L^{2}(\mathbb{R}^{2})$ verifying (1.4) then there
exist a constant $C_{\omega_{0}}\geq 1$ such that

$\frac{1}{C_{\omega_{0}}}\left\|f\right\|_{\omega_{0}}\leq\sup_{k\geq
0}\frac{1}{dr_{\omega_{0}(2^{-k})}}\left\|\Delta_{k}f\right\|_{L^{2}}:=\left|f\right|_{\omega_{0}}\leq
C_{\omega_{0}}\left\|f\right\|_{\omega_{0}}.$

where $\Delta_{k}f$ are the Littlewood-Paley projectors given in Definition
A.1.

###### Proof.

We introduce the norm

$\sup_{k\geq
0}\frac{1}{dr_{\omega_{0}(2^{-k})}}\sum^{+\infty}_{i=k}\left\|\Delta_{i}f\right\|_{L^{2}}=\left\|f\right\|^{\prime}_{\omega_{0}}$

then by the slow varying hypothesis on $dr_{\omega_{0}}$ given by (1.4),
$\left\|f\right\|^{\prime}_{\omega_{0}}$ and $\left\|f\right\|_{\omega_{0}}$
are equivalent (we refer to Chapter 1 of [27] for a more general treatment of
spaces defined with a slow varying moduli). Next we have the immediate
inequality
$\left|f\right|_{\omega_{0}}\leq\left\|f\right\|^{\prime}_{\omega_{0}}$ and
conversely

$\frac{1}{dr_{\omega_{0}(2^{-k})}}\sum^{+\infty}_{i=k}\left\|\Delta_{i}f\right\|_{L^{2}}\leq\left|f\right|_{\omega_{0}}\frac{1}{dr_{\omega_{0}(2^{-k})}}\sum^{+\infty}_{i=k}dr_{\omega_{0}}(2^{-i}).$

Now by Proposition 2.1

$\sum^{+\infty}_{i=k}dr_{\omega_{0}}(2^{-i})=\sum^{+\infty}_{i=0}dr_{\omega_{0}}(2^{-i-k})\leq
C_{\omega_{0}}dr_{\omega_{0}}(2^{-k})\underbrace{\sum^{+\infty}_{i=0}2^{-\alpha_{\omega_{0}}i}}_{<+\infty},$

which gives the desired result. ∎

###  2.2 Actions of operators on $\left\|\cdot\right\|_{\omega_{0}}$

We first give a lemma on the action of multiplication on
$\left\|\cdot\right\|_{\omega_{0}}$.

######  Lemma 2.4.

Consider $f,g\in H^{s}\left(\mathbb{R}^{2}\right)$ for $s>1$ and
$\omega_{0}\in H^{\epsilon}\left(\mathbb{R}^{2}\right)$ with $\epsilon>0$ and
$\omega_{0}$ verifying hypothesis (1.4) and then there exists $C_{\omega_{0}}$
such that

$\left\|fg\right\|_{\omega_{0}}\leq\frac{C_{\omega_{0}}}{\epsilon}\left\|f\right\|_{L^{\infty}}\left\|g\right\|_{\omega_{0}}+\frac{C_{\omega_{0}}}{\epsilon}\left\|g\right\|_{L^{\infty}}\left\|f\right\|_{\omega_{0}}.$

More generally we have for the paraproduct and residual paraproduct given in
Theorem A.13:

$\left\|T_{f}g\right\|_{\omega_{0}}\leq
C_{\omega_{0}}\left\|f\right\|_{L^{\infty}}\left\|g\right\|_{\omega_{0}}\text{
and
}\left\||D|^{r}R(f,g)\right\|_{\omega_{0}}\leq\frac{C_{\omega_{0},r}}{\epsilon}\left\|f\right\|_{W^{r,\infty}}\left\|g\right\|_{\omega_{0}},$

and for a paradifferential operator with symbol
$a\in\Gamma^{0}_{0}(\mathbb{R}^{2})$ given in Definition A.6

$\left\|T_{a}f\right\|_{\omega_{0}}\leq
C_{\omega_{0}}M_{0}^{0}(a;2)\left\|g\right\|_{\omega_{0}}.$

###### Proof.

We follow the classical approach in [4, 27]. For the product estimate we
decompose $fg$ as

$fg=T_{g}f+T_{f}g+R(f,g),$

thus it suffices to prove the estimates on $T_{f}g$ and $R(f,g)$. For $T_{f}g$
we compute

$\Delta_{k}(T_{f}g)=\sum^{k+N}_{\begin{subarray}{c}i\geq 0\\\
i=k-N\end{subarray}}\Delta_{k}\left(P_{\leq i-1}(D)f\Delta_{i}g\right),$

for a fixed $N$, thus by the slow varying property of $dr_{\omega_{0}}$ we get
the desired bound on $|T_{f}g|_{\omega_{0}}$. Next for $R(f,g)$ we write for

$R(f,g)=\sum_{q}R_{q}\text{ with
}R_{q}=\sum^{1}_{i=-1}\Delta_{q-i}f\Delta_{q}g$

and we note that $R_{q}$ is supported in a ball $2^{q}B(0,\lambda)$ for a
fixed $\lambda$. We now need the following standard lemma which follows in
verbatim from the proof of Lemmas 2.49 and 2.84 of [4] and the slow varying
property (1.4) of $dr_{\omega_{0}}$.

######  Lemma 2.5.

Consider $\omega_{0}\in H^{\epsilon}(\mathbb{R}^{2})$ with $\epsilon>0$
verifying (1.4) then there exist a constant $C_{\omega_{0}}\geq 1$ such that

$\frac{\epsilon}{C_{\omega_{0}}}\left\|f\right\|_{\omega_{0}}\leq\sup_{k\geq
0}\frac{1}{dr_{\omega_{0}(2^{-k})}}\left\|P_{\leq
k}(D)f\right\|_{L^{2}}\leq\frac{C_{\omega_{0}}}{\epsilon}\left\|f\right\|_{\omega_{0}}.$

where $P_{k}(D),\ P_{\leq k}(D)$ are Littlewood-Paley projectors are the
Littlewood-Paley projectors given in Definition A.1.

Applying the previous Lemma to $R_{q}$,

$\left\||D|^{r}R(f,g)\right\|_{L^{2}}\leq
2^{qr}\left\|\Delta_{q}g\right\|_{L^{2}}\sum^{1}_{i=-1}\left\|\Delta_{q-i}f\right\|_{L^{\infty}},$

giving again the stated bound on $|R(f,g)|_{\omega_{0}}$. For a $0$th order
paradifferential operators following [7, 3], $T_{a}$ can be written as a sum
of a rapidly decreasing sequence of “elementary symbols”, where an elementary
symbol is of the form

$q(x,\xi)=\sum^{+\infty}_{k=0}T_{Q_{k}(x)}P_{k}(\xi),$

where $P_{k}$ are the Littlewood-Paley projector given in Definition A.1 and
$Q_{k}(x)$ are uniformly bounded in $L^{\infty}$ which again gives the stated
bound from the previous computations. ∎

Next we need a lemma on the action of composition on
$\left\|\cdot\right\|_{\omega_{0}}$. We will give it through the operator of
paracomposition given in Theorem A.14 as those estimates will be needed for
the proof of Theorem 1.6.

######  Lemma 2.6.

Consider $\omega_{0}\in L^{2}\left(\mathbb{R}^{2}\right)$ with $\omega_{0}$
verifying hypothesis (1.4). Let $\chi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$
be a $C^{1}\left(\mathbb{R}^{2}\right)$ diffeomorphism with $D\chi\in
W^{r,\infty}$, $r>0,r\notin\mathbb{N}$, take $u\in H^{s}(\mathbb{R}^{2})$ and
define

$\chi^{*}u=\sum_{k\geq 0}\sum_{\begin{subarray}{c}l\geq 0\\\ k-N\leq l\leq
k+N\end{subarray}}P_{l}(D)(\Delta_{k}\omega_{0})\circ\chi,$

where $N\in\mathbb{N}^{*}$ is such that
$2^{N}>\sup_{k,\mathbb{R}^{d}}\left|\Phi_{k}D\chi\right|^{-1}$ and
$2^{N}>\sup_{k,\mathbb{R}^{2}}\left|\Phi_{k}D\chi\right|$. Then there exists
an increasing function $C_{\omega_{0}}$ (increasing at most polynomially fast)
such that

$\left\|\chi^{*}u\right\|_{\omega_{0}}\leq
C_{\omega_{0}}(\left\|D\chi\right\|_{L^{\infty}},\left\|D\chi^{-1}\right\|_{L^{\infty}})\left\|u\right\|_{\omega_{0}}.$

Now the composition $\omega_{0}\circ\chi(x)$ can be decomposed as

$\omega_{0}\circ\chi(x)=\chi^{*}\omega_{0}(x)+T_{\nabla\omega_{0}\circ\chi}\cdot\chi(x)+R(x),$

and the remainder verifies the estimate for an increasing function
$C_{\omega_{0}}$ (increasing at most polynomially fast)

$\left\||D|^{r}R\right\|_{\omega_{0}}\leq
C_{\omega_{0},r}(\left\|D\chi\right\|_{W^{r,\infty}},\left\|D\chi^{-1}\right\|_{L^{\infty}})\left\|\omega_{0}\right\|_{\omega_{0}}.$

###### Proof.

The estimate on $\chi^{*}u$ follows exactly from the one on $T_{f}g$ as

$\Delta_{j}\left(\chi^{*}\omega_{0}\right)=\sum_{\begin{subarray}{c}l\geq 0\\\
j-N\leq l\leq j+N\end{subarray}}P_{l}(D)(\Delta_{k}\omega_{0})\circ\chi.$

Next for the estimates on $R$ we will follow the presentation in Appendix A
and B of chapter 2 of [27] and write

$R=R_{1}+R_{2}+R_{3},$

with

$R_{1}=\sum_{k\geq 0}\sum_{\begin{subarray}{c}l\geq 0\\\ k-N\leq l\leq
k+N\end{subarray}}P_{l}(D)\left((\Delta_{k}\omega_{0})\circ\chi-(\Delta_{k}\omega_{0})\circ\left(P_{\leq
k}(D)\chi\right)\right),$ $R_{2}=\sum_{k\geq N}P_{\leq
k-N}(D)[(\Delta_{k}\omega_{0})\circ\left(P_{\leq k}(D)\chi\right)],$

and

$R_{3}=\sum_{k\geq N}\left(Id-P_{\leq
k+N}(D)\right)[(\Delta_{k}\omega_{0})\circ\left(P_{\leq k}(D)\chi\right)].$

We note that $R_{1}$ can be treated similarly to $\chi^{*}\omega_{0}$. For
$R_{2}$ and $R_{3}$ we recall estimates $(B.15)$ and $(B.16)$ page 133 of
[27]: there exists $N$ depending on $\left\|D\chi\right\|_{L^{\infty}}$ and K
depending on $\left\|D\chi\right\|_{W^{r,\infty}}$ and
$\left\|D\chi^{-1}\right\|_{L^{\infty}}$ such that for $k\geq K$, $j\geq
k+N,\nu\geq r$

$\left\|P_{j}(D)\left((\Delta_{k}\omega_{0})\circ\left(P_{\leq
k}(D)\chi\right)\right)\right\|_{L^{2}}\leq
C_{\nu}\left(\left\|D\chi\right\|_{W^{r,\infty}},\left\|D\chi^{-1}\right\|_{L^{\infty}}\right)2^{-j\nu}2^{k(\nu-r)}\left\|(\Delta_{k}\omega_{0})\right\|_{L^{2}},$

and

$\left\|P_{\leq k-N}(D)\left((\Delta_{k}\omega_{0})\circ\left(P_{\leq
k}(D)\chi\right)\right)\right\|_{L^{2}}\leq
C_{\nu}\left(\left\|D\chi\right\|_{W^{r,\infty}},\left\|D\chi^{-1}\right\|_{L^{\infty}}\right)2^{-kr}\left\|(\Delta_{k}\omega_{0})\right\|_{L^{2}}.$

which again give the claim. ∎

Now we turn to the estimate on $\left\|D\Phi_{t}-Id\right\|_{\omega_{0}}$. For
this we start from

$\frac{d}{dt}\Phi_{t}=u\circ\Phi_{t},$

and use the paracomposition operator to write

$\frac{d}{dt}\Phi_{t}=\Phi_{t}^{*}u+T_{Du\circ\Phi_{t}}\Phi_{t}+R.$

Applying Lemmas 2.4 and 2.6 we get

$\frac{d}{dt}\left\|D\Phi_{t}-Id\right\|_{\omega_{0}}\leq
C_{\omega_{0}}\left\|Du\right\|_{L^{\infty}}\left\|D\Phi_{t}-Id\right\|_{\omega_{0}}+C_{\omega_{0}}\left(\left\|D\Phi_{t}\right\|_{L^{\infty}},\left\|D\Phi_{t}^{-1}\right\|_{L^{\infty}}\right)\left\|Du\right\|_{\omega_{0}},$

giving again the stated bound.

##  3 Convergence of semi-classical measures

While not stated exactly in this form, Shnirelman observation behind the
“microlocal scalar product” is on the effect of regularity on the convergence
rate of semi-classical measures. First let us introduce those notions and the
convergence result on $L^{2}\left(\mathbb{R}^{2}\right)$.

###  3.1 Convergence in $L^{2}(\mathbb{R}^{2})$

Consider a pseudodifferential operator $a\in S^{0}\left(\mathbb{R}^{2}\right)$
and $u,v\in L^{2}\left(\mathbb{R}^{2}\right)$, then by the continuity of
pseudodifferential operators on $L^{2}$ (see Theorem A.4) we see that

$\left(a(x,\epsilon D)u,v\right)_{L^{2}(\mathbb{R}^{2})}\underset{\epsilon\to
0}{\to}\int_{\mathbb{R}^{2}}a(x,0)u(x)\bar{v}(x)dx=\lambda_{u,v}(a),$

and $\lambda_{u,v}$ can be seen as measure on the space
$S^{0}(\mathbb{R}^{2})$. This measure was first introduced simultaneously by
two independent procedures in [12] and [18]. The above construction method is
the procedure introduced in [12]. Note that $\lambda_{u,u}$ is always a
positive measure, thus if $a(x,\xi)\geq 0$ and $\lambda_{u,u}(a)>0$ then for
$\epsilon$ sufficiently small

$\left(a(x,\epsilon
D)u,u\right)_{L^{2}}=\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}a(x,\epsilon\xi)e^{ix\cdot\xi}\mathscr{F}(u)(\xi)u(x)dxd\xi>0.$

Note that the only property on $a$ we used in the proof of the previous
theorem is the continuity of $a(x,D)=\mbox{Op}(a)$ on $L^{2}(\mathbb{R}^{2})$
which holds more generally, see Theorem A.10, for paradifferential operators
with symbol $a\in\Gamma_{0}^{0}(\mathbb{R}^{2})$ with regularised symbol
$\sigma_{a}$ (see Definitions A.6 and A.7). The cut-off used in the definition
of paradifferential operators ensures that $\sigma_{a}(x,\xi)=0$ for $\xi\in
B(0,b)$ for some $b>0$ fixed thus we have proved the following.

######  Theorem 3.1 (From [12] and [18]).

For $u,v\in L^{2}\left(\mathbb{R}^{2}\right)$ and $a\in
S^{0}\left(\mathbb{R}^{2}\right)$ and
$a^{\prime}\in\Gamma^{0}_{0}(\mathbb{R}^{2})$ then

$\left(a(x,\epsilon
D)u,v\right)_{L^{2}(\mathbb{R}^{2})},\left(\sigma_{a^{\prime}}(x,\epsilon
D)u,v\right)_{L^{2}(\mathbb{R}^{2})},$

converge respectively to $\lambda_{u,v}(a)$ and $0$ in the limit $\epsilon\to
0$.

###  3.2 The rate of convergence

A first observation is that the rates of convergence of sequences given in
Theorems 3.1 depend on the regularity of $u,v$ more precisely by Remark
$III.9$ of [18] if $u\in H^{s}(\mathbb{R}^{2}),s>0$ then for $a\in
S^{0}\left(\mathbb{R}^{2}\right)$, $\left(a(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2}\times\mathbb{R}^{2})}-\lambda_{u,u}(a)=O(\epsilon^{2s})$.
Shnirelman then showed in [24] that the same rate holds for $u\in
B^{s}_{2,\infty}$ and he insight-fully observed that if $u\notin
B^{s+\delta}_{2,\infty}(\mathbb{R}^{2})$ for $\delta>0$ then the previous rate
of convergence is essentially exact. While he did not state the second
observation explicitly he uses it to prove Theorem 7.1 of [24]. We will extend
the decay rate observations of [18] and [24] to $L^{2}$ functions in general.
We recall the definition for $u\in L^{2}\left(\mathbb{R}^{2}\right)$

$dr_{u}(\epsilon)=\left(\int_{\mathbb{R}^{2}\setminus
B\left(0,\frac{1}{\epsilon}\right)}\left|\mathscr{F}(u)(\xi)\right|^{2}d\xi\right)^{\frac{1}{2}},$

which is an increasing function of $\epsilon$ with
$\displaystyle\lim_{\epsilon\to 0}dr_{u}(\epsilon)=0$ and
$\displaystyle\lim_{\epsilon\to+\infty}dr_{u}(\epsilon)=\left\|u\right\|_{L^{2}(\mathbb{R}^{2})}$.

######  Theorem 3.2.

Consider $u,v\in L^{2}(\mathbb{R}^{2})$, then there exists universal constants
$C_{1},C_{2},C_{3}>0$ such that for $a\in\Gamma_{0}^{0}(\mathbb{R}^{2})$ and
$\epsilon>0$

$\left|\left(\sigma_{a}(x,\epsilon
D)u,v\right)_{L^{2}(\mathbb{R}^{2})}\right|\leq
C_{1}M^{0}_{0}(a;2)dr_{u}(C_{2}\epsilon)dr_{v}(C_{3}\epsilon),$

where the semi-norms $M^{\cdot}_{\cdot}(\cdot;\cdot)$ are defined as in
Definition A.6. If there exists a $0$-homogeneous symbol
$a_{0}\in\Gamma^{0}_{0}(\mathbb{R}^{2})$ such that
$a-a_{0}\in\Gamma^{-\alpha}_{0}(\mathbb{R}^{2})$ for some $\alpha>0$ then
$a_{0}$ is called the principal symbol of $a$ and moreover we have

$\left|\left(\sigma_{a-a_{0}}(x,\epsilon
D)u,v\right)_{L^{2}(\mathbb{R}^{2})}\right|\leq
C_{1}\epsilon^{\alpha}M^{-\alpha}_{0}(a-a_{0};2)dr_{u}(C_{2}\epsilon)dr_{v}(C_{3}\epsilon).$

######  Remark 3.3.

The rates given in the previous theorem are optimal in the sense that by
definition there exists a $C$ such that

$\left|\left(\sigma_{1}(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2})}\right|\gtrsim
C_{1}dr_{u}(C_{2}\epsilon)dr_{u}(C_{3}\epsilon).$

Thus the previous theorem shows that the exact rate of convergence of semi-
classical is an exact bilinear test of smoothness.

###### Proof.

First noting that by the polarization formula for a sesquilinear form $s$

$s(u,v)=\frac{1}{4}\left(s(u+v,u+v)-s(u-v,u-v)+is(u+iv,u+iv)-is(u-iv,u-iv)\right),$

it suffices to work with $u=v$. First we study $\left(\sigma_{a}(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2})}$ and write by Parseval’s identity

$\displaystyle\left|\left(\sigma_{a}(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2})}\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{2}}\mathscr{F}\left(\sigma_{a}(x,\epsilon
D)u\right)(\xi)\mathscr{F}(u)(\xi)d\xi\right|$ using the Cauchy-Schwarz
inequality combined with the spectral localisation property of
paradifferential operators given in Proposition A.9 we get that there exists a
universal constant $C_{1},C_{2}>0$ such that
$\displaystyle\left|\left(\sigma_{a}(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2})}\right|$ $\displaystyle\leq
C_{1}\left\|\mathscr{F}\left(\sigma_{a}(x,\epsilon
D)u\right)\right\|_{L^{2}\left(\mathbb{R}^{2}\setminus
B\left(0,\frac{C_{2}}{\epsilon}\right)\right)}\left\|\mathscr{F}(u)\right\|_{L^{2}\left(\mathbb{R}^{2}\setminus
B\left(0,\frac{C_{2}}{\epsilon}\right)\right)}$ by the boundness of
paradifferential operators on $L^{2}$, see Theorem A.10 we get, and again the
spectral localisation property of paradifferential operators
$\displaystyle\left|\left(\sigma_{a}(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2})}\right|$ $\displaystyle\leq
C_{1}M^{0}_{0}(a;2)dr_{u}(C_{2}\epsilon)dr_{u}(C_{3}\epsilon).$

To get the estimate on $\left|\left(\sigma_{a-a_{0}}(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2})}\right|$ we start from

$\left|\left(\sigma_{a-a_{0}}(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2})}\right|\leq\left\|\mathscr{F}\left(\sigma_{a-a_{0}}(x,\epsilon
D)u\right)\right\|_{L^{2}\left(\mathbb{R}^{2}\setminus
B\left(0,\frac{C}{\epsilon}\right)\right)}\left\|\mathscr{F}(u)\right\|_{L^{2}\left(\mathbb{R}^{2}\setminus
B\left(0,\frac{C}{\epsilon}\right)\right)},$

by the boundness of paradifferential operators in
$\Gamma^{-\alpha}_{0}(\mathbb{R}^{2})$ from $L^{2}(\mathbb{R}^{2})$ to
$H^{\alpha}(\mathbb{R}^{2})$, see Theorem A.10 and again the spectral
localisation property of paradifferential operators we get

$\left|\left(\sigma_{a-a_{0}}(x,\epsilon
D)u,u\right)_{L^{2}(\mathbb{R}^{2})}\right|\leq
C_{1}M^{-\alpha}_{0}(a-a_{0};2)\epsilon^{\alpha}dr_{u}(C_{2}\epsilon)dr_{u}(C_{3}\epsilon).$

∎

The previous proof adapts in verbatim to give the following pseudodifferential
version of the previous theorem.

######  Theorem 3.4.

Consider $u,v\in L^{2}(\mathbb{R}^{2})$, $0<\delta<\epsilon$ and a function
$\chi\in C^{\infty}_{0}\left(\mathbb{R}^{2}\setminus B(0,1)\right)$. Then
there exists a universal constant $C$ and constant $C_{\chi}$, depending only
on $\chi$, such that for $a\in S^{0}\left(\mathbb{R}^{2}\right)$ and
$\epsilon>0$

$\left|\left(\chi(\epsilon D)a(x,D)\chi(\epsilon
D)u,v\right)_{L^{2}(\mathbb{R}^{2})}\right|\leq
CM^{0}_{0}(a;2)dr_{u}(C_{\chi}\epsilon)dr_{v}(C_{\chi}\epsilon),$

where the semi-norms $M^{\cdot}_{\cdot}(\cdot;\cdot)$ are defined as in
Definition A.3. If there exists a zero homogeneous symbol $a_{0}$ such that
$a_{0}\chi\in S^{0}\left(\mathbb{R}^{2}\right)$ and $(a-a_{0})\chi\in
S^{-\alpha}(\mathbb{R}^{2})$ for some $\alpha>0$ then $a_{0}$ is called the
principal symbol of $a$ and moreover we have

$\left|\left(\chi(\epsilon D)(a-a_{0})(x,D)\chi(\epsilon
D)u,v\right)_{L^{2}(\mathbb{R}^{2})}\right|\leq
C\epsilon^{\alpha}M^{-\alpha}_{0}(a-a_{0};2)dr_{u}(C_{\chi}\epsilon)dr_{v}(C_{\chi}\epsilon).$

######  Remark 3.5.

Theorem $3.1$ in [24] corresponds to the case in the previous theorem where
$dr_{\epsilon}(u),dr_{\epsilon}(v)$ are of order $\epsilon^{s}$ for some
$s\in\mathbb{R}$.

##  4 The monotone quantity in 2d Euler

With Theorems 3.4 and 3.2 as well as Theorem 1.4 we transform the qualitative
computation in the proof of Theorem $5.1$ of [24] into the quantitative
results given in Theorems 1.6.

###  4.1 Proof of Theorem 1.6

We will prove a more general form of Theorem 1.6 where $\chi(\epsilon D)$ is
replaced by a paradifferential operator $\sigma_{a}(x,\epsilon D)$ with
$a\in\Gamma_{1}^{0}$. We start by computing the time derivative of
$T_{D\Phi_{t}}\left(\Phi_{t}-Id\right)$ where the paraproduct here should be
understood component by component. We compute

$\partial_{t}\left(T_{\left[D\Phi_{t}\right]^{-1}}\Phi_{t}\right)=T_{\partial_{t}\left[D\Phi_{t}\right]^{-1}}\Phi_{t}+T_{\left[D\Phi_{t}\right]^{-1}}u\circ\Phi_{t}.$

For a matrix valued function $A_{t}$ we have
$\partial_{t}\left(A_{t}^{-1}\right)=-A_{t}^{-1}\partial_{t}A_{t}A_{t}^{-1}$,
thus

$\displaystyle\partial_{t}\left[D\Phi_{t}\right]^{-1}$
$\displaystyle=-\left[D\Phi_{t}\right]^{-1}\left(\partial_{t}D\Phi_{t}\right)\left[D\Phi_{t}\right]^{-1}$
$\displaystyle=-\left[D\Phi_{t}\right]^{-1}\left(D\left(u\circ\Phi_{t}\right)\right)\left[D\Phi_{t}\right]^{-1}$
$\displaystyle=-\left[D\Phi_{t}\right]^{-1}Du\circ\Phi_{t}.$

Which gives

$\partial_{t}\left(T_{\left[D\Phi_{t}\right]^{-1}}\Phi_{t}\right)=-T_{\left[D\Phi_{t}\right]^{-1}Du\circ\Phi_{t}}\Phi_{t}+T_{\left[D\Phi_{t}\right]^{-1}}u\circ\Phi_{t}.$

Using the paracomposition operator defined in A.14 we define $R_{1}$ by

$u\circ\Phi_{t}=\Phi_{t}^{*}u+T_{Du\circ\Phi_{t}}\Phi_{t}+R_{1}=\Phi_{t}^{*}u+T_{Du\circ\Phi_{t}}\Phi_{t}+R_{1}.$
(4.1)

Note that the paracomposition operator should also be understood component by
component. Thus we get the key cancellation in the second term on the left
hand side

$\partial_{t}\left(T_{\left[D\Phi_{t}\right]^{-1}}\Phi_{t}\right)=T_{\left[D\Phi_{t}\right]^{-1}}\Phi_{t}^{*}u+\left(T_{\left[D\Phi_{t}\right]^{-1}}T_{Du\circ\Phi_{t}}-T_{\left[D\Phi_{t}\right]^{-1}Du\circ\Phi_{t}}\right)\Phi_{t}+T_{\left[D\Phi_{t}\right]^{-1}}R_{1}.$

Now we write

$u=\nabla^{\perp}\Delta^{-1}\left(\omega_{0}\circ\Phi_{t}^{-1}\right),$

and again using the paracomposition operator we define $R_{2}$ by

$\omega_{0}\circ\Phi_{t}^{-1}={\Phi_{t}^{-1}}^{*}\omega_{0}+T_{\left(\nabla\omega_{0}\right)\circ\Phi^{-1}_{t}}\cdot\Phi^{-1}_{t}+R_{2}.$
(4.2)

Next we define $R_{3}$ by

$\nabla\times
T_{\left[D\Phi_{t}\right]^{-1}}\Phi_{t}^{*}\left(\nabla^{\perp}\Delta^{-1}\left({\Phi_{t}^{-1}}^{*}\omega_{0}\right)\right)=T_{\frac{\left|\xi\right|^{2}}{\left|[D\Phi_{t}]^{-1}\xi\right|^{2}}}\omega_{0}+R_{3}.$
(4.3)

Putting the previous computations together we finally get the paralinearised
evolution equation on the flow

$\partial_{t}\left(\nabla\times\left(T_{\left[D\Phi_{t}\right]^{-1}}\Phi_{t}\right)\right)=T_{\frac{\left|\xi\right|^{2}}{\left|[D\Phi_{t}]^{-1}\xi\right|^{2}}}\omega_{0}+\boldsymbol{R}_{1}+\boldsymbol{R}_{2}+R_{3},$
(4.4)

with

$\boldsymbol{R}_{1}=\nabla\times
T_{\left[D\Phi_{t}\right]^{-1}}R_{1}+\nabla\times
T_{\left[D\Phi_{t}\right]^{-1}}\nabla^{\perp}\Delta^{-1}R_{2},$

and

$\boldsymbol{R}_{2}=\nabla\times
T_{\left[D\Phi_{t}\right]^{-1}}\nabla^{\perp}\Delta^{-1}T_{\left(\nabla\omega_{0}\right)\circ\Phi^{-1}_{t}}\Phi^{-1}_{t}+\nabla\times\left(T_{\left[D\Phi_{t}\right]^{-1}}T_{Du\circ\Phi_{t}}-T_{\left[D\Phi_{t}\right]^{-1}Du\circ\Phi_{t}}\right)\Phi_{t}.$

We now proceed to estimate each term in $\left(\sigma_{a}(x,\epsilon
D)(\boldsymbol{R}_{1}+\boldsymbol{R}_{2}+R_{3}),\sigma_{a}(x,\epsilon
D)\omega_{0}\right)_{L^{2}(\mathbb{R}^{2})}$, note that it suffices to
estimate $dr_{\boldsymbol{R}_{1}}(\epsilon),\
dr_{\boldsymbol{R}_{2}}(\epsilon)$ and $dr_{R_{3}}(\epsilon)$. Henceforth we
fix $s-1>\delta>0$. For $R_{3}$ we note that by the pull-back formula of
paradifferential operators by the paracomposition operator given in (A.7) as
well as the composition property of paradifferential operators in Theorem A.11
that by construction
$T_{\frac{\left|\xi\right|^{2}}{\left|[D\Phi_{t}]^{-1}\xi\right|^{2}}}$ is the
principal symbol of

$\nabla\times
T_{\left[D\Phi_{t}\right]^{-1}}\Phi_{t}^{*}\left(\nabla^{\perp}\Delta^{-1}\left({\Phi_{t}^{-1}}^{*}\omega_{0}\right)\right).$

Thus by the spectral localisation property of paradifferential operators in
Proposition A.9, the continuity of paraproducts in Theorem A.13 we have for a
universal increasing function $C_{\delta}$ growing at most polynomially fast

$dr_{R_{3}}(\epsilon)\leq\epsilon^{\min(s-1-\delta,1)}C_{\delta}\left(\left\|D\Phi_{t}\right\|_{L^{\infty}},\left\|[D\Phi_{t}]^{-1}\right\|_{L^{\infty}}\right)\left\|D\Phi_{t}\right\|_{W^{\min(s-1-\delta,1),\infty}}dr_{\omega_{0}}(\lambda\epsilon).$

Next $\boldsymbol{R}_{2}$ is treated in the same fashion as $R_{3}$. First
$T_{\left(\nabla\omega_{0}\right)\circ\Phi_{t}^{-1}}$ is treated as an
operator of order $\max(2-s,0)$ to give

$dr_{\nabla\times
T_{\left[D\Phi_{t}\right]^{-1}}\nabla^{\perp}\Delta^{-1}T_{\left(\nabla\omega_{0}\right)\circ\Phi^{-1}_{t}}\cdot\Phi^{-1}_{t}}(\epsilon)\leq
C_{\delta}\epsilon^{\min(s-1-\delta,1)}\left\|[D\Phi_{t}]^{-1}\right\|_{L^{\infty}}\left\|\omega_{0}\right\|_{H^{s}}dr_{D\Phi^{-1}_{t}}(\lambda\epsilon),$

and
$T_{\left[D\Phi_{t}\right]^{-1}}T_{Du\circ\Phi_{t}}-T_{\left[D\Phi_{t}\right]^{-1}Du\circ\Phi_{t}}$
is of order $-\max(s-1-\delta,1)$ which gives

$dr_{\nabla\times\left(T_{\left[D\Phi_{t}\right]^{-1}}T_{Du\circ\Phi_{t}}-T_{\left[D\Phi_{t}\right]^{-1}Du\circ\Phi_{t}}\right)\Phi_{t}}(\epsilon)\\\
\leq
C_{\delta}\epsilon^{\min(s-1-\delta,1)}\left\|D\Phi_{t}\right\|_{W^{\min(s-1-\delta,1),\infty}}\left\|Du\right\|_{W^{\min(s-1-\delta,1),\infty}}dr_{D\Phi_{t}}(\lambda\epsilon).$

Finally for $\boldsymbol{R}_{1}$ we have the terms that bring into play the
residual term of the paracomposition operator thus we have the gain of
$\min(s-1-\delta,1)$ derivative and thus the $\epsilon^{\max(s-1-\delta,1)}$
factor by Lemma 2.6 to get

$dr_{\nabla\times T_{\left[D\Phi_{t}\right]^{-1}}R_{1}}(\epsilon)\leq
C_{\delta}\left(\left\|D\Phi_{t}\right\|_{W^{\min(s-1-\delta,1),\infty}},\left\|[D\Phi_{t}]^{-1}\right\|_{L^{\infty}}\right)\epsilon^{\min(s-1-\delta,1)}dr_{Du}(\epsilon),$

and

$dr_{\nabla\times
T_{\left[D\Phi_{t}\right]^{-1}}\nabla^{\perp}\Delta^{-1}R_{2}}(\epsilon)\leq
C_{\delta}\left(\left\|D\Phi_{t}\right\|_{W^{\min(s-1-\delta,1),\infty}},\left\|[D\Phi_{t}]^{-1}\right\|_{L^{\infty}}\right)\epsilon^{\min(s-1-\delta,1)}dr_{\omega_{0}}(\epsilon).$

Plugging the estimates given in Theorem 1.3 and 1.4 we get the desired result
on $\boldsymbol{R}_{1},\boldsymbol{R}_{2}$ and $R_{3}$. Finally to get the
result in the form stated in Theorem 1.6 we see that

$\left[T_{a},T_{\frac{\left|\xi\right|^{2}}{\left|[D\Phi_{t}]^{-1}\xi\right|^{2}}}\right],\
T_{\frac{\left|\xi\right|^{2}}{\left|[D\Phi_{t}]^{-1}\xi\right|^{2}}}-T_{\frac{\left|\xi\right|}{\left|[D\Phi_{t}]^{-1}\xi\right|}}T_{\frac{\left|\xi\right|}{\left|[D\Phi_{t}]^{-1}\xi\right|}}\text{
and
}T_{\frac{\left|\xi\right|}{\left|[D\Phi_{t}]^{-1}\xi\right|}}-\left(T_{\frac{\left|\xi\right|}{\left|[D\Phi_{t}]^{-1}\xi\right|}}\right)^{t}$

are all smoothing operators of order $-\min(s-1-\delta,1)$ where $T_{a}^{t}$
is the adjoint of $T_{a}$.

###  4.2 Proof of Corollary 1.9

We suppose without loss of generality that
$\left\|D\Phi_{t}\right\|_{L^{\infty}}<+\infty$ thus there exists $K>0$ such
that for all $t\geq 0$

$\frac{\left|\xi\right|}{\left|[D\Phi_{t}]^{-1}\xi\right|}\geq K.$

Thus for all $t$ there exists an $a$ such that

$\left\|T_{\frac{\left|\xi\right|}{\left|[D\Phi_{t}]^{-1}\xi\right|}}a(x,\epsilon)\omega_{0}\right\|^{2}_{L^{2}(\mathbb{R}^{2})}\geq
Cdr_{\omega_{0}}(\epsilon)^{2},$

thus

$\sup_{\epsilon\geq 0}\sup_{\begin{subarray}{c}a\in
C_{0}^{\infty}\left(\mathbb{R}^{2}\right)\\\ M^{0}_{1}(a)\leq
1\end{subarray}}\frac{\left(a(x,\epsilon D)\nabla\times
T_{[D\Phi_{t}]^{-1}}\Phi_{t},\chi(\epsilon
D)\omega_{0}\right)}{dr_{\omega_{0}}(\epsilon)^{2}}\geq Ct.$

Thus $\left\|D\Phi_{t}-Id\right\|_{\omega_{0}}$ blows up at least linearly
when $t\to+\infty$. To get the more general $t^{\frac{1}{3}}$ we note that
with an analogous reasoning we have the differential inequality

$\left\|D\Phi_{t}\right\|_{L^{\infty}}\left\|D\Phi_{t}-Id\right\|_{\omega_{0}}\geq
C\int_{0}^{t}\frac{1}{\left\|D\Phi_{s}\right\|_{L^{\infty}}}ds.$

##  Appendix A Notions of mircolocal analysis

At the heart of the construction in Theorem 1.6 is the use of microlocal
analysis, in particular Shnirelman’s construction can be seen as a consequence
of the algebra property of pseudodifferential and paradifferential operators
combined with the fact that restricted to principal symbols this algebra
becomes commutative. In this section we will give a simple yet complete
recount of all the concepts and classical results that are needed to carry out
the construction. All of the results are taken from the excellent monographs
[2, 14, 20, 27]

###  A.1 The Littlewood-Paley decomposition

######  Definition A.1 (Littlewood-Paley decomposition).

Pick $P_{0}\in C^{\infty}_{0}(\mathbb{R}^{2})$ so that:

$P_{0}(\xi)=1\text{ for }\left|\xi\right|<1\text{ and }P_{0}(\xi)=0\text{ for
}\left|\xi\right|>2.$

We define a dyadic decomposition of unity by:

$\text{for }k\geq 1,\ P_{\leq k}(\xi)=P_{0}(2^{-k}\xi),\ P_{k}(\xi)=P_{\leq
k}(\xi)-P_{\leq k-1}(\xi).$

Thus,

$P_{\leq k}(\xi)=\sum^{k}_{j=0}P_{j}(\xi)\text{ and
}1=\sum_{j=0}^{\infty}P_{j}(\xi).$

Introduce the operator acting on $\mathscr{S}^{\prime}(\mathbb{R}^{2})$:

$P_{\leq k}(D)u=\mathscr{F}^{-1}(P_{\leq k}(\xi)u)\text{ and
}u_{k}=P_{k}(D)u=\mathscr{F}^{-1}(P_{k}(\xi)u).$

Thus,

$u=\sum^{+\infty}_{k=0}u_{k}.$

Finally put for $k\geq 1,C_{k}=\mbox{supp}\ P_{k}$ the set of rings associated
to this decomposition.

An interesting property of the Littlewood-Paley decomposition is that even if
the decomposed function is merely a distribution the terms of the
decomposition are regular, indeed they all have compact spectrum and thus are
entire functions. On classical functions spaces this regularization effect can
be “measured” by the following inequalities due to Bernstein.

######  Proposition A.2 (Bernstein’s inequalities).

Suppose that $a\in L^{p}(\mathbb{R}^{2})$ has its spectrum contained in the
ball $\\{\left|\xi\right|\leq\lambda\\}$.

Then $a\in C^{\infty}$ and for all $\alpha\in\mathbb{N}^{d}$ and $1\leq p\leq
q\leq+\infty$, there is $C_{\alpha,p,q}$ (independent of $\lambda$) such that

$\left\|\partial_{x}^{\alpha}a\right\|_{L^{q}}\leq
C_{\alpha,p,q}\lambda^{\left|\alpha\right|+\frac{d}{p}-\frac{d}{q}}\left\|a\right\|_{L^{p}}.$

In particular,

$\left\|\partial_{x}^{\alpha}a\right\|_{L^{q}}\leq
C_{\alpha}\lambda^{\left|\alpha\right|}\left\|a\right\|_{L^{p}},\text{ and for
$p=2$, $p=\infty$}$ $\left\|a\right\|_{L^{\infty}}\leq
C\lambda^{\frac{d}{2}}\left\|a\right\|_{L^{2}}.$

If moreover a has its spectrum included in
$\\{0<\mu\leq\left|\xi\right|\leq\lambda\\}$ then:

$C_{\alpha,q}^{-1}\mu^{\left|\alpha\right|}\left\|a\right\|_{L^{q}}\leq\left\|\partial_{x}^{\alpha}a\right\|_{L^{q}}\leq
C_{\alpha,q}\lambda^{\left|\alpha\right|}\left\|a\right\|_{L^{q}}.$

###  A.2 Pseudodifferential calculus

We introduce here the basic definitions and symbolic calculus results. We
first introduce the classes of regular symbols.

######  Definition A.3.

Given $m\in\mathbb{R}$ we denote the symbol class $S^{m}(\mathbb{R}^{2})$ as
the set of all $a\in C^{\infty}(\mathbb{R}^{2}\times\mathbb{R}^{2})$ such that
for all $\alpha,\beta\in\mathbb{N}^{2}$ we have the estimate:

$\left|\partial^{\alpha}_{x}\partial^{\beta}_{\xi}a(x,\xi)\right|\leq
C_{\alpha,\beta}(1+\left|\xi\right|)^{m-\beta}.$

$S^{m}(\mathbb{R}^{2})$ is a Fréchet space with the topology defined by the
family of semi-norms:

$M^{m}_{\alpha}(a;\beta)=\sup_{i\leq\left|\alpha\right|,j\leq\left|\beta\right|}\sup_{\mathbb{R}^{2}\times\mathbb{R}^{2}}\left|\partial^{i}_{x}\partial^{j}_{\xi}a(x,\xi)(1+\left|\xi\right|)^{j-m}\right|.$

Given a symbol $a\in S^{m}(\mathbb{R}^{2})$, we define the pseudodifferential
operator:

$\mbox{Op}(a)u(x)=a(x,D)u(x)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^{ix.\xi}a(x,\xi)\mathscr{F}(u)(\xi)d\xi.$

######  Theorem A.4.

Consider $m\in\mathbb{R}$ then for $a\in S^{m}\left(\mathbb{R}^{2}\right)$,
$\mbox{Op}(a)$ is of order $m$, more precisely for $s\in\mathbb{R}$ there
exists a constant $C_{s}$ such that:

$\left\|\mbox{Op}(a)\right\|_{H^{s}\rightarrow H^{s-m}}\leq C_{s}M^{m}_{\lceil
s\rceil}(a,2).$

We will now present the main results in symbolic calculus associated to
pseudodifferential operators.

######  Theorem A.5.

Consider two real numbers $m,m^{\prime}\in\mathbb{R}$ and two symbols $a\in
S^{m}\left(\mathbb{R}^{2}\right)$and $b\in S^{m^{\prime}}(\mathbb{R}^{2})$
then we have the following.

  * •

Composition: $\mbox{Op}(a)\circ\mbox{Op}(b)$ is a pseudodifferential operator
of order $m+m^{\prime}$ with symbol $a\otimes b$ defined by:

$a\otimes
b(x,\xi)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}e^{i(x-y)\cdot(\xi-\eta)}a(x,\eta)b(y,\xi)dyd\eta.$

Moreover there exists a constant $C_{s}>0$ such that for $k\in\mathbb{N}$,
$s\in\mathbb{R}$ and $(p,q)\in[1,+\infty]$,

$\left\|\mbox{Op}(a)\circ\mbox{Op}(b)(x,\xi)-\mbox{Op}\left(\sum_{\left|\alpha\right|<k}\frac{1}{i^{\left|\alpha\right|}\alpha!}(\partial^{\alpha}_{\xi}a(x,\xi))(\partial^{\alpha}_{x}b(x,\xi))\right)\right\|_{H^{s}\rightarrow
H^{s-m+k}}\\\ \leq C_{s}(M^{m}_{k+\lceil s\rceil}(a;2)M^{m^{\prime}}_{\lceil
s\rceil}(b;k+2)+M^{m}_{\lceil s\rceil}(a;k+2)M^{m^{\prime}}_{k+\lceil
s\rceil}(b;2)).$

  * •

Adjoint: The adjoint operator of $\mbox{Op}(a)$, that we will denote with
$\mbox{Op}(a)^{t}$ to avoid confusion with the pullback operator defined in
this work, is a pseudodifferential operator of order m with symbol $a^{t}$
defined by:

$a^{t}(x,\xi)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}e^{-iy\cdot\eta}\bar{a}(x-y,\xi-\eta)dyd\eta$

Moreover there exists a constant $K>0$ such that for $k\in\mathbb{N}$,
$s\in\mathbb{R}$ and $(p,q)\in[1,+\infty]$,

$\left\|\mbox{Op}(a^{t})(x,\xi)-\mbox{Op}\left(\sum_{\left|\alpha\right|<k}\frac{1}{i^{\left|\alpha\right|}\alpha!}(\partial^{\alpha}_{\xi}\partial^{\alpha}_{x}\bar{a}(x,\xi))\right)\right\|_{H^{s}\rightarrow
H^{s-m+k}}\leq C_{s}M^{m}_{k+\lceil s\rceil}(a;k+2).$

  * •

Change of variables: consider a smooth diffeomorphism
$\chi:\mathbb{R}^{2}\to\mathbb{R}^{2}$ such that $D\chi\in
W^{k,\infty}\left(\mathbb{R}^{2}\right)$ for all $k\in\mathbb{N}$. Then
defining

$a^{*}(x,\xi)=e^{-ix\cdot\xi}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}a(\chi(x),\eta)e^{i(\chi(x)-\chi(y))\cdot\eta+iy\cdot\xi}\left|D\chi(y)\right|dyd\eta\in
S^{m}(\mathbb{R}^{2}\times\mathbb{R}^{2}),$

we have for all $u\in\mathscr{S}(\mathbb{R}^{2})$,
$\left(\mbox{Op}(a)\left(u\circ\chi^{-1}\right)\right)\circ\chi=\mbox{Op}(a^{*})(u)$.
Moreover for $k\in\mathbb{N}$ there exists an increasing function $C_{k}$ such
that for all $s\in\mathbb{R}$ and $(p,q)\in[1,+\infty]$

$\left\|\mbox{Op}(a^{*})(x,\xi)-\mbox{Op}\left(\sum_{\left|\alpha\right|<k}\frac{1}{\alpha!}\partial^{\alpha}a(\chi(x),([D\chi(x)]^{-1})^{t}\xi)Q_{\alpha}(\chi(x),\xi)\right)\right\|_{H^{s}\rightarrow
H^{s-m+k}}\\\ \leq
C_{k}\left(\left\|D\chi\right\|_{L^{\infty}},\left\|D\chi^{-1}\right\|_{L^{\infty}},\left\|D\chi\right\|_{W^{(k-1)^{+},\infty}}\right)M^{m}_{k+\lceil
s\rceil}(a;k+2),$

where,

$Q_{\alpha}(x^{\prime},\xi)=D^{\alpha}_{y^{\prime}}(e^{i(\chi^{-1}(y^{\prime})-\chi^{-1}(x^{\prime})-D\chi^{-1}(x^{\prime})(y^{\prime}-x^{\prime}))\cdot\xi})_{|y^{\prime}=x^{\prime}}$

and $Q_{\alpha}$ is polynomial in $\xi$ of degree
$\leq\frac{\left|\alpha\right|}{2}$, with $Q_{0}=1,Q_{1}=0$.

###  A.3 Paradifferential calculus

We start by the definition of symbols with limited spatial regularity.

######  Definition A.6.

Given $m\in\mathbb{R}$, $\Gamma^{m}_{\rho}(\mathbb{R}^{2})$ denotes the space
of locally bounded functions $a(x,\xi)$ on
$\mathbb{R}^{2}\times(\mathbb{R}^{2}\setminus 0)$, which are $C^{\infty}$ with
respect to $\xi$ for $\xi\neq 0$ and such that, for all
$\alpha\in\mathbb{N}^{2}$ and for all $\xi\neq 0$, the function
$x\mapsto\partial^{\alpha}_{\xi}a(x,\xi)$ belongs to $W^{\rho,\infty}$ and
there exists a constant $C_{\alpha}$ such that, for all $\epsilon>0$:

$\forall\left|\xi\right|>\epsilon,\left\|\partial^{\alpha}_{\xi}a(\cdot,\xi)\right\|_{W^{\rho,\infty}}\leq
C_{\alpha,\epsilon}(1+\left|\xi\right|)^{m-\left|\alpha\right|}.$ (A.1)

The spaces $\Gamma^{m}_{\rho}(\mathbb{R}^{2})$ are equipped with their natural
Fréchet topology induced by the semi-norms defined by the best constants in
(A.1) (see also [20]):

$M^{m}_{\rho}(a;n)=\sup_{\left|\alpha\right|\leq n}\
\sup_{\left|\xi\right|\geq\frac{1}{2}}\left\|(1+\left|\xi\right|)^{m-\left|\alpha\right|}\partial^{\alpha}_{\xi}a(.,\xi)\right\|_{W^{\rho,\infty}},\text{
for }n\in\mathbb{N}.$

######  Definition A.7.

Define an admissible cutoff function as a function $\psi^{B,b}\in
C^{\infty}(\mathbb{R}^{4})$, $B>1,b>0$ that verifies:

  1. 1.

$\psi^{B,b}(\xi,\eta)=0\text{ when
}\left|\xi\right|<B\left|\eta\right|+b,\text{ and
}\psi^{B,b}(\xi,\eta)=1\text{ when }\left|\xi\right|>B\left|\eta\right|+b+1.$

  2. 2.

For all $(\alpha,\beta)\in\mathbb{N}^{4},$ there exists $C_{\alpha_{\beta}}$,
with $C_{0,0}\leq 1$, such that:

$\forall(\xi,\eta):\left|\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}\psi^{B,b}(\xi,\eta)\right|\leq
C_{\alpha,\beta}(1+\left|\xi\right|)^{-\left|\alpha\right|-\left|\beta\right|}.$
(A.2)

######  Definition A.8.

Consider a real numbers $m\in\mathbb{R}$, a symbol
$a\in\Gamma^{m}_{\rho}(\mathbb{R}^{2})$ and an admissible cutoff function
$\psi^{B,b}$ define the paradifferential operator $T_{a}$ by

$T_{a}u=\mbox{Op}\ (\sigma_{a})u,\text{ where
}\mathscr{F}_{x}(\sigma_{a})(\xi,\eta)=\mathscr{F}_{x}(\sigma^{B,b}_{a})(\xi,\eta)=\psi^{B,b}(\xi,\eta)\mathscr{F}_{x}(a)(\xi,\eta).$

Let
$G_{\psi^{B,b}}(x,\eta)=\mathscr{F}^{-1}_{x}\left(\psi^{B,b}(\cdot,\eta)\right)$
then $\sigma_{a}(\cdot,\eta)=G_{\psi^{B,b}}(\cdot,\eta)\ast a(\cdot,\eta)$.

An important property of paradifferential operators is their action on
functions with localised spectrum.

######  Proposition A.9.

Consider two real numbers $m\in\mathbb{R}$, $\rho\geq 0$, a symbol
$a\in\Gamma^{m}_{0}(\mathbb{R}^{2})$, an admissible cutoff function
$\psi^{B,b}$ and $u\in\mathscr{S}(\mathbb{R}^{2})$.

  * •

For $R>>b$, if $\mbox{supp}\ \mathscr{F}(u)\subset\\{\left|\xi\right|\leq
R\\},$ then:

$\mbox{supp}\
\mathscr{F}\left(T_{a}u\right)\subset\left\\{\left|\xi\right|\leq\left(1+\frac{1}{B}\right)R-\frac{b}{B}\right\\},$
(A.3)

  * •

For $R>>b$, if $\mbox{supp}\ \mathscr{F}(u)\subset\left\\{\left|\xi\right|\geq
R\right\\},$ then:

$\mbox{supp}\
\mathscr{F}\left(T_{a}u\right)\subset\left\\{\left|\xi\right|\geq\left(1-\frac{1}{B}\right)R+\frac{b}{B}\right\\},$
(A.4)

The main features of symbolic calculus for paradifferential operators are
given by the following theorems taken from [20], [23] and [27].

######  Theorem A.10.

Consider $m\in\mathbb{R}$ then for $a\in\Gamma^{m}_{0}(\mathbb{R}^{2})$, then
$T_{a}$ is of order $m$, more precisely for $s\in\mathbb{R}$ there exists a
constant $C_{s}$ such that:

$\left\|T_{a}\right\|_{H^{s}\rightarrow H^{s-m}}\leq C_{s}M^{m}_{0}(a,2).$

######  Theorem A.11.

Let $m,m^{\prime}\in\mathbb{R}$, and $\rho>0$,
$a\in\Gamma^{m}_{\rho}(\mathbb{R}^{2})$and
$b\in\Gamma^{m^{\prime}}_{\rho}(\mathbb{R}^{2})$.

  * •

Composition: Then $T_{a}T_{b}$ is a paradifferential operator with symbol:

$a\otimes b\in\Gamma^{m+m^{\prime}}_{\rho}(\mathbb{R}^{2}),\text{ more
precisely,}$
$T^{\psi^{B,b}}_{a}T^{\psi^{B^{\prime},b}}_{b}=T^{\psi^{\frac{BB^{\prime}}{B+B^{\prime}+1},b}}_{a\otimes
b}.$

Moreover $T_{a}T_{b}-T_{a\\#b}$ is of order $m+m^{\prime}-\rho$ where $a\\#b$
is defined by:

$a\\#b=\sum_{\left|\alpha\right|<\rho}\frac{1}{i^{\left|\alpha\right|}\alpha!}\partial^{\alpha}_{\xi}a\partial^{\alpha}_{x}b,$

and there exists $r\in\Gamma^{m+m^{\prime}-\rho}_{0}(\mathbb{R})$ such that:

$M^{m+m^{\prime}-\rho}_{0}(r)\leq
C_{\rho}(M^{m}_{\rho}(a;2)M^{m^{\prime}}_{0}(b;\lceil\rho\rceil+2)+M^{m}_{0}(a;\lceil\rho\rceil+2)M^{m^{\prime}}_{\rho}(b;2)),$

and we have

$T^{\psi^{B,b}}_{a}T^{\psi^{B^{\prime},b}}_{b}-T^{\psi^{\frac{BB^{\prime}}{B+B^{\prime}+1},b}}_{a\\#b}=T^{\psi^{\frac{BB^{\prime}}{B+B^{\prime}+1},b}}_{r}.$

  * •

Adjoint: The adjoint operator of $T_{a}$, $T_{a}^{t}$ is a paradifferential
operator of order m with symbol $a^{t}$ defined by:

$a^{t}=\sum_{\left|\alpha\right|<\rho}\frac{1}{i^{\left|\alpha\right|}\alpha!}\partial^{\alpha}_{\xi}\partial^{\alpha}_{x}\bar{a}.$

Moreover, for all $s\in\mathbb{R}$ there exists a constant $C_{s,\rho}$ such
that

$\left\|T_{a}^{t}-T_{a^{t}}\right\|_{H^{s}\rightarrow H^{s-m+\rho}}\leq
C_{s,\rho}M^{m}_{\rho}(a).$

If $a=a(x)$ is a function of $x$ only then the paradifferential operator
$T_{a}$ is called a paraproduct. It follows from Theorem A.11 and the Sobolev
embedding that:

  * •

If $a\in H^{\alpha}(\mathbb{R}^{2})$ and $b\in H^{\beta}(\mathbb{R}^{2})$ with
$\alpha,\beta>1$, then

$T_{a}T_{b}-T_{ab}\text{ is of order
}-\bigg{(}\min\\{\alpha,\beta\\}-1\bigg{)}.$

  * •

If $a\in H^{\alpha}(\mathbb{R}^{2})$ with $\alpha>1$, then

$T_{a}^{*}-T_{\bar{a}}\text{ is of order }-\bigg{(}\alpha-1\bigg{)}.$

  * •

If $a\in W^{r,\infty}(\mathbb{R}^{2})$ and $r\geq 0$ then:

$\left\|au-T_{a}u\right\|_{H^{r}(\mathbb{R}^{2})}\leq
C\left\|a\right\|_{W^{r,\infty}}\left\|u\right\|_{L^{2}}.$

An important feature of paraproducts is that they are well defined for
function $a=a(x)$ which are not $L^{\infty}$ but merely in some Sobolev space
$H^{1-m}$, see [4].

######  Proposition A.12.

Let $m>0$. If $a\in H^{1-m}(\mathbb{R}^{2})$ and $u\in H^{\mu}(\mathbb{R})$
then $T_{a}u\in H^{\mu-m}(\mathbb{R})$. Moreover there exists $C_{\mu}$ such
that

$\left\|T_{a}u\right\|_{H^{\mu-m}}\leq
C_{\mu}\left\|a\right\|_{H^{1-m}}\left\|u\right\|_{H^{\mu}}.$

A main feature of paraproducts is the existence of paralinearisation theorems
which allow us to replace nonlinear expressions by paradifferential
expressions, at the price of error terms which are smoother than the main
terms.

######  Theorem A.13 (From [4]).

Let $\alpha,\beta,\kappa\in\mathbb{R}$ be such that $\alpha,\beta>1$ and
$\kappa\geq 0$, then

Bony’s Linearization Theorem: For all $C^{\infty}$ function F, if $a\in
H^{\alpha}(\mathbb{R})$ then

$F(a)-F(0)-T_{F^{\prime}(a)}a\in H^{2\alpha-1}(\mathbb{R}^{2}).$

If $a\in H^{\alpha}(\mathbb{R}^{2})$, $b\in H^{\beta}(\mathbb{R}^{2})$ and
$c\in W^{\kappa,\infty}(\mathbb{R}^{2})$, then $R(a,b)=ab-T_{a}b-T_{b}a\in
H^{\alpha+\beta-1}(\mathbb{R}^{2})$ and $R(a,c)=ac-T_{a}c-T_{c}a\in
H^{\alpha+\kappa}(\mathbb{R}^{2})$. Moreover there exists a positive constant
$C$ independent of $a$, $b$ and $c$ such that:

$\begin{cases}\left\|R(a,b)\right\|_{H^{\alpha+\beta-1}}=\left\|ab-
T_{a}b-T_{b}a\right\|_{H^{\alpha+\beta-1}}\leq
C\left\|a\right\|_{H^{\alpha}}\left\|b\right\|_{H^{\beta}},\\\
\left\|R(a,c)\right\|_{H^{\alpha+\kappa}}\leq
C\left\|a\right\|_{H^{\alpha}}\left\|c\right\|_{W^{\kappa,\infty}}.\end{cases}$
(A.5)

The residual term verifies for $\lambda>>b$, if $\mbox{supp}\
\mathscr{F}(a)\subset\left\\{\left|\xi\right|\geq\lambda\right\\}$ and
$\mbox{supp}\
\mathscr{F}(b)\subset\left\\{\left|\xi\right|\geq\lambda\right\\}$then:

$\mbox{supp}\
\mathscr{F}\left(R(a,b)\right)\subset\left\\{\left|\xi\right|\geq\left(1-\frac{1}{B}\right)\lambda+\frac{b}{B}\right\\}.$
(A.6)

###  A.4 Paracomposition

We recall the main properties of the paracomposition operator first introduced
by S. Alinhac in [1] to treat low regularity change of variables. Here we
present the results we reviewed and generalised in some cases in [23, 27].

######  Theorem A.14.

Let $\chi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be a
$C^{1}\left(\mathbb{R}^{2}\right)$ diffeomorphism with $D\chi\in
W^{r,\infty}$, $r>0,r\notin\mathbb{N}$ and take $s\in\mathbb{R}$ then the
following map is continuous:

$\displaystyle H^{s}(\mathbb{R}^{2})$ $\displaystyle\rightarrow
H^{s}(\mathbb{R}^{2})$ $\displaystyle u$
$\displaystyle\mapsto\chi^{*}u=\sum_{k\geq 0}\sum_{\begin{subarray}{c}l\geq
0\\\ k-N\leq l\leq k+N\end{subarray}}P_{l}(D)u_{k}\circ\chi,$

where $N\in\mathbb{N}^{*}$ is such that
$2^{N}>\sup_{k,\mathbb{R}^{d}}\left|\Phi_{k}D\chi\right|^{-1}$ and
$2^{N}>\sup_{k,\mathbb{R}^{2}}\left|\Phi_{k}D\chi\right|$. For two distinct
choices of $N$ and $\hat{N}$ defining two operators $\chi^{*}$ and
$\hat{\chi}^{*}$ we have

$\left\|\chi^{*}u-\hat{\chi}^{*}u\right\|_{H^{s+r}}\leq
C_{s+r}(\left\|D\chi\right\|_{L^{\infty}},\left\|D\chi^{-1}\right\|_{L^{\infty}})\left\|D\chi\right\|_{W^{r,\infty}}\left\|u\right\|_{H^{s}}.$

Taking $\tilde{\chi}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ a
$C^{1,\tilde{r}}\left(\mathbb{R}^{2}\right)$ diffeomorphism with $D\chi\in
W^{\tilde{r},\infty}$ map with $\tilde{r}>0,r\notin\mathbb{N}$, then the
previous operation has the natural fonctorial property:

$\forall u\in
H^{s}(\mathbb{R}^{2}),\chi^{*}\tilde{\chi}^{*}u=({\chi\circ\tilde{\chi}})^{*}u+Ru,$
$\text{with, }R:H^{s}(\mathbb{R}^{2})\rightarrow
H^{s+min(r,\tilde{r})}(\mathbb{R}^{2})\text{ continuous}.$

We now give the key paralinearisation theorem taking into account the
paracomposition operator.

######  Theorem A.15.

Consider $s\in\mathbb{R}$, $u\in H^{s}(\mathbb{R}^{2})$ and let
$\chi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be a
$C^{1}\left(\mathbb{R}^{2}\right)$ diffeomorphism with $D\chi\in
W^{r,\infty}(\mathbb{R}^{2})$, $r>0,r\notin\mathbb{N}$. Then:

$u\circ\chi(x)=\chi^{*}u(x)+T_{Du\circ\chi}\cdot\chi(x)+R(x)$

where the paracomposition given in the previous theorem verifies the
estimates:

$\left\|\chi^{*}u\right\|_{H^{s}}\leq
C_{s}(\left\|D\chi\right\|_{L^{\infty}},\left\|D\chi^{-1}\right\|_{L^{\infty}})\left\|u\right\|_{H^{s}},$
$Du\circ\chi\in\Gamma^{0}_{0}(\mathbb{R}^{2})\text{ for $u$ Lipschitz,}$

and the remainders verify the estimate if $r+s>0$

$\left\|R\right\|_{H^{s+r}}\leq
C_{s+r}(\left\|D\chi\right\|_{W^{r,\infty}},\left\|D\chi^{-1}\right\|_{L^{\infty}})\left\|u\right\|_{H^{s}}.$

Finally the commutation between a paradifferential operator
$a\in\Gamma^{m}_{\beta}(\mathbb{R}^{d})$ and a paracomposition operator
$\chi^{*}$ is given by the following

$\chi^{*}T_{a}u=T_{a^{*}}\chi^{*}u+T_{{q}^{*}}\chi^{*}u\text{ with
}q\in\Gamma^{m-\min(r,\beta)}_{0}(\mathbb{R}^{d}),$

where $a^{*}$ has the asymptotic expansion

$a^{*}(x,\xi)=\sum_{\begin{subarray}{c}\left|\alpha\right|\leq\lfloor
min(r,\beta)\rfloor\end{subarray}}\frac{1}{\alpha!}\partial^{\alpha}a(\chi(x),([D\chi(x)]^{-1})^{t}\xi)Q_{\alpha}(\chi(x),\xi),$
(A.7)

where,

$Q_{\alpha}(x^{\prime},\xi)=D^{\alpha}_{y^{\prime}}(e^{i(\chi^{-1}(y^{\prime})-\chi^{-1}(x^{\prime})-D\chi^{-1}(x^{\prime})(y^{\prime}-x^{\prime}))\cdot\xi})_{|y^{\prime}=x^{\prime}},$

and $Q_{\alpha}$ is polynomial in $\xi$ of degree
$\leq\frac{\left|\alpha\right|}{2}$, with $Q_{0}=1,Q_{1}=0$.

######  Remark A.16.

A close inspection of the proofs in [1, 27, 23] shows that the increasing
functions $C_{s}$ and $C_{s+r}$ increase at most polynomially fast.

## References

  * [1] Alinhac, S. Paracomposition et opérateurs paradifférentiels. Communications In Partial Differential Equations. 11, 87-121 (1986)
  * [2] Alinhac, S. & Gérard, P. Pseudo-differential operators and the Nash-Moser theorem. (American Mathematical Soc.,2007)
  * [3] Auscher, P. & Taylor, M. Paradifferential operators and commutator estimates: paradifferential operators. Communications In Partial Differential Equations. 20, 1743-1775 (1995)
  * [4] Bahouri, H., Chemin, J. & Danchin, R. Fourier analysis and nonlinear partial differential equations. (Springer,2011)
  * [5] Majda, A., Bertozzi, A. & Ogawa, A. Vorticity and incompressible flow. Cambridge texts in applied mathematics. Appl. Mech. Rev.. 55, B77-B78 (2002)
  * [6] Chemin, J. Fluides parfaits incompressibles. (Société mathématique de France,1995)
  * [7] Coifman, R. & Meyer, Y. Au dela des opérateurs pseudo-différentiels. Astérisque. 57 (1978)
  * [8] Drivas, T. & Elgindi, T. Singularity formation in the incompressible Euler equation in finite and infinite time. ArXiv Preprint ArXiv:2203.17221. (2022)
  * [9] Drivas, T., Elgindi, T. & Jeong, I. Twisting in Hamiltonian Flows and Perfect Fluids. ArXiv Preprint ArXiv:2305.09582. (2023)
  * [10] Denisov, S. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. ArXiv Preprint ArXiv:0908.3466. (2009)
  * [11] Elgindi, T., Murray, R. & Said, A. On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications. ArXiv Preprint ArXiv:2211.08418. (2022)
  * [12] Gérard, P. Mesures semi-classiques et ondes de Bloch. Séminaire Équations Aux Dérivées Partielles (Polytechnique) Dit Aussi” Séminaire Goulaouic-Schwartz”. pp. 1-19 (1991)
  * [13] Gérard, P. Résultats récents sur les fluides parfaits incompressibles bidimensionnels. Séminaire Bourbaki. 757 pp. 411-424 (1992)
  * [14] Hörmander, L. Lectures on nonlinear hyperbolic differential equations. (Springer Science & Business Media,1997)
  * [15] Kiselev, A. & Šverák, V. Small scale creation for solutions of the incompressible two-dimensional Euler equation. Annals Of Mathematics. 180, 1205-1220 (2014)
  * [16] Koch, H. Transport and instability for perfect fluids. Mathematische Annalen. 323, 491-523 (2002)
  * [17] Jeong, I. & Said, A. Logarithmic spirals in 2d perfect fluids. ArXiv Preprint ArXiv:2302.09447. (2023)
  * [18] Lions, P. & Paul, T. Sur les mesures de Wigner. Revista Matemática Iberoamericana. 9, 553-618 (1993)
  * [19] Marchioro, C. & Pulvirenti, M. Mathematical theory of incompressible nonviscous fluids. (Springer Science & Business Media,2012)
  * [20] Métivier, G. Para-differential calculus and applications to the Cauchy problem for nonlinear systems. (2008)
  * [21] Morgulis, A., Shnirelman, A. & Yudovich, V. Loss of smoothness and inherent instability of 2D inviscid fluid flows. Communications In Partial Differential Equations. 33, 943-968 (2008)
  * [22] Nadirashvili, N. Wandering solutions of Euler’s D-2 equation. Functional Analysis And Its Applications. 25, 220-221 (1991)
  * [23] Said, A. On paracomposition and change of variables in paradifferential operators. Journal Of Pseudo-Differential Operators And Applications. 14, 25 (2023)
  * [24] Shnirelman, A. Evolution of singularities, generalized Liapunov function and generalized integral for an ideal incompressible fluid. American Journal Of Mathematics. 119, 579-608 (1997)
  * [25] Shnirelman, A. Microglobal analysis of the Euler equations. Journal Of Mathematical Fluid Mechanics. 7 pp. S387-S396 (2005)
  * [26] Khesin, B., Misiołek, G. & Shnirelman, A. Geometric hydrodynamics in open problems. Archive For Rational Mechanics And Analysis. 247, 15 (2023)
  * [27] Taylor, M. Tools for PDE: pseudodifferential operators, paradifferential operators, and layer potentials. (American Mathematical Soc.,2000)
  * [28] Yudovich, V. The loss of smoothness of the solutions of Euler equations with time. Dinamika Splosn. Sredy Vyp. 16 pp. 71-78 (1974)
  * [29] Yudovich, V. On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid. Chaos: An Interdisciplinary Journal Of Nonlinear Science. 10, 705-719 (2000)
  * [30] Zlatoš, A. Exponential growth of the vorticity gradient for the Euler equation on the torus. Advances In Mathematics. 268 pp. 396-403 (2015)