# Non-uniqueness of Leray solutions to the hypodissipative Navier-Stokes
equations in two dimensions

Dallas Albritton  School of Mathematics, Institute for Advanced Study, 1
Einstein Dr., Princeton, NJ 08540, USA<EMAIL_ADDRESS>and  Maria
Colombo  EPFL SB, Station 8, CH-1015 Lausanne, Switzerland
<EMAIL_ADDRESS>
###### Abstract.

We exhibit non-unique Leray solutions of the forced Navier-Stokes equations
with hypodissipation in two dimensions. Unlike the solutions constructed in
[ABC22b], the solutions we construct live at a supercritical scaling, in which
the hypodissipation formally becomes negligible as $t\to 0^{+}$. In this
scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik
[Vis18a, Vis18b] to the hypodissipative setting at the nonlinear level. Our
perturbation argument is quasilinear in spirit and circumvents the spectral
theoretic approach to incorporating the dissipation in [ABC22b].

###### Key words and phrases:

Fractional Navier–Stokes equation, non-uniqueness, self-similarity

###### 1991 Mathematics Subject Classification:

35Q30, 35Q35, 35R11

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Linear theory
  4. 4 Nonlinear theory

##  1\. Introduction

We consider the two-dimensional fractional Navier-Stokes equations

$\left\\{\begin{aligned} \partial_{t}u+(u\cdot\nabla)u+\nabla
p+\Lambda^{\beta}u&=f\\\ \operatorname{div}u&=0\end{aligned}\right.\quad\text{
in }\mathbb{R}^{2}\times\mathbb{R}_{+}$ (NSβ)

where $\beta\in(0,2]$ and $\Lambda^{\beta}=(-\Delta)^{\beta/2}$ is the
fractional Laplacian. We consider the Cauchy problem for (NSβ) with
divergence-free initial velocity

$u(\cdot,0)=u_{0}\in L^{2}(\mathbb{R}^{2}).$ (1.1)

It is not difficult to adapt the seminal work [Ler34] of Leray to demonstrate
the existence of a global-in-time _Leray-Hopf solution_ for each divergence-
free initial velocity $u_{0}\in L^{2}(\mathbb{R}^{2})$ and body force $f\in
L^{1}_{t}L^{2}_{x}(\mathbb{R}^{2}\times\mathbb{R}_{+})$. These are
distributional solutions $u\in L^{\infty}_{t}L^{2}_{x}\cap
L^{2}_{t}\dot{H}^{\frac{\beta}{2}}_{x}(\mathbb{R}^{2}\times\mathbb{R}_{+})$ to
(NSβ) satisfying $\|u(\cdot,t)-u_{0}\|_{L^{2}(\mathbb{R}^{2})}\to 0$ as $t\to
0^{+}$ and the _energy inequality_ for all $t>0$:

$\frac{1}{2}\int_{\mathbb{R}^{2}}|u(x,t)|^{2}\,dx+\int_{0}^{t}\int_{\mathbb{R}^{2}}|\Lambda^{\frac{\beta}{2}}u|^{2}\,dx\,ds\leq\frac{1}{2}\int_{\mathbb{R}^{3}}|u_{0}(x)|^{2}\,dx+\int_{0}^{t}\int_{\mathbb{R}^{2}}f\cdot
u\,dx\,ds\,.$ (1.2)

It is well known that Leray’s solutions are unique in the classical setting
$\beta=2$. In this paper, we demonstrate non-uniqueness when $\beta<2$:

######  Theorem 1.1 (Non-uniqueness of Leray-Hopf solutions).

For all $\beta\in(0,2)$, there exist a force $f\in
L^{1}_{t}L^{2}_{x}(\mathbb{R}^{2}\times\mathbb{R}_{+})$ and two distinct
Leray-Hopf solutions, $u$ and $\bar{u}$, with zero initial velocity
$u_{0}\equiv 0$ and force $f$:

$u\neq\bar{u}\text{ on }\mathbb{R}^{2}\times(0,T)\text{ for all }T>0\,.$ (1.3)

The solutions are constructed explicitly and satisfy many desirable
properties, among which is smoothness on $\mathbb{R}^{2}\times(0,T)$ for $T\ll
1$ and energy equality in (1.2).

In a recent work [ABC22b] with Brué, we constructed non-unique Leray solutions
of the three-dimensional Navier-Stokes equations. While the method therein is
also applicable to (NSβ), at least when $\beta>1$, our goal is to highlight a
different but equally natural way of incorporating viscosity into the scheme.

The main idea of the proof is to perturb the examples of non-uniqueness
discovered by Vishik in [Vis18a, Vis18b] to the viscous setting. To begin, we
recall the basic dimensional analysis for the Euler equations

$\partial_{t}u+u\cdot\nabla u+\nabla p=f\,,\quad\operatorname{div}u=0\,,$
(1.4)

which can be equivalently rewritten in vorticity formulation as

$\partial_{t}\omega+u\cdot\nabla\omega=g\,,\quad
u=\Delta^{-1}\nabla^{\perp}\omega\,,$ (1.5)

where $\omega=\operatorname{curl}u$ is the vorticity and
$g=\operatorname{curl}f$. They have a two-parameter scaling symmetry

$u_{\mu,\eta}=\frac{\eta}{\mu}u(\mu x,\eta t)\,,\qquad\omega_{\mu,\eta}=\eta
u(\mu x,\eta t)$ (1.6)

corresponding to the physical dimensions $[x]=L$, $[t]=T$, $[u]=L/T$,
$[\omega]=1/T$, $[f]=L/T^{2}$, and $[g]=1/T^{2}$. We suppose a relationship
$T=L^{\alpha}$ between the dimensions. This yields a one-parameter scaling
symmetry:

$u_{\lambda}=\lambda^{\alpha-1}u(\lambda
x,\lambda^{\alpha}t)\,,\qquad\omega_{\lambda}=\lambda^{\alpha}u(\lambda
x,\lambda^{\alpha}t)\,.$ (1.7)

One may seek special solutions $\bar{\omega}$ which are invariant under the
above symmetry and necessarily have the form

$\bar{\omega}(x)=\frac{1}{t}\bar{\Omega}(\xi)\,,$ (1.8)

where $\xi=x/t^{\frac{1}{\alpha}}$. In fact, we may analyze an arbitrary Euler
solution $\omega$ in _similarity variables_ $(\xi,\tau)$, where $\tau=\log t$
and the _similarity profile_ $\Omega$ is defined by

$\omega(x,t)=\frac{1}{t}\Omega(\xi,\tau)\,,\qquad
g(x,t)=\frac{1}{t^{2}}G(\xi,\tau)\,.$ (1.9)

The Euler equations (1.5) in similarity variables are

$\partial_{\tau}\Omega+(-1-\frac{\xi}{\alpha}\cdot\nabla)\Omega+U\cdot\nabla\Omega=G\,,\quad
U=\nabla^{\perp}\Delta^{-1}\Omega\,.$ (1.10)

Self-similar solutions $\bar{\omega}$ correspond precisely to steady states
$\bar{\Omega}$ in the new variables.

If $\bar{\Omega}$ is an unstable steady state of (1.10), then it is natural to
seek its _unstable manifold_. Suppose that $\Omega^{\rm lin}$ is an
exponentially growing solution of the linearization of (1.10) around
$\bar{\Omega}$. One seeks a trajectory

$\Omega=\bar{\Omega}+\Omega^{\rm lin}+\Omega^{\rm per}\,,$ (1.11)

where $\Omega^{\rm per}=o_{\tau\to-\infty}(|\Omega^{\rm lin}|)$ ensures that
$\Omega$ is not identically equal to $\bar{\Omega}$. On the other hand,
$\Omega\overset{\tau\to-\infty}{\longrightarrow}\bar{\Omega}$. This is the
non-uniqueness scenario in [Vis18a], which demonstrated the sharpness of the
Yudovich class [Yud63] in the ‘forced category’.

How does the above non-uniqueness scenario look with viscosity? The fractional
Navier-Stokes equations (NSβ) in self-similarity variables are

$\partial_{\tau}\Omega+(-1-\xi\cdot\nabla_{\xi}/\alpha)\Omega+U\cdot\nabla\Omega+e^{\tau\gamma}\Lambda^{\beta}\Omega=G\,,\quad
U=\nabla^{\perp}\Delta^{-1}\Omega\,,$ (1.12)

where

$\gamma=1-\frac{\beta}{\alpha}\,.$ (1.13)

The traditional choice of exponent is $\alpha=\beta$, so that the PDE (1.12)
is $\tau$-autonomous. (The choice $\alpha=\beta=2$ in [JŠ14, ABC22b] is the
classical Navier-Stokes scaling.) This choice is determined by the dimensions
of the viscosity: $[\nu]=L^{\beta}/T$. With this choice, it is possible to
follow the strategy in [ABC22b] to prove Theorem 1.1 when $\beta\in(1,2)$. The
same strategy would be valid in dimension three with $\beta\in(1,5/2)$, that
is, up to the Lions exponent. The most delicate part of the analysis in
[ABC22b] is an eigenvalue perturbation argument which shows that not only the
linearized $2$D Euler equations, but also the linearized $3$D Navier-Stokes
equations admit an unstable eigenvalue around a well chosen background
solution.

In this paper, we instead consider $\alpha>\beta$, so that $\gamma>0$ and the
dissipation term $e^{\tau\gamma}\Lambda^{\beta}$ becomes perturbative
(formally) as $\tau\to-\infty$. Then we consider the ansatz (1.11) exactly as
in the inviscid setting, in particular, choosing $\Omega^{\rm lin}$ as an
unstable mode for $2$D Euler rather than Navier-Stokes, and study $\Omega^{\rm
per}$. We take the existence of an unstable inviscid self-similar solution for
granted.111Indeed, the majority of the present work was actually written
before we realized that the spectral problem in [ABC22b] was manageable. This
work is independent from [ABC22b] except for the observation that Vishik’s
vortex can be truncated. In this way, we turn our attention away from the
spectral problem at the heart of [ABC22b] and toward the nonlinear arguments,
which become quasilinear in nature and inherently more difficult. The
nonlinear scheme will be outlined in Section 2.1.

Our strategy can be compared to analogous strategies in the singularity
formation literature, which seek _stable manifolds_ as $\tau\to+\infty$ of
(potentially unstable) steady states in backward self-similarity variables. In
contrast, we seek _unstable manifolds_ as $\tau\to-\infty$ of unstable steady
states in forward self-similarity variables. Notably, the three-dimensional
compressible Navier-Stokes ‘implosion’ singularity in [MRRS19] is a
perturbation of a compressible Euler singularity; in similarity variables, the
viscous term appears with a factor $e^{-\tau\gamma}$. See [OP21, CMVP21] for
perturbative dissipation in the fractional Burgers equation.

The above comparison highlights the following point: In the absence of
forcing, the desired self-similar solutions and corresponding exponents are
_chosen for you_. When the scaling of the self-similar solution is
supercritical with respect to the scaling of the viscous equation, the formal
calculations give hope to perturb inviscid solutions to viscous solutions. For
this reason, we choose to work with general (supercritical) exponents below.
The present work provides another general method to incorporate a
supercritical viscosity into non-uniqueness scenarios; we hope that this will
inform future studies on non-uniqueness scenarios and continuation past
singularities of various PDEs in fluid mechanics.

_Previous work_. The important works [JŠ14, JŠ15, GŠ17] proposed a route,
accompanied by significant numerical evidence, to the (yet unproven) non-
uniqueness of Leray solutions _without force_. The unstable manifold in
similarity variables is already in [JŠ15, Theorem 4.1]. A related program for
the two-dimensional Euler equations was initiated in [BS21, BM20]. Non-
uniqueness of weak Navier-Stokes solutions was demonstrated via convex
integration in [BV19]. With hypoviscosity $\beta<2/3$, convex integration was
able to achieve the Leray class [CDLDR18, DR19]. See [ABC22b] for a more
comprehensive review.

Finally, we recently demonstrated non-uniqueness of Leray solutions to the
forced Navier-Stokes solutions in bounded domains via a gluing method
[ABC22a].

##  2\. Preliminaries

Let $0<\beta<\alpha<2$. Given a compactly supported, smooth velocity field
$\bar{U}\in C^{\infty}_{0}(B_{2};\mathbb{R}^{2})$,
$\bar{U}(x)=V(r)e_{\theta}$, with corresponding vorticity
$\bar{\Omega}=\operatorname{curl}\bar{U}$, we define the linearized operator
around $\bar{\Omega}$,

$-\bm{L}_{{\rm
ss}}\Omega=\left(-1-\frac{\xi}{\alpha}\cdot\nabla\right)\Omega+\bar{U}\cdot\nabla\Omega+U\cdot\nabla\bar{\Omega}\,,$
(2.1)

where $U=\nabla^{\perp}\Delta^{-1}\Omega$. The above operator is considered
$\bm{L}_{\rm ss}\colon D(\bm{L}_{\rm ss})\subset L^{2}_{m}\to L^{2}_{m}$,
where $L^{2}_{m}\subset L^{2}$ consists of $m$-fold rotationally symmetric
$L^{2}$ functions and $D(\bm{L}_{\rm ss}):=\\{\Omega\in L^{2}_{m}:\bm{L}_{\rm
ss}\Omega\in L^{2}\\}$.

The following proposition was essentially demonstrated in [Vis18a, Vis18b,
ABC+21, ABC22b]

######  Proposition 2.1 (Unstable vortex).

Let $a_{0}\geq 0$. There exists a compactly supported, smooth velocity field
$\bar{U}\in C^{\infty}_{0}(B_{2};\mathbb{R}^{2})$,
$\bar{U}(x)=V(r)e_{\theta}$, with corresponding vorticity
$\bar{\Omega}=\operatorname{curl}\bar{U}$ and satisfying the following
properties.

1\. Linear instability. There exists an integer $m\geq 2$ such that
$\bm{L}_{{\rm ss}}$ has an unstable eigenvalue. More specifically, there exist
$\lambda\in\mathbb{C}$ with $\operatorname{Re}\lambda=:a\geq a_{0}$ and
$\eta\in L^{2}_{m}\setminus\\{0\\}$ satisfying $\bm{L}_{\rm
ss}\eta=\lambda\eta$.

2\. Semigroup estimate. The operator $\bm{L}_{{\rm ss}}$ generates a
continuous semigroup in $L^{2}_{m}$, and for all $\delta>0$, we have the
semigroup estimate

$\lVert e^{\tau\bm{L}_{\rm ss}}\rVert_{L^{2}_{m}\to
L^{2}}\lesssim_{\delta}e^{(a+\delta)\tau}\text{ for all }\tau\geq 0\,.$ (2.2)

3\. Eigenfunction estimates. Any eigenfunction $\eta$ with unstable eigenvalue
$\lambda$ is compactly supported in $B_{1}$. For any $k\in\mathbb{N}_{0}$ with
$\operatorname{Re}\lambda>k/\alpha-1$, the above eigenfunction belongs to
$H^{k}$, and its velocity field ${\rm BS}[\eta]$ belongs to $H^{k+1}$.

###### Proof.

1\. and 2. above are contained in [ABC+21, Theorem 2.4.2] except for the
compact support property of $\bar{U}$, which was pointed out in [ABC22b,
Proposition 2.2]. Therefore, we focus on 3\. Eigenfunction estimates. We
present an alternative to the approach in [ABC+21, Lemma 5.0.3], where it was
shown that $\eta\in W^{2,\infty}$. Suppose that $\eta\in D(\bm{L}_{\rm ss})$
is an eigenfunction with unstable eigenvalue $\lambda=\lambda^{\prime}-1$:

$\left(\lambda^{\prime}-\frac{\xi}{\alpha}\cdot\nabla+\bar{U}\cdot\nabla\right)\eta+{\rm
BS}[\eta]\cdot\nabla\bar{\Omega}=0\,.$ (2.3)

First, we demonstrate that $\eta$ is compactly supported. When $|\xi|\geq 2$
is beyond the support of $\bar{U}\in C^{\infty}_{0}(B_{2})$, we have

$\frac{r}{\alpha}\partial_{r}\eta=\lambda^{\prime}\eta\,,\quad r\geq 2\,,$
(2.4)

whose only decaying solutions are identically zero in $\mathbb{R}^{2}\setminus
B_{2}$.

Second, we prove that $\eta\in H^{k}$ provided that
$\operatorname{Re}\lambda^{\prime}>k/\alpha$. We consider ${\rm
BS}[\eta]\cdot\nabla\bar{\Omega}$ to be a ‘forcing term’ in the resolvent
problem, and we rewrite the above equation via the Laplace transform
characterization of the resolvent: $R(\lambda^{\prime},\bm{L}_{\rm
ss})=\int_{0}^{+\infty}e^{-\lambda^{\prime}t}e^{t\bm{L}_{\rm ss}}\,dt$. Let
$X_{t}$ be the flow map associated to the autonomous vector field
$V(\xi):=\xi/\alpha-\bar{U}(\xi)$. Then

$\eta=-\int_{0}^{+\infty}e^{-\lambda^{\prime}t}({\rm
BS}[\eta]\cdot\nabla\bar{\Omega})\circ X_{t}\,dt\,.$ (2.5)

To estimate $\eta$, we recall basic properties of $X_{t}$, which is explicitly

$X_{t}(r_{0}e^{i\theta_{0}})=r(t,r_{0})e^{i\theta(t,\theta_{0},r_{0})}\,,$
(2.6)
$r(t,r_{0})=e^{t/\alpha}r_{0}\,,\quad\theta(t,\theta_{0},r_{0})=\theta_{0}-\int_{0}^{t}\bar{U}_{\theta}(r(s,r_{0}))\,ds\,.$
(2.7)

By the general formulae

$(\nabla X_{t}e_{r})(r_{0}e^{i\theta_{0}})=\partial_{r_{0}}r\;(e_{r}\circ
X_{t})+r\partial_{r_{0}}\theta\;(e_{\theta}\circ X_{t})\,,$ (2.8) $(\nabla
X_{t}e_{\theta})(r_{0}e^{i\theta_{0}})=\frac{1}{r_{0}}\partial_{\theta_{0}}r\;(e_{r}\circ
X_{t})+\frac{r}{r_{0}}\partial_{\theta_{0}}\theta\;(e_{\theta}\circ X_{t})\,,$
(2.9)

it is not difficult to demonstrate that $|\nabla X_{t}|\lesssim e^{t/\alpha}$.
A different way of saying this is that solutions of the vector-valued ODE
$\dot{\vec{y}}=(\nabla V)(X_{t})\vec{y}$ satisfy $|\vec{y}(t)|\lesssim
e^{t/\alpha}|\vec{y}(0)|$. By taking spatial derivatives of the ODE
$\dot{X}_{t}=V(X_{t})$, this is enough to bootstrap the estimate on $\nabla
X_{t}$ to $|\nabla^{k}X_{t}|\lesssim_{k}e^{kt/\alpha}$ for all $k\geq 0$. In
particular, we have

$\|\eta\|_{H^{k}}\lesssim\int_{0}^{+\infty}e^{(k/\alpha-\lambda^{\prime})t}\,dt\;\|{\rm
BS}[\eta]\cdot\nabla\bar{\Omega}\|_{H^{k}}\lesssim\|\eta\|_{H^{k-1}}\,,$
(2.10)

where the implied constant depends on various quantities. Hence, one can
bootstrap $\eta\in L^{2}$ to $\eta\in H^{k}$ provided that
$\operatorname{Re}\lambda>k/\alpha-1$. Finally, since $\eta$ is compactly
supported and mean zero, we have $|{\rm BS}[\eta]|\lesssim|\xi|^{-2}$ for
$|\xi|\geq 2$. In particular, ${\rm BS}[\eta]\in L^{2}$. ∎

######  Remark 2.2.

It is natural that the unstable eigenfunctions should have smoothness related
to $\operatorname{Re}\lambda$ since solutions to the ODE $\lambda
f=r\partial_{r}f$, $\operatorname{Re}\lambda>0$, are $f=\text{const}.\times
r^{\lambda}$. In Proposition 2.1, once it is known that the eigenfunctions are
$C^{\beta}$, it is possible to bootstrap them in the $C^{k,\beta}$ scale, but
we do not require this here.

Recall the similarity variables $\xi=x/t^{1/\alpha}$ and $\tau=\log t$ defined
above (1.9). Given a similarity profile $\bar{\Omega}$ as in Proposition 2.1,
we define the self-similar solution $\bar{\omega}$ according to (1.8). Its
velocity field is $\bar{u}(x,t)=t^{-1+1/\alpha}\bar{U}(\xi)$. More generally,
given a vorticity $\omega$, we extract its similarity profile $\Omega$
according to (1.9). Its velocity field is $u(x,t)=t^{-1+1/\alpha}U(\xi,\tau)$.
We make the convention that, given a lowercase variable representing vorticity
or velocity, its similarity profile is represented by the corresponding
uppercase variable, and vice versa.

We now state basic properties of the force, which is induced entirely by the
time derivative and fractional Laplacian of $\bar{\omega}$.

######  Lemma 2.3 (Finite energy forcing).

Define $h:=\partial_{t}\bar{\omega}+\Lambda^{\beta}\bar{\omega}$ and
$f:=\Delta^{-1}\nabla^{\perp}h=\partial_{t}\bar{u}+\Lambda^{\beta}\bar{u}$.
Then the background velocity $\bar{u}$ solves the fractional Navier-Stokes
equations with velocity forcing $f$, which belongs to
$L^{1}_{t}L^{2}_{x}(\mathbb{R}^{2}\times(0,T))$ for all $T>0$. Additionally,
$f\in L^{\infty}_{t}H^{k}_{x}(\mathbb{R}^{2}\times(\varepsilon,T))$ for all
$k\geq 0$ and $0<\varepsilon<T<+\infty$.

This is a simple consequence of scaling.

###  2.1. Strategy

We revisit the ansatz (1.11):

$\Omega=\bar{\Omega}+\Omega^{\rm lin}+\Omega^{\rm per}$ (2.11)

where

$\Omega^{\rm lin}(\xi,\tau):=\operatorname{Re}(e^{\lambda\tau}\eta)\,,$ (2.12)

$\lambda$ is a maximally unstable eigenvalue of $\bm{L}_{\rm ss}$, and $\eta$
is an associated non-trivial eigenfunction. Then $\Omega^{\rm lin}$ is a
solution to the linearized evolution equation

$\partial_{\tau}\Omega^{\rm lin}-\bm{L}_{\rm ss}\Omega^{\rm lin}=0\,,$ (2.13)

and our goal is to solve for the higher-order correction $\Omega^{\rm per}$.
The PDE satisfied by $\Omega^{\rm per}$ is

$\displaystyle\partial_{\tau}\Omega^{\rm per}-\bm{L}_{\rm ss}\Omega^{\rm
per}+(U^{\rm lin}+U^{\rm per})\cdot\nabla\Omega^{\rm per}+U^{\rm
per}\cdot\nabla\Omega^{\rm lin}+e^{\tau\gamma}\Lambda^{\beta}\Omega^{\rm per}$
(2.14) $\displaystyle\quad=-U^{\rm lin}\cdot\nabla\Omega^{\rm
lin}-e^{\tau\gamma}\Lambda^{\beta}\Omega^{\rm lin}\,,$

where the right-hand side is considered a source term. The fractional
Laplacian $e^{\tau\gamma}\Lambda^{\beta}$ will be treated perturbatively.
Hence, our approach is essentially quasilinear in nature, and it will be
necessary to incorporate the transport $(U^{\rm lin}+U^{\rm per})\cdot\nabla$
into the operator. Define

$-\bm{L}[W]=-\bm{L}_{\rm ss}+(U^{\rm
lin}+W)\cdot\nabla+e^{\tau\gamma}\Lambda^{\beta}\,,$ (2.15)

the ‘main operator’ in the PDE, and the associated (formal) solution operator
$S[W]=S[W](\tau,s)$. We seek solutions satisfying Duhamel’s formula:

$\Omega^{\rm per}(\cdot,\tau)=-\int_{-\infty}^{\tau}S[U^{\rm
per}](\tau,s)[(U^{\rm lin}+U^{\rm per})\cdot\nabla\Omega^{\rm
lin}+e^{s\gamma}\Lambda^{\beta}\Omega^{\rm lin}](\cdot,s)\,ds\,.$ (2.16)

It will be necessary to develop the (strong) solution theory for $\Omega^{\rm
per}$ and the solution operator $S$ in high regularity spaces, whose
corresponding velocity fields are Lipschitz continuous, e.g., $\Omega\in
H^{s}$, $s>d/2$. It will be more convenient to estimate $S$ in a function
space $X\subset W^{1,4}$ which will be defined in Section 3.

There are two main difficulties, which are already present in the Euler
equations and have been encountered in previous work, e.g., [BGS02]; the
difficulties will only be exacerbated by the dissipation
$e^{\tau\gamma}\Lambda^{\beta}$.

First, there is the apparent derivative loss in adding the transport operator
$(U^{\rm lin}+U^{\rm per})\cdot\nabla$ onto $\bm{L}_{\rm ss}$. On the one
hand, it is necessary to use the spectral information on $\bm{L}_{\rm ss}$
through the semigroup estimate in $L^{2}$. On the other hand, it is too naïve
to write Duhamel’s formula, containing expressions like
$\int_{0}^{\tau}e^{(\tau-s)\bm{L}_{\rm ss}}(U^{\rm lin}\cdot\nabla\Omega^{\rm
per})\,ds$, and hope to close an estimate at the $L^{2}$ level. Moreover, in
our setting, adding the dissipation operator $e^{\tau\gamma}\Lambda^{\beta}$
further exhibits a loss of $\beta$ derivatives. The key observation for
handling the loss is that the compact term $U^{\rm
per}\cdot\nabla\bar{\Omega}$, responsible for the creation of the unstable
eigenvalues, gains regularity. Hence, the strategy will be to lose derivatives
while applying the semigroup estimate in Duhamel’s formula and regain them in
a bootstrapping procedure based on energy estimates.222A different approach is
to work with the Lagrangian formulation, in which there is no derivative loss,
see the construction of unstable manifolds in [LZ13, LZ14].

The second difficulty concerns the operator $\bar{U}\cdot\nabla$ in high
regularity spaces. One way to see the difficulty is to pass a derivative into
the equation: The PDE for $\partial_{i}\Omega^{\rm per}$ contains the term
$\partial_{i}\bar{U}\cdot\nabla\Omega^{\rm per}$, which in principle can
change the growth rate. In [BGS02], it was observed that it is enough assume
that $\operatorname{Re}\lambda$ is greater than the maximal fluid Lyapunov
exponent. Our approach will be Eulerian rather than Lagrangian. It is an
energy method specific to shear flows and vortices (in short, the idea is to
apply $\partial_{\theta}$ first), and crucially, it is well behaved even with
the term $e^{\tau\gamma}\Lambda^{\beta}$. In fact, when $\beta>1$, the term
$e^{\tau\gamma}\Lambda^{\beta}$ will help us in the bootstrapping procedure:
Once we have propagated one derivative, the other $\beta-1$ derivatives needed
to close the argument are regained by smoothing. Heuristically, a gain of
$\beta$ derivatives costs a factor $e^{\tau\gamma}$, which is why the
dissipation term cannot be used to control everything. This is seen in the
tracking of the dissipation norm $D$ below.

##  3\. Linear theory

In the following, we will assume that $a_{0}=8/\alpha$ and
$\delta\leq\delta_{0}:=\min\\{\gamma/4,1/8,a_{0}/2\\}$. We also fix
$\bar{\Omega}$, $\lambda=a+ib$, $\eta$ and $m$ as in Proposition 2.1. We
denote by $C$ constants whose values depend only on

${\rm data}=(\alpha,\beta,\bar{\Omega},a,b,\eta,m,\delta)\,.$ (3.1)

The dependence on $\bar{\Omega}$ and $\eta$ can be made more precise through
their regularity and size of the support, but this is not needed for the rest
of our arguments, and we do not pursue it here. The value of $C$ may change
from line to line. When we need to recall a specific constant of this type, we
denote it with $C_{1},C_{2}$, …

This section is devoted to estimates on the solution operator $S[W]$ to the
PDE

$\partial_{\tau}\Omega+(-1-\xi\cdot\nabla/\alpha)\Omega+\bar{U}\cdot\nabla\Omega+U\cdot\nabla\bar{\Omega}+(U^{\rm
lin}+W)\cdot\nabla\Omega+e^{\tau\gamma}\Lambda^{\beta}\Omega=0$ (3.2)

with initial data $\Omega(\cdot,\tau_{0})=\Omega_{0}$ and divergence-free $W$
satisfying the property (3.7) below. We assume that $\Omega_{0}$ and $W$ are
$m$-fold rotationally symmetric, so that the semigroup estimate in Proposition
2.1 can be applied. The remaining terms in the PDE (2.14) will be incorporated
in Section 4 by Duhamel’s formula.

Let us fix a parameter $Q\in[1,3/2)$. In view of the precise structure of the
terms in our equation, it turns out that a natural space to perform such
estimates is given by the norm

$\|\Omega\|_{X}=\|\Omega\|_{L^{Q}\cap
L^{2}}+\|\partial_{\theta}\Omega\|_{L^{Q}\cap
L^{4}}+\|\nabla\Omega\|_{L^{2}\cap L^{4}}\,,$ (3.3)

which is equivalent to
$\|\Omega\|_{L^{Q}}+\|\partial_{\theta}\Omega\|_{L^{Q}\cap
L^{4}}+\|\nabla\Omega\|_{L^{2}\cap L^{4}}$, but the description in (3.3) is
perhaps more indicative of the estimates we perform. We also keep track of the
quantity

$\|\Omega\|_{D(\tau,\tau_{0})}^{2}=\int_{\tau_{0}}^{\tau}e^{s\gamma}\|\Lambda^{1+\frac{\beta}{2}}\Omega\|_{L^{2}}^{2}\,ds$
(3.4)

related to the dissipation.

The space $X$ has some useful properties listed below. Let us consider $W\in
L^{2}$ such that $\operatorname{curl}W\in X$. Since
$K_{2}=1_{B_{1}}K_{2}+1_{B_{1}^{c}}K_{2}\in L^{p}+L^{q}$ for every $p<2,q>2$,
we have by Young’s convolution inequality that

$\|W\|_{L^{\infty}}=\|K_{2}\ast\operatorname{curl}W\|_{L^{\infty}}\leq
C(\|\operatorname{curl}W\|_{L^{Q}}+\|\operatorname{curl}W\|_{L^{4}})\leq
C\|\operatorname{curl}W\|_{X}\,,$ (3.5)

and similarly,

$\|\partial_{\theta}W\|_{L^{\infty}}=\|K_{2}\ast\partial_{\theta}\operatorname{curl}W\|_{L^{\infty}}\leq
C(\|\partial_{\theta}\operatorname{curl}W\|_{L^{Q}}+\|\partial_{\theta}\operatorname{curl}W\|_{L^{4}})\leq
C\|\operatorname{curl}W\|_{X}\,.$ (3.6)

The main result of this section is

######  Proposition 3.1 (Estimates for $S$).

Let $W\in L^{\infty}((-\infty,0];L^{2}(\mathbb{R}^{2}))$ be such that

$\sup_{\tau\in(-\infty,0)}e^{-a\tau}\lVert\operatorname{curl}W\rVert_{X}\leq
1.$ (3.7)

Then for every $\delta\leq\delta_{0}$, there exist constants $T_{\rm
max}:=T_{\rm max}({\rm data})\leq 0$ and $\bar{C}:=\bar{C}({\rm data})>1$ such
that for every $\tau_{0}\in(-\infty,T_{\rm max}]$ and $\tau\in[\tau_{0},T_{\rm
max}]$, we have

$\|S[W](\tau,\tau_{0})\Omega_{0}\|_{X}+\|S[W](\cdot,\tau_{0})\Omega_{0}\|_{D(\tau_{0},\tau)}\leq\bar{C}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.$
(3.8)

###### Proof of Proposition 3.1.

The constants $T_{\rm max}\leq 0$ and $\bar{C}>1$ will be chosen later.

For $\tau\in[\tau_{0},T_{\rm max}]$, we denote
$\Omega(\cdot,\tau)=S[W](\tau,\tau_{0})\Omega_{0}$ the solution of (3.2). The
proof of Proposition 3.1 will be established through the following claim.

_Let $\tau_{1}\in[\tau_{0},T_{\rm max}]$ satisfying the following bootstrap
assumption:_

For all $\tau\in[\tau_{0},\tau_{1}]$, the estimate (3.8) holds. (3.9)

_Then for all $\tau\in[\tau_{0},\tau_{1}]$ we have_

$\|\Omega(\cdot,\tau)\|_{X}+\|\Omega\|_{D(\tau_{0},\tau)}\leq
C_{1}(1+\bar{C}^{1/2}e^{\zeta T_{\rm
max}})\bar{C}^{1/2}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}\,,$ (3.10)

_where $C_{1}:=C_{1}({\rm data})>1$ and $\zeta:=\zeta({\rm data})>0$._

Once this claim is established, we easily conclude the proof of Proposition
3.1 by the following argument. We fix $\bar{C}=(4C_{1})^{2}$ and $T_{\rm max}$
in terms of ${\rm data}$, but otherwise independent of $\tau_{0}$, such that
$\bar{C}^{1/2}e^{\zeta T_{\rm max}}\leq 1$. Next, we consider $\bar{\tau}_{1}$
to be the largest value of $\tau_{1}\in(\tau_{0},T_{\rm max})$ such that (3.8)
holds. If $\bar{\tau}_{1}=T_{\rm max}$, our main claim is proved. The case
$\bar{\tau}_{1}<T_{\rm max}$ is ruled out by the continuity of
$\tau\to\|\Omega(\cdot,\tau)\|_{X}$ and by (3.10), which implies with our
choice of $T_{\rm max}$ that the bound (3.8) is not saturated in
$\bar{\tau}_{1}$, namely,

$\|\Omega(\cdot,\bar{\tau}_{1})\|_{X}+\|\Omega\|_{D(\tau_{0},\bar{\tau}_{1})}\leq\frac{\bar{C}}{2}e^{(a+\delta)(\bar{\tau}_{1}-\tau_{0})}\|\Omega_{0}\|_{X}\,.$
(3.11)

Hence, the proof of Proposition 3.1 is complete provided that (3.10) is
established.

Step 1. [Baseline estimate: Improved $L^{2}$ bound] Suppose that the bootstrap
assumption (3.9) holds. Then, for all $\tau\in[\tau_{0},\tau_{1}]$,

$\|\Omega(\cdot,\tau)\|_{L^{2}}\leq C(1+\bar{C}e^{2\zeta T_{\rm
max}}e^{(a+\delta)(\tau-\tau_{0})})\|\Omega_{0}\|_{X}.$ (3.12)

Indeed, by Duhamel’s formula:

$\Omega(\cdot,\tau)=e^{(\tau-\tau_{0})L_{\rm
ss}}\Omega_{0}-\int_{\tau_{0}}^{\tau}e^{(\tau-s)L_{\rm ss}}[(W+U^{\rm
lin})\cdot\nabla\Omega+e^{s\gamma}\Lambda^{\beta}\Omega](\cdot,s)\,ds.$ (3.13)

According to the semigroup estimate (2.2), we have333 Crucially, this is where
we use assumptions on the semigroup $e^{\tau\bm{L}_{\rm ss}}$ generated by the
linearized operator. While the above Duhamel formula ‘loses derivatives’ in
the sense that we are estimating in $L^{2}$ according to ‘higher’ quantities,
e.g., $\|\nabla\Omega\|_{L^{2}}$, it is acceptable to lose derivatives at this
level, though it will not be acceptable at the ‘top tier’
($\|\nabla\Omega\|_{L^{4}}$). This is common in quasi-linear perturbation
arguments and can already be seen in standard proofs of local-in-time
existence for the Euler equations.

$\begin{split}\|\Omega(\cdot,\tau)\|_{L^{2}}&\leq\|e^{(\tau-\tau_{0})L_{\rm
ss}}\Omega_{0}\|_{L^{2}}+\int_{\tau_{0}}^{\tau}\|e^{(\tau-s)L_{\rm
ss}}[(W+U^{\rm
lin})\cdot\nabla\Omega-e^{s\gamma}\Lambda^{\beta}\Omega]\|_{L^{2}}(\cdot,s)\,ds\\\
&\leq
Ce^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{L^{2}}+\int_{\tau_{0}}^{\tau}e^{(a+\delta)(\tau-s)}[\|(W+U^{\rm
lin})\cdot\nabla\Omega(\cdot,s)\|_{L^{2}}(\cdot,s)\,ds\\\
&\qquad+\int_{\tau_{0}}^{\tau}e^{(a+\delta)(\tau-s)}e^{s\gamma}\|\Lambda^{\beta}\Omega\|_{L^{2}}(\cdot,s)]\,ds\,.\end{split}$
(3.14)

Next, we estimate the first integrand in the right-hand side of (3.14). By
(3.5), the smoothness of $\eta$ in Proposition 2.1, the assumption on $W$
(3.7) and the bootstrap assumption (3.9), we obtain that

$\displaystyle\|(W+U^{\rm lin})\cdot\nabla\Omega(\cdot,s)\|_{2}$
$\displaystyle\leq\|W\|_{\infty}\|\nabla\Omega\|_{2}+\|U^{\rm
lin}\|_{\infty}\|\nabla\Omega\|_{2}$ (3.15) $\displaystyle\leq
e^{as}\|\Omega(\cdot,s)\|_{X}\lesssim
e^{2s\zeta}e^{(a+\delta)(s-\tau_{0})}\|\Omega_{0}\|_{X}.$ (3.16)

Plugging this estimate in the first integral in (3.14), we can estimate such
integral with $C\bar{C}e^{2\zeta T_{\rm
max}}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}$.

Next, we consider the dissipation term. First, we interpolate between $L^{2}$
and $\dot{H}^{1+\frac{\beta}{2}}$ and by Young inequality we have

$\displaystyle\Big{(}\int_{\tau_{0}}^{\tau}e^{s\gamma}\|\Lambda^{\beta}\Omega\|_{L^{2}}^{2}\Big{)}^{1/2}\,ds$
$\displaystyle\leq\Big{(}\int_{\tau_{0}}^{\tau}e^{s\gamma}\|\Lambda^{1+\frac{\beta}{2}}\Omega\|_{L^{2}}^{2\frac{2-\beta}{2+\beta}}\|\Omega\|_{L^{2}}^{\frac{4\beta}{2+\beta}}\,ds\Big{)}^{1/2}$
(3.17) $\displaystyle\leq
C\Big{(}\int_{\tau_{0}}^{\tau}e^{s\gamma}\big{(}\|\Lambda^{1+\frac{\beta}{2}}\Omega\|_{L^{2}}^{2}+\|\Omega\|_{L^{2}}^{2}\big{)}\,ds\Big{)}^{1/2}$
$\displaystyle\leq
C\|\Omega\|_{D(\tau_{0},\tau)}+C\bar{C}\Big{(}\int_{\tau_{0}}^{\tau}e^{2s\gamma}e^{2(a+\delta)(s-\tau_{0})}\|\Omega_{0}\|_{X}^{2}\,ds\Big{)}^{1/2}$
$\displaystyle\leq C\bar{C}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X},$

where we used the bootstrap assumption both on $\|\Omega\|_{D(\tau_{0},\tau)}$
and on $\|\Omega\|_{L^{2}}$. Then, using the Cauchy-Schwarz in time and
(3.17),

$\displaystyle\int_{\tau_{0}}^{\tau}e^{(\tau-s)(a+\delta)}e^{s\gamma/2}\|e^{s\gamma/2}\Lambda^{\beta}\Omega(\cdot,s)\|_{L^{2}}\,ds$
$\displaystyle\leq
Ce^{\tau\gamma/2}(\bar{C}+1)e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.$
(3.18)

This completes the proof of (3.12).

We notice that from an interpolation inequality, (3.12) and the bootstrap
assumption it follows also an improved $L^{4}$ bound of the following form:

$\begin{split}\|\Omega\|_{L^{4}}&\leq
C\|\Omega\|_{L^{2}}^{1/2}\|\nabla\Omega\|_{L^{2}}^{1/2}\\\
&\leq[C(1+\bar{C}e^{2\zeta T_{\rm
max}})e^{(a+\delta)(\tau-\tau_{0})})]^{1/2}\bar{C}^{1/2}e^{\frac{a+\delta}{2}(\tau-\tau_{0})}\|\Omega_{0}\|_{X}\\\
&\leq C(1+\bar{C}^{1/2}e^{\zeta T_{\rm
max}})\bar{C}^{1/2}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.\end{split}$
(3.19)

Notice that this estimate is worse than (3.12) due to the presence of
$\bar{C}^{1/2}$; however, being its power strictly less than $1$, the estimate
is still good enough.

Step 2. [Improved $L^{Q}$ bound] Suppose that the bootstrap assumption (3.9)
holds. Then for all $\tau\in[\tau_{0},\tau_{1}]$

$\|\Omega(\cdot,\tau)\|_{L^{Q}}\leq C(C_{\delta_{0}}+C\bar{C}e^{2\zeta T_{\rm
max}}e^{(a+\delta)(\tau-\tau_{0})})\|\Omega_{0}\|_{X}.$ (3.20)

We perform an $L^{Q}$ energy estimate on the equation for $\Omega$, namely we
multiply the equation by $Q|\Omega|^{Q-2}\Omega$ and integrate in space. We
obtain

$\frac{d}{dt}\int|\Omega|^{Q}\,d\xi\leq\Big{(}-Q+\frac{2}{\alpha}\Big{)}\int|\Omega|^{Q}\,d\xi+Q\int|U\cdot\nabla\bar{\Omega}||\Omega|^{Q-1}\,d\xi-Q\int
e^{\tau\gamma}\Lambda^{\beta}\Omega|\Omega|^{Q-2}\Omega\,d\xi$ (3.21)

The last term, related to the dissipation, is known to be negative via the
Córdoba-Córdoba inequality (see, for instance, [CC04, Lemma 2.4]). The main
term of interest is the ‘forcing’ $U\cdot\nabla\bar{\Omega}$, which we
estimate thanks to (3.5) and the fact that $\bar{\Omega}$ is compactly
supported:

$\|U\cdot\nabla\bar{\Omega}\|_{L^{Q}}\leq\|U\|_{L^{\infty}}\|\nabla\bar{\Omega}\|_{L^{Q}}\leq
C\|\Omega\|_{X}\,.$ (3.22)

We conclude, via a Gronwall estimate on the inequality (3.21), that (3.20)
holds. Here we take advantage of the fact that $a_{0}>\frac{2}{\alpha}$ to see
that the term $\big{(}-Q+\frac{2}{\alpha}\big{)}\int|\Omega|^{Q}\,d\xi$ can be
neglected in this estimate, since it does not change the exponential in time
behavior of the solution.

Step 3. [Improved $\partial_{\theta}\Omega$ bound] Suppose that the bootstrap
assumption (3.9) holds. Then for all $\tau\in[\tau_{0},\tau_{1}]$ and for
$p=Q,2,4$, we have

$\|\partial_{\theta}\Omega(\cdot,\tau)\|_{L^{p}}\leq
C(1+\bar{C}^{1/2}e^{2\zeta T_{\rm
max}})\bar{C}^{1/2}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.$ (3.23)

We first observe that

$\partial_{\theta}\Lambda^{\beta}=\Lambda^{\beta}\partial_{\theta}\,,$ (3.24)

which follows from differentiating the rotation symmetry ${\rm
Rot}_{\theta}\Lambda^{\beta}=\Lambda^{\beta}{\rm Rot}_{\theta}$ in the angle
$\theta\in\mathbb{R}$ at $\theta=0$. Similarly
$\partial_{\theta}\operatorname{curl}W=\operatorname{curl}\partial_{\theta}W$
and notice that $\partial_{\theta}$ also commutes through the Biot-Savart law.

Commuting $\partial_{\theta}$ into the PDE, we have

$\partial_{\tau}\partial_{\theta}\Omega+(-1-\frac{\xi}{\alpha}\cdot\nabla)\partial_{\theta}\Omega+(\bar{U}+W+U^{\rm
lin})\cdot\nabla\partial_{\theta}\Omega+e^{\tau\gamma}\Lambda^{\beta}\partial_{\theta}\Omega=F,$
(3.25)

where

$-F=\partial_{\theta}U\cdot\nabla\bar{\Omega}+(\partial_{\theta}W+\partial_{\theta}U^{\rm
lin})\cdot\nabla\Omega.$ (3.26)

As regards the first term in the force, we estimate it in $L^{p}$ via
Calderón-Zygmund as follows

$\|\partial_{\theta}U\cdot\nabla\bar{\Omega}\|_{L^{p}}\leq\|\frac{1}{|x|}\partial_{\theta}U\|_{L^{p}}\||x||\nabla\bar{\Omega}|\|_{L^{\infty}}\leq
C\|\nabla U\|_{L^{p}}\leq C\|\Omega\|_{L^{p}}$

Using the baseline estimate (3.12) if $p=Q$ or $p=2$ and the improved bound
(3.20) if $p=4$, we get

$\begin{split}\|\partial_{\theta}U\cdot\nabla\bar{\Omega}\|_{L^{4}}&\leq
C\|\Omega\|_{L^{4}}\leq C(1+\bar{C}e^{\zeta T_{\rm
max}})\bar{C}^{1/2}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.\end{split}$

By (3.5), (3.6), (3.7) and the bootstrap assumption we have, for $p=2,4$,

$\begin{split}\|(\partial_{\theta}W+\partial_{\theta}U^{\rm
lin})\nabla\Omega\|_{L^{p}}&\leq
C(\|\partial_{\theta}W\|_{L^{\infty}}+\|\partial_{\theta}U^{\rm
lin}\|_{L^{\infty}})\|\nabla\Omega\|_{L^{p}}\\\ &\leq
C\bar{C}e^{a\tau}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.\end{split}$
(3.27)

For $p=Q$ we modify the previous computation as follows

$\begin{split}\|(\partial_{\theta}W+\partial_{\theta}U^{\rm
lin})\nabla\Omega\|_{L^{Q}}&\leq
C(\|\partial_{\theta}W\|_{L^{\frac{2Q}{2-Q}}}+\|\partial_{\theta}U^{\rm
lin}\|_{L^{\frac{2Q}{2-Q}}})\|\nabla\Omega\|_{L^{2}}\\\ &\leq
C(\|\partial_{\theta}\operatorname{curl}W\|_{L^{Q}}+\|\partial_{\theta}U^{\rm
lin}\|_{L^{\frac{2Q}{2-Q}}})\|\nabla\Omega\|_{L^{2}}\\\ &\leq
C\bar{C}e^{a\tau}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.\end{split}$
(3.28)

We multiply the equation by $p(\partial_{\theta}\Omega)^{p-1}$ if $p=Q,2,4$
and integrate by parts. After rewriting the fractional dissipation term and
using Holder inequality to estimate the term which includes $F$, we get

$\frac{d}{d\tau}\int|\partial_{\theta}\Omega|^{p}\,d\xi+\Big{(}\frac{2}{\alpha}-p)\int|\partial_{\theta}\Omega|^{p}\,d\xi+ce^{\tau\gamma}\int|\Lambda^{\frac{\beta}{2}}|\partial_{\theta}\Omega|^{p/2}|^{2}\,d\xi\leq
C\|F\|_{L^{p}}\|\partial_{\theta}\Omega\|_{L^{p}}^{p-1}.$ (3.29)

We neglect the last term in the left-hand side, which has the right sign,
divide by $\|\partial_{\theta}\Omega\|_{L^{p}}^{p-1}$ and sum over $p=Q,2,4$.

Step 4. [Improved $\nabla\Omega$ bound] Suppose that the bootstrap assumption
(3.9) holds. Then for all $\tau\in[\tau_{0},\tau_{1}]$ and for $p=2,4$, we
have

$\|\nabla\Omega\|_{L^{p}}\leq C(1+\bar{C}^{1/2}e^{2\zeta T_{\rm
max}})e^{(a+\delta)(\tau-\tau_{0})}\bar{C}^{1/2}\|\Omega_{0}\|_{X}\,,$ (3.30)
$\|\Omega\|_{D(\tau_{0},\tau)}\leq C(1+\bar{C}^{1/2}e^{2\zeta T_{\rm
max}})e^{(a+\delta)(\tau-\tau_{0})}\bar{C}^{1/2}\|\Omega_{0}\|_{X}\,.$ (3.31)

Let $i=1,2$ and consider the equation for $\partial_{i}\Omega$:

$\displaystyle\partial_{\tau}\partial_{i}\Omega+(-1-1/\alpha-\xi\cdot\nabla/\alpha)\partial_{i}\Omega+(\bar{U}+U^{\rm
lin}+W)\cdot\nabla\partial_{i}\Omega$ (3.32)
$\displaystyle\quad+\partial_{i}\bar{U}\cdot\nabla\Omega+\partial_{i}(U^{\rm
lin}+W)\cdot\nabla\Omega-e^{\tau\gamma}\Lambda^{\beta}\partial_{i}\Omega=F_{i},$

where

$-F_{i}=\partial_{i}U\cdot\nabla\bar{\Omega}+U\cdot\nabla\partial_{i}\bar{\Omega}.$
(3.33)

We multiply by $|\nabla\Omega|^{p-2}\partial_{i}\Omega$, sum over $i=1,2$ and
integrate by parts. The last term in the first line of
(LABEL:eq:piomegaequation) vanishes, after this computation, due to the
transport structure. We have

$\displaystyle\frac{1}{p}\frac{d}{d\tau}\int|\nabla\Omega|^{p}\,d\xi+\Big{(}\frac{2}{\alpha
p}-1-\frac{1}{\alpha}\Big{)}\int|\nabla\Omega|^{p}\,d\xi-\int
e^{\tau\gamma}\Lambda^{\beta}\partial_{i}\Omega|\nabla\Omega|^{p-2}\partial_{i}\Omega\,d\xi$
(3.34)
$\displaystyle\quad+\sum_{i=1}^{2}\int\partial_{i}\bar{U}\cdot\nabla\Omega\partial_{i}\Omega|\nabla\Omega|^{p-2}\,d\xi+\partial_{i}(U^{\rm
lin}+W)\cdot\nabla\Omega\partial_{i}\Omega|\nabla\Omega|^{p-2}d\xi$
$\displaystyle\quad=\sum_{i=1}^{2}\int
F_{i}\partial_{i}\Omega|\nabla\Omega|^{p-2}d\xi\,.$

We remark that

$\partial_{i}\bar{U}\cdot\nabla\Omega=\zeta^{\prime}(|x|)\frac{x_{i}}{|x|}\partial_{\theta}\Omega+\zeta(|x|)\partial_{i}x^{\perp}\cdot\nabla\Omega=\zeta^{\prime}(|x|)\frac{x_{i}}{|x|}\partial_{\theta}\Omega-\zeta(|x|)\nabla^{\perp}\Omega$
(3.35)

and we exploit the cancellation of the last term in (3.35) when multiplied by
$\nabla\Omega$. Therefore (3.36) rewrites as

$\displaystyle\frac{1}{p}\frac{d}{d\tau}\int|\nabla\Omega|^{p}\,d\xi+\Big{(}\frac{2}{\alpha
p}-1-\frac{1}{\alpha}\Big{)}\int|\nabla\Omega|^{p}\,d\xi-\int
e^{\tau\gamma}\Lambda^{\beta}\partial_{i}\Omega|\nabla\Omega|^{p-2}\partial_{i}\Omega\,d\xi$
(3.36)
$\displaystyle\quad=-\sum_{i=1}^{2}\int\Big{(}\zeta^{\prime}(|x|)\frac{x_{i}}{|x|}\partial_{\theta}\Omega+\partial_{i}(U^{\rm
lin}+W)\cdot\nabla\Omega+F_{i}\Big{)}\partial_{i}\Omega|\nabla\Omega|^{p-2}d\xi$
$\displaystyle\leq
C\|\zeta^{\prime}(|x|)\frac{x_{i}}{|x|}\partial_{\theta}\Omega+\partial_{i}(U^{\rm
lin}+W)\cdot\nabla\Omega+F_{i}\|_{L^{p}}^{p}+\frac{1}{2}\int|\nabla\Omega|^{p}\,d\xi\,.$

We estimate each term in the $L^{p}$ norm in the right-hand side. By (3.23)
and since $\zeta^{\prime}$ is bounded, we have

$\|\zeta^{\prime}(|x|)\frac{x_{i}}{|x|}\partial_{\theta}\Omega\|_{L^{p}}\leq\|\zeta^{\prime}(|x|)\|_{L^{\infty}}\|\partial_{\theta}\Omega\|_{L^{p}}\leq
C(1+\bar{C}^{1/2}e^{2\zeta T_{\rm
max}})e^{(a+\delta)(\tau-\tau_{0})}\bar{C}^{1/2}\|\Omega_{0}\|_{X}.$ (3.37)

By the explicit expression of $\Omega^{\rm lin}$, the regularity of $\eta$ in
Proposition 2.1, and (3.5), we have $\|\nabla U^{\rm
lin}\|_{L^{\infty}}\lesssim e^{a\tau}$. Using the Gagliardo-Nirenberg
interpolation inequality, Calderón-Zygmund and (3.7), we have

$\|\nabla W\|_{L^{\infty}}\leq C\|\nabla^{2}W\|_{L^{4}}^{\theta}\|\nabla
W\|_{L^{Q}}^{1-\theta}\leq
C\|\nabla\operatorname{curl}W\|_{L^{4}}^{\theta}\|\operatorname{curl}W\|_{L^{Q}}^{1-\theta}\leq
e^{a\tau}$

for a suitable choice of $\theta\in(0,1)$. We estimate

$\begin{split}\|\partial_{i}(U^{\rm
lin}+W)\cdot\nabla\Omega\|_{L^{p}}&\leq\big{(}\|\nabla U^{\rm
lin}\|_{L^{\infty}}+\|\nabla
W\|_{L^{\infty}}\big{)}\|\nabla\Omega\|_{L^{p}}\\\ &\leq
e^{a\tau}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.\end{split}$

Finally, as regards the force appearing in (3.33), we use the baseline
estimate (3.12) and (3.19). For the first term in the force, also by Calderón-
Zygmund, we have

$\begin{split}\|\partial_{i}U\cdot\nabla\bar{\Omega}\|_{L^{p}}&\leq\|\nabla\bar{\Omega}(\cdot,\tau)\|_{\infty}\|\nabla
U(\cdot,\tau)\|_{L^{p}}\leq C\|\Omega(\cdot,\tau)\|_{L^{p}}\\\ &\leq
C(C_{\delta_{0}}+\bar{C}e^{\zeta T_{\rm
max}})\bar{C}^{1/2}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.\end{split}$

For the second term in the force, given that $D^{2}\bar{\Omega}\in L^{1}$
since $\Omega$ is smooth and compactly supported, and by the $L^{\infty}$
bound on $U$ in (3.5) we obtain

$\displaystyle\|U\cdot\nabla\partial_{i}\bar{\Omega}\|_{L^{p}}$
$\displaystyle\leq\|U\|_{L^{\infty}}\|D^{2}\bar{\Omega}(\cdot,\tau)\|_{L^{p}}\leq
C(\|\Omega\|_{L^{Q}}+\|\Omega\|_{L^{4}})$ (3.38) $\displaystyle\leq
C(1+\bar{C}^{1/2}e^{\zeta T_{\rm
max}})\bar{C}^{1/2}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}.$ (3.39)

By the previous estimates, (3.36) rewrites for $p=2$ as

$\displaystyle\frac{1}{2}\frac{d}{d\tau}\int|\nabla\Omega|^{2}\,d\xi-\int|\nabla\Omega|^{2}\,d\xi+\int
e^{\tau\gamma}|\Lambda^{\beta/2}\nabla\Omega|^{2}\,d\xi$ (3.40)
$\displaystyle\leq C[(1+\bar{C}^{1/2}e^{2\zeta T_{\rm
max}})\bar{C}^{1/2}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}]^{2}.$

We now observe that, when applying the formula (3.36) for $p=4$, the term
related to the fractional dissipation (last term in first line) is
nonnegative. This is a technical computation that can be seen, for instance,
through the Caffarelli extension $(\partial_{i}\Omega)^{*}$ of
$\partial_{i}\Omega$ in
$\\{(\xi,\xi_{3})\in\mathbb{R}^{2}\times\mathbb{R}:y\geq 0\\}$ and through the
characterization of the fractional laplacian in terms of the extension as
follows. By means of the divergence theorem, we compute

$\displaystyle-$
$\displaystyle\sum_{i=1}^{2}\int_{\mathbb{R}^{2}}\Lambda^{\beta}\partial_{i}\Omega(\xi,\tau)|\nabla\Omega|^{2}(\xi,\tau)\partial_{i}\Omega(\xi,\tau)\,dx$
$\displaystyle=-C\sum_{i=1}^{2}\lim_{\xi_{3}\to
0^{+}}\int_{\mathbb{R}^{2}}\xi_{3}^{b}\partial_{\xi_{3}}(\partial_{i}\Omega)^{*}(x,\xi_{3},t)|(\nabla\Omega)^{*}|^{2}(\xi,\tau)(\partial_{i}\Omega)^{*}(\xi,\tau)\,dx$
$\displaystyle=C\sum_{i=1}^{2}\int_{\mathbb{R}^{3}_{+}}\overline{\operatorname{div}}(\xi_{3}^{b}\overline{\nabla}(\partial_{i}\Omega)^{*}(x,\xi_{3},t)|(\nabla\Omega)^{*}|^{2}(\xi,\tau)(\partial_{i}\Omega)^{*}(\xi,\tau))\,dx\,d\xi_{3}$
$\displaystyle=C\sum_{i=1}^{2}\int_{\mathbb{R}^{3}_{+}}(\xi_{3}^{b}\overline{\nabla}\frac{(\partial_{i}\Omega)^{*}}{2}\cdot\overline{\nabla}[|(\nabla\Omega)^{*}|^{2}(\xi,\tau)])+\xi_{3}^{b}|\overline{\nabla}(\partial_{i}\Omega)^{*}|^{2}|(\nabla\Omega)^{*}|^{2})\,d\xi\,d\xi_{3}\,,$
$\displaystyle=C\int_{\mathbb{R}^{3}_{+}}(\xi_{3}^{b}\overline{\nabla}\frac{|({\nabla}\Omega)^{*}|^{2}}{2}\cdot\overline{\nabla}[|(\nabla\Omega)^{*}|^{2}]+\xi_{3}^{b}\Big{(}\sum_{i=1}^{2}|\overline{\nabla}(\partial_{i}\Omega)^{*}|^{2}\Big{)}|(\nabla\Omega)^{*}|^{2})\,d\xi\,d\xi_{3}\geq
0\,,$

where the constant depends only on $\beta$ and in the last two equalities we
omitted to indicate that all integrands are evaluated at $(\xi,\xi_{3},\tau)$.
Thanks to the previous estimates, (3.36) implies then

$\displaystyle\frac{1}{4}\frac{d}{d\tau}\int|\nabla\Omega|^{4}\,d\xi-\Big{(}1+\frac{1}{2\alpha}\Big{)}\int|\nabla\Omega|^{4}\,d\xi\leq
C[(1+\bar{C}^{1/2}e^{2\zeta T_{\rm
max}})\bar{C}^{1/2}e^{(a+\delta)(\tau-\tau_{0})}\|\Omega_{0}\|_{X}]^{4}.$
(3.41)

Integrating the previous inequality and (3.41) between $\tau_{0}$ and $\tau$,
and taking advantage of the fact that the second term in the right-hand side
does not influence the final estimate via Gronwall, we deduce that (3.30)
holds and, with $p=2$, that also (3.31) holds. ∎

##  4\. Nonlinear theory

In this section, we complete the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Let $\bar{\Omega},a,b,\eta,m,\delta_{0}$ be fixed as in the beginning of
Section 3.

_Step 1. Construction of two distinct solutions via a limiting procedure_. For
$k\in\mathbb{N}$, consider $t_{k}:=e^{-k}$. We solve the Cauchy problem for
the fractional Navier-Stokes equations (NSβ) with smooth ‘initial’ vorticity
$u^{(k)}(\cdot,t_{k})=\bar{u}(\cdot,t_{k})+u^{\rm lin}(\cdot,t_{k})$ and
smooth forcing term $f$ defined in Lemma 2.3. For each $k$, a short-time
solution $u^{(k)}$ on $\mathbb{R}^{2}\times(t_{k},T_{k})$ exists and is unique
in $C([t_{k},T_{k}];L^{2}\cap W^{1,4})$, among other function spaces,
including $X$.444When $\beta\leq 1$, one must argue existence and uniqueness
in a quasilinear fashion, whereas when $\beta\in(1,2)$ it is possible to
develop the well-posedness theory in a semilinear fashion. Furthermore, these
solutions are regular enough to demonstrate energy conservation:

$\frac{1}{2}\int|u^{(k)}|^{2}(x,t^{\prime})\,dx+\int_{t}^{t^{\prime}}\int|\Lambda^{\frac{\beta}{2}}u^{(k)}|^{2}\,dx\,ds=\frac{1}{2}\int|u^{(k)}|^{2}(x,t)\,dx+\int_{t}^{t^{\prime}}\int
f\cdot u^{(k)}\,dx\,ds.$ (4.1)

Our desired second solution, violating uniqueness, should arise as
$k\to+\infty$. For this purpose, it is necessary to continue the solutions up
to some time $\bar{t}$ independent of $k$. In particular, we will demonstrate
that there exists a time $\bar{t}>0$, independent of $k$, such that the
solutions $u^{(k)}$ can be continued to $\mathbb{R}^{2}\times(t_{k},\bar{t})$,
and writing $u^{(k)}=\bar{u}+u^{\rm lin}+u^{\rm per,(k)}$, the associated
vorticity in self-similar variables $\Omega^{\rm per,(k)}$ solves (2.16) in
$[-k,\bar{\tau}]$ and satisfies

$\|\Omega^{{\rm per},(k)}(\cdot,\tau)\|_{X}\leq
e^{\tau(a+\delta_{0})},\quad\tau\in[-k,\bar{\tau}],$ (4.2)

where $\bar{\tau}=\log\bar{t}$.

Once we have the above estimate, we may employ standard weak compactness
arguments to take the limit $k\to\infty$ and obtain a solution
$u=\bar{u}+u^{\rm lin}+u^{\rm per}$ satisfying the estimate (4.2) with
$\Omega^{\rm per}$ instead of $\Omega^{{\rm per},(k)}$ on the time interval
$(-\infty,\bar{\tau}]$. In particular, we will have

$\|\Omega(\cdot,\tau)-\bar{\Omega}(\cdot,\tau)\|_{X}\geq\|\Omega^{\rm
lin}(\cdot,\tau)\|_{X}-\|\Omega^{{\rm per}}(\cdot,\tau)\|_{X}\geq Ce^{\tau
a}-e^{\tau(a+\delta_{0})}>0\,,$ (4.3)

when $\tau\ll 0$, which implies non-uniqueness. To demonstrate that $u$ is a
Leray solution of (NSβ), we use the energy conservation (4.1) with initial
time $t=t_{k}$. The first term on the right-hand side of (4.1) is simply
$\|\bar{u}(\cdot,t_{k})+u^{\rm lin}(\cdot,t_{k})\|^{2}_{L^{2}}/2$, which
converges to zero as $t_{k}\to 0^{+}$. Hence, we have the energy equality

$\frac{1}{2}\int|u|^{2}(x,t^{\prime})\,dx+\int_{0}^{t}\int|\Lambda^{\frac{\beta}{2}}u|^{2}\,dx\,ds=\int_{0}^{t}\int
f\cdot u\,dx\,ds\,.$ (4.4)

_Step 2. The estimate ( 4.2) via Duhamel’s formula_. It remains to verify
(4.2). By continuity in $X$, the estimate (4.2) holds on a small neighborhood
of $-k$. We define $\bar{\tau}_{k}>-k$ to be the largest non-positive time
such that (4.2) holds and we claim that there exists $\bar{\tau}$ independent
of $k$ (which will be fixed at the end of the proof) such that
$\bar{\tau}_{k}\geq\bar{\tau}$. Therefore, in $[-k,\bar{\tau}_{k}]$ we are in
the position to apply Duhamel’s formula and the assumptions of Proposition 3.1
applied with $\delta=\delta_{0}$ and $W=U^{\rm per}$ satisfying the
requirement (3.7). We conclude that for every $\tau\in[-k,\bar{\tau}_{k}]$,

$\Omega^{\rm per}(\cdot,\tau)=\int_{-k}^{\tau}S[U^{\rm per}](\tau,s)[-(U^{\rm
per}+U^{\rm lin})\cdot\nabla\Omega^{\rm
lin}+e^{s\gamma}\Lambda^{\beta}\Omega^{\rm lin}](\cdot,s)\,ds\,,$ (4.5)

and that

$\displaystyle\|\Omega^{\rm per}(\cdot,\tau)\|_{X}$
$\displaystyle\leq\int_{-k}^{\tau}\|S[U^{\rm per}](\tau,s)[-(U^{\rm
per}+U^{\rm lin})\cdot\nabla\Omega^{\rm
lin}+e^{s\gamma}\Lambda^{\beta}\Omega^{\rm lin}](\cdot,s)\|_{X}\,ds$
$\displaystyle\leq\int_{-k}^{\tau}e^{(a_{0}+\delta_{0})(\tau-s)}\|-(U^{\rm
per}+U^{\rm lin})\cdot\nabla\Omega^{\rm
lin}+e^{s\gamma}\Lambda^{\beta}\Omega^{\rm lin}\|_{X}(\cdot,s)\,ds\,.$ (4.6)

We claim that the integrands in the right-hand side satisfy

$\|(U^{\rm per}+U^{\rm lin})\cdot\nabla\Omega^{\rm
lin}(\cdot,s)\|_{X}+\|e^{s\gamma}\Lambda^{\beta}\Omega^{\rm
lin}(\cdot,s)\|_{X}\lesssim e^{s(a+2\delta_{0})}\,.$ (4.7)

Indeed, by the eigenfunction properties in Proposition 2.1 as well as the
structure of $\eta$ in the variable $\theta$ and elementary interpolation, we
have

$\|e^{s\gamma}\Lambda^{\beta}\Omega^{\rm lin}(\cdot,s)\|_{X}\leq
Ce^{s(a+\gamma)}(\|\eta\|_{C^{3}({\rm
supp}\eta)}+\|\partial_{\theta}\eta\|_{C^{2}({\rm supp}\eta)})\leq
Ce^{s(a+\gamma)}.$ (4.8)

By (3.5) and (3.6) applied to $U^{\rm per}$, the bound (4.2) which by
assumption holds in $[-k,\bar{\tau}_{k}]$, and the bounds on $\Omega^{\rm
lin}$ in terms of $e^{a_{0}s}$, we estimate all the terms composing $\|(U^{\rm
per}+U^{\rm lin})\cdot\nabla\Omega^{\rm lin}(\cdot,s)\|_{X}$ as

$\begin{split}\|(U^{\rm per}+U^{\rm lin})\cdot\nabla\Omega^{\rm
lin}\|_{L^{p}}&\leq\big{(}\|U^{\rm per}\|_{L^{\infty}}+\|U^{\rm
lin}\|_{L^{\infty}}\big{)}\|\Omega^{\rm lin}\|_{L^{p}}\\\ &\leq
C\big{(}\|\Omega^{\rm per}\|_{X}+\|\Omega^{\rm lin}\|_{X}\big{)}\|\Omega^{\rm
lin}\|_{C^{2}({\rm supp}\eta)}\leq Ce^{2a_{0}s}\end{split}$
$\|\big{(}\partial_{\theta}U^{\rm per}+\partial_{\theta}U^{\rm
lin}\big{)}\cdot\nabla\Omega^{\rm
lin}\|_{L^{p}}\leq\big{(}\|\partial_{\theta}U^{\rm
per}\|_{L^{\infty}}\|+\|\partial_{\theta}U^{\rm
lin}\|_{L^{\infty}}\big{)}\|\nabla\Omega^{\rm lin}\|_{L^{p}}\leq Ce^{2a_{0}s}$
$\|\big{(}U^{\rm per}+U^{\rm
lin}\big{)}\cdot\partial_{\theta}\nabla\Omega^{\rm
lin}\|_{L^{p}}\leq\big{(}\|U^{\rm per}\|_{L^{\infty}}+\|U^{\rm
lin}\|_{L^{\infty}}\big{)}\|\partial_{\theta}\nabla\Omega^{\rm
lin}\|_{L^{p}}\leq Ce^{2a_{0}s}$

for $p=Q,2,4$ and

$\begin{split}\|\big{(}\nabla&U^{\rm per}+\nabla U^{\rm
lin}\big{)}\cdot\nabla\Omega^{\rm lin}\|_{L^{p}}+\|\big{(}U^{\rm per}+U^{\rm
lin}\big{)}\cdot\nabla^{2}\Omega^{\rm lin}\|_{L^{p}}\\\ &\leq\big{(}\|\nabla
U^{\rm per}\|_{L^{p}}+\|\nabla U^{\rm lin}\|_{L^{p}}\big{)}\|\nabla\Omega^{\rm
lin}\|_{L^{\infty}}+\big{(}\|U^{\rm per}\|_{L^{\infty}}+\|U^{\rm
lin}\|_{L^{\infty}}\big{)}\|\nabla^{2}\Omega^{\rm lin}\|_{L^{p}}\\\ &\leq
Ce^{2a_{0}s}\end{split}$

for $p=2,4$.

Therefore, since our choice of $\delta_{0}$ satisfies
$\delta_{0}\leq\frac{1}{4}\min\\{\gamma,a_{0}\\}$, the previous inequalities
imply the validity of (4.7). We obtain then from (4) and these estimates that

$\|\Omega^{\rm per}(\cdot,\tau)\|_{X}\leq
Ce^{(a_{0}+\delta_{0})\bar{\tau}}\int_{-\infty}^{\tau}e^{\delta_{0}s}\,ds\leq
Ce^{(a_{0}+2\delta_{0})\bar{\tau}}$

for every $\tau\in[-k,\bar{\tau}_{k}]$. Choosing finally $\bar{\tau}$ so that
$Ce^{\delta_{0}\bar{\tau}}<1/2$, where $C$ is the constant appearing in the
right-hand side of the previous formula, we find that
$\bar{\tau}_{k}\geq\bar{\tau}$. ∎

####  4.0.1. Acknowledgments

DA was supported by NSF Postdoctoral Fellowship Grant No. 2002023 and Simons
Foundation Grant No. 816048. MC was supported by the SNSF Grant 182565.

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