A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation
Ángel Castro Daniel Faraco Francisco Mengual Marcos Solera

	



We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class L^1∩ L^p with 2<p<∞. The main simplification is an alternative construction of a smooth and compactly supported unstable vortex, which is split into two steps: Firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This simpler structure of the unstable vortex yields a simplification of the other parts of Vishik's proof.

BeSound: Bluetooth-Based Position Estimation Enhancing with Cross-Modality Distillation
    Hymalai Bello
DFKI and RPTU 

Kaiserslautern, Germany 


Sungho Suh
DFKI and RPTU 

Kaiserslautern, Germany 


Bo Zhou
DFKI and RPTU  

Kaiserslautern, Germany 


Paul Lukowicz
DFKI and RPTU  

Kaiserslautern, Germany 




    April 29, 2024
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§ INTRODUCTION


We consider the Euler equation in vorticity form

    ∂_tω + v·∇ω=f,

posed on _+×^2,
for some fixed external force f and initial datum ω^∘

    ω|_t=0
    =ω^∘.

The velocity v is recovered from the vorticity ω through the Biot-Savart law

    v=∇^⊥Δ^-1ω.

The aim of this work is to simplify the proof of Vishik's nonuniqueness Theorem <cit.>:


For any 2<p<∞ there exists a force

    f∈ L_t^1(L^1∩ L^p)
      with  
    g∈ L_t^1L^2,

where g=∇^⊥Δ^-1 f, 
with the property that there are two different solutions

    ω_0,ω_1∈ L_t^∞ (L^1∩ L^p)
      with  
    v_0,v_1∈ L_t^∞ L^2,

where v_j=∇^⊥Δ^-1ω_j, to the Euler equation starting from  ω^∘=0.


Theorem <ref> shows sharpness of Yudovich's uniqueness Theorem for the forced Euler equation. More precisely, Yudovich's theory implies that if p=∞ in (<ref>)(<ref>), then necessarily ω_0=ω_1.


In fact, in the construction of Theorem <ref>, both the force (f,g) and the main solution (ω_0,v_0) belong to C_c^∞ for positive times. Moreover, for any given k∈, it is not difficult to see that the second solution (ω_1,v_1) can be upgraded to be in H^k× H^k+1 for positive times. These estimates inevitably deteriorate as t→ 0. As this information is not relevant in Yudovich's proof of uniqueness, we have chosen not to include it in  the statement of Theorem <ref>.

In a nutshell, Vishik's proof is split into three steps: (1) Construction of an unstable vortex, which is then shown to be  (2) Self-similarly unstable, and (3) Nonlinearly unstable as well.

Step (1) is carried out in <cit.> by an intricate modification of the power-law vortex ω^∘(x)=β|x|^-α. As noticed in <cit.>, by decoupling the parameters governing the self-similar scaling from the
decay rate, Vishik's argument leads to nonuniqueness for the zero initial data ω^∘=0, which showcases the primary role played by the forcing f.

The observation in <cit.> that decay does not play a significant role in Vishik's Theorem
suggested us the possibility of obtaining  the proof by considering compactly supported vortices. Even more, inspired by simple examples for shear flows (see e.g. <cit.>) and vortex patches, we initially aimed to construct a piecewise constant unstable vortex.
Remarkably, Step (1) for piecewise constant vortices reduces to an elementary computation. In order to upgrade the piecewise constant vortex to a smooth vortex, we need to apply a fixed point argument in suitable rescaled variables. This step is inspired in <cit.> but needs of new ideas to be carried out here.

With  the new unstable vortex available, the proofs of Steps (2) and (3)  from Vishik or <cit.> would work verbatim. However, since the new vortex is structurally simple, both Steps (2) and (3) admit also several simplifications leading to a straightforward self-contained proof of the entire theorem, which is presented in this work. In any case, the last two steps follow the presentation from book <cit.> by Albritton, Brué, Colombo, De Lellis, Giri, Janisch, and Kwon, simplifying the arguments whenever possible. 

As highlighted in Vishik's works <cit.> and the book <cit.>, the rigorous construction of the unstable vortex is a key intermediate step, which is of independent physical and mathematical interest. 

Indeed,  Albritton, Brué and Colombo  <cit.> proved the nonuniqueness of Leray solutions for the forced 3D Navier-Stokes equation by carefully adapting Vishik's unstable vortex into the cross section of an axisymmetric vortex ring. We remark that, although our vortex differs from Vishik's one, it fits into the requirements of the paper <cit.> leading to  an alternative example of nonuniqueness for the forced 3D Navier-Stokes equation. 

We finally remark that our unstable vortices appear robust enough to be found in more general settings. Indeed, our investigation started because it was not obvious how Vishik's strategy  could be modified to prove nonuniqueness in the more singular (and nonlocal) context of generalized SQG. In this case, the Biot-Savart law (<ref>) is replaced by v=-∇^⊥(-Δ)^α/2-1ω.
The construction and regularization of piecewise constant unstable vortices presented in this paper are however rather flexible and have the potential to work even for α all the way up to 1.  
Tackling the SQG case (α=1) poses even more challenges, notably because v is not obtained through a smoothing operator. Both cases, generalized SQG and SQG will be the matter of a forthcoming paper.





 §.§ Brief background
 

The global existence and uniqueness of classical solutions to the 2D Euler equation has been known for almost a century, starting with the works of Wolibner <cit.> and Hölder <cit.> in C^k,γ spaces for k=0,1,… and γ>0. 
Local well-posedness was previously established by Lichtenstein <cit.> and Gunther <cit.>.
The global well-posedness in 2D sharply contrasts with the 3D case due to vortex stretching, as demonstrated by Elgindi <cit.>, who showed finite-time singularities
with ω^∘∈ C^γ for some 0<γ≪ 1 in the unforced case. 
Recently, Córdoba, Martínez-Zoroa and Zheng <cit.> simplified the proof of this blow-up with a different strategy.
Additionally, Córdoba and Martínez-Zoroa <cit.> upgrade the regularity in the forced case with
f∈ L_t^∞ C^1/2-ε.
We refer to the work <cit.> by Kiselev and Šverák for the related issue on small scale creation in two dimensions.

In the celebrated paper <cit.>, Yudovich extended the 2D global well-posedness into the vorticity class L^1∩ L^∞. 
The existence of global solutions in L^1∩ L^p for 1<p≤∞ was proved by DiPerna and Majda <cit.> at the late eighties.
Since then, a long-standing open question in the field is whether uniqueness fails for p<∞. 


In the pioneering works <cit.>, Jia and Šverák introduced, within the context of the 3D Navier-Stokes equation, the idea that nonuniqueness could potentially arise from a self-similar instability. 
Given that uniqueness is known to be satisfied for small data (in suitable spaces, see e.g. <cit.>), they conjectured that nonuniqueness could occur due to bifurcations within solutions of the form βω̅, for some self-similar steady state. Specifically, they proved  in the celebrated paper <cit.> the existence of self-similar solutions for large data. Later,  in <cit.>, the authors speculated that some eigenvalues of the linearization in self-similar coordinates might cross the imaginary axis as β increases, leading to the emergence of multiple solutions. While there are numerical evidences, due to Guillod and Šverák <cit.>, supporting the existence of such solutions, the rigorous proof appears to be elusive, primarily because the Jia-Šverák  self-similar  solutions are not explicit.

In the recent groundbreaking works <cit.>, Vishik was able to construct a self-similarly unstable vortex for the 2D Euler equation, but at the cost of introducing a force. 
One of the innovative ideas in these works is Vishik's key observation that, by taking β sufficiently large,  in order to find a self-similar unstable vortex it suffices to find an unstable vortex   
ω̅ in the original Eulerian coordinates. This  simplifies the analysis considerably. In the book <cit.>, upon which Sections <ref> and <ref> of the present work are based, the authors reviewed Vishik's nonuniqueness Theorem <ref>, providing more details, offering some simplifications and clarifying certain subtle points.

To the best of our knowledge, the question of nonuniqueness without forcing, both for the 2D Euler equation below the Yudovich class and for the 3D Navier-Stokes equation in the Leray class, remains unresolved to date.

In <cit.> Bressan, Murray, and Shen 
presented numerical evidences of the nonuniqueness for the unforced 2D Euler equation. 
Specifically, they introduced two different ways of regularizing a cleverly designed initial vorticity, leading to either one or two algebraic spirals. 
Recent research on these spirals by García and Gómez-Serrano <cit.>, and by Shao, Wei and Zhang <cit.>, based on the earlier work of Elling <cit.>, could be relevant in the potential proof of nonuniqueness without force. 

The first examples of nonuniqueness for weak solutions can be attributed to Scheffer <cit.> and Shnirelman <cit.>. Specifically, they constructed weak solutions v∈ L_t,x^2 with compact support in time, to the Euler equation in velocity form.
In their seminal work <cit.>, De Lellis and Székelyhidi introduced the convex integration method in hydrodynamics, constructing non-trivial Euler velocities within the energy space L_t^∞ L^2 for any space dimension d≥ 2. Over the last years, there has been a significant increase in research intensity focused on the method,
showcasing its remarkable robustness and flexibility.
As a pivotal landmark, this method allowed constructing Euler velocities in C_t,x^1/3-ε 
exhibiting nonuniqueness <cit.> by Isett, and dissipating the kinetic energy <cit.> by Buckmaster, De Lellis, Székelyhidi, and Vicol, thus solving the dissipative part of the 3D Onsager conjecture. See also the work <cit.> by Novack and Vicol on an intermittent Onsager theorem, based on their joint work <cit.> with Buckmaster and Masmoudi, as well as the recent work <cit.> on the L^3-based strong Onsager conjecture by Giri, Kwon and Novack. 
The conservative part of the Onsager conjecture in any dimension was partially proved by Eyink <cit.> and later proven in full by Constantin, E, and Titi <cit.>. See also the work <cit.> by Cheskidov, Constantin, Friedlander, and Shvydkoy on critical regularity.
The dissipative part of the 2D Onsager conjecture was solved recently in <cit.> by Giri and Radu by means of convex integration. 
We refer to the recent
work <cit.> by De Rosa and Park for the related issue on anomalous dissipation in two dimensions.
In the case with force, Bulut, Huynh and Palasek proved in <cit.> that the regularity in 3D can be upgraded to C^1/2-ε.
For other applications of convex integration to forced equations, we refer to the recent work <cit.> by Dai and Friedlander,  and the references therein.

As already mentioned in <cit.>, Theorem <ref> implies that, for any γ<1,
there exist two different Euler velocities v_1,v_2∈ L_t^∞(L^2∩ C^γ) with force g∈ L_t^1(L^2∩ C^γ) starting from v^∘=0.
To the best of our knowledge, the problem of nonuniqueness for the unforced Euler equation in the velocity class L^2∩ C^γ remains open in the regime 1/3≤γ<1.

Unfortunately, it seems not possible with the current convex integration techniques to construct
solutions, neither with v∈ C^γ for 1/3≤γ <1, nor with ω∈ L^p for p≥ 1 in two dimensions.
Despite this inconvenience, some partial results 
have been obtained in the last years. In <cit.>
the third author proved the existence, for any 2<p<∞, of initial data v^∘∈ L^2 with ω^∘∈ L^1∩ L^p for which there are infinitely many admissible solutions v∈ C_t L^2. This result shows sharpness of Yudovich's proof of uniqueness, but with the drawback that the vorticity information is lost for positive times. In <cit.> Brué and Colombo constructed a Cauchy sequence ω_k in the Lorentz space L^1,∞, whose velocities v_k converge to an anomalous weak solution v. 
In <cit.> Buck and Modena adapted the previous construction for the Hardy space H^p for 2/3<p<1. This space is also weaker than L^1 but, in contrast to L^1,∞, it already embeds into the space of distributions.

Finally, we emphasize that the convex integration method allowed for proving nonuniqueness of Navier-Stokes solutions <cit.> by Buckmaster and Vicol, and sharpness of the Ladyzhenskaya-Prodi-Serrin criteria <cit.> by Cheskidov and Luo. Furthermore, the method has been also applied to the construction of non-unique dissipative Euler flows with concrete rough initial data such as vortex sheets <cit.> by Székelyhidi and the third author, and recently with C^1/3-ε regularity <cit.> by Enciso, Peñafiel-Tomás and Peralta-Salas, vortex filaments <cit.> by Gancedo, Hidalgo-Torné and the third author, and also to
the inhomogeneous case <cit.> by
Gebhard, Kolumbán and Székelyhidi. These later developments are inspired by related constructions for IPM <cit.>.



 §.§ Organization of the paper
 In Section <ref> we outline the proof of Theorem <ref>. In Section <ref> we construct a piecewise constant unstable vortex. In Section <ref> we find a regularization that is also unstable. Finally, in Sections <ref> and <ref> we prove that the vortex is also self-similarly and nonlinearly unstable, respectively. 



§ SKETCH OF THE PROOF


In this section we summarize the main ideas for the [sec:proofVishik]Proof of Theorem <ref>. The missing  technical details will be dealt rigorously in the next sections.

Before going further let us introduce concisely Vishik's strategy to prove nonuniqueness <cit.>.
Firstly, following the ideas of Jia and Šverák <cit.>, the construction of a self-similar unstable vortex ω_0 gives hope to find other solutions ω_1 deviating from ω_0 as

    (ω_1-ω_0)(t)_L^2/a∼ t^λ/ab,

as t→ 0,
where 0<a,b≤ 1 are two parameters governing the self-similar scaling, and λ∈ is the unstable eigenvalue (λ>0). However, it is not clear how to directly construct a self-similar unstable vortex. Remarkably, Vishik realized and proved
that  some spectral properties of the linearization in self-similar coordinates can be recovered from the linearization in the original Eulerian coordinates. 

Thus, the [sec:proofVishik]Proof of Theorem <ref> divides into 
  three steps:  (1) Eulerian, (2) Self-similar, and (3) Nonlinear instability. As discussed in the intro, in our approach,
Step (1) is split into two intermediate steps: (1.1) Construction of a piecewise constant unstable vortex, and (1.2) its Regularization.



 §.§ Step 1. Eulerian instability

We start by recalling how linear stability for the Euler equation leads to the Rayleigh stability equation, which we found convenient to express in vorticity form.













Given a background (steady) vorticity ω̅, a perturbation  ω̅+ϵω is a second solution
to the Euler equation if the deviation ω  satisfies the equation

    (∂_t-L)ω
    +ϵ (v·∇ω)=0.

Here,  L=L_ω̅ is the linearization of the Euler equation (<ref>) around ω̅. The linearization can be written as

    L=T+K,

where T is a transport operator,  K is  a compact operator, and they are defined by

    Tω   =-v̅·∇ω,
    
    Kω   =-v·∇ω̅.

Here, v and v̅ are recovered from ω and ω̅ respectively through the Biot-Savart law (<ref>).

As usual in Stability theory, as ϵ→ 0 one is lead to study the linear equation

    (∂_t-L)ω^lin=0,

and seek for solutions that grow exponentially in time. That is, for λ strictly positive, 

    ω^lin(t,x)=e^λ tw(x).

For solutions of this form, the equation (<ref>) is equivalent to the eigenvalue problem

    Lw=λ w.

In other words, instability, at this linear level, translates into the existence of eigenvalues λ in the right half plane

    _+={λ∈ : λ>0},

and eigenfunctions w in some suitable Hilbert space, of the linear operator L=L_ω̅ 


It is useful to consider the eigenvalue problem (<ref>) for complex-valued vorticities. Since the operator L is real, the real-valued solutions to (<ref>) will be recovered by replacing (<ref>) with

    ω^lin(t,x)=(e^λ tw(x)).

Notice that the imaginary part is also a real-valued solution to (<ref>). In particular, if λ=0, both the real and imaginary parts of w are eigenfunctions.
Finally, notice that if w is an eigenfunction with eigenvalue λ, then its complex conjugate w^* is also an eigenfunction with eigenvalue λ^*.
This remark will become relevant in the [sec:proofVishik]Proof of Theorem <ref>.


The caveat of  this approach is that the spectral analysis of L is  quite complicated, even for simple steady states ω̅. Thus,  we restrict ourselves to the case of radially symmetric ω̅'s called vortices, which in polar coordinates 
x=re^iθ reads as

    ω̅(x)=w̅(r).


Next, we choose on which  function space the eigenvalue problem will be solved.  Given 0≠ n∈, we seek for eigenfunctions in the subspace of L^2 formed by purely n-fold symmetric vorticities 

    U_n={w∈ L^2 : w(x)=w_n(r)e^inθ}.

Since U_-n=U_n^*, we will consider without loss of generality the case n∈.

 
We say that the vortex ω̅ is unstable if, for some n∈, there exists 0≠ w∈ U_n satisfying (<ref>) with λ∈_+. 






As we will see in Section <ref>,
the space U_n is invariant under the operator L. Namely, for w ∈ U_n,

    Tw
       =-ine^inθv̅_θ(r)/rw_n(r),
    
    Kw
       =-ie^inθ∂_rw̅(r)∫_0^∞ K_n(r/s)w_n(s) s,

where v̅_θ is the angular component of the velocity v̅, and K_n is the Biot-Savart kernel acting on U_n (see (<ref>) and (<ref>) respectively).

Finally, the Rayleigh stability equation is nothing but the eigenvalue problem (<ref>) for w∈ U_n  

    v̅_θ/rw_n +∂_rw̅/n∫_0^∞ K_n(r/s)w_n(s) s
    =zw_n,

where we have replaced λ=-inz.
Observe that λ>0 translates to z>0.
Notice that (<ref>) implies that w_n must vanish wherever ∂_rw̅=0. Thus, we can write

    w_n=h∂_rw̅,

in terms of some profile h.
For that h the equation (<ref>) reads as

    v̅_θ(r)/rh(r) +1/n∫_0^∞ K_n(r/s)(h∂_rw̅)(s) s
    =zh(r),
        
    r∈supp(∂_rw̅).

Thus, Eulerian instability reduces to  finding a vortex that admits nontrivial  solutions (h,z) of (<ref>) with z>0. We will show that this is indeed the
case and therefore land in our first main result:



There exists an unstable vortex
Ω̅∈ C_c^∞(^2) with zero mean.


As anticipated in the intro, our proof of Theorem <ref> is split into two steps:



 §.§ Step 1.1. Piecewise constant unstable vortex

In Section <ref> we construct an unstable vortex of the form

    w̅
    =c 1_[0,r_1) - 1_[r_1,r_2),

in terms of some parameters 0<r_1<r_2<∞ and c>0, to be determined.
Firstly, we choose c making the mean of ω̅ equals zero

    ∫_^2ω̅(x) x
    =2π∫_0^∞w̅(r)r r
    =0.

This condition guarantees that v̅∈ L^2.
Moreover, v̅ is Lipschitz and compactly supported with
supp(v̅)=supp(ω̅)=B_r_2.
Secondly, since

    supp(∂_rw̅)={r_1,r_2},

the equation (<ref>) turns into two conditions for the vector h=(h(r_1),h(r_2)) that can be written as a linear system

    A h=zh,

where the matrix A∈^2× 2 depends on the parameters r_1,r_2, and the frequency n.
Thus, there is an eigenvector h≠ 0 if and only if z is a root of the characteristic polynomial (A-z). Finally, we show in Proposition <ref> that for any n≥ 2 we can choose r_1,r_2 such that the roots z,z^* of (A-z) satisfy z>0.





 §.§ Step 1.2. Regularization
 In Section <ref> we prove that there exists a smooth vortex ω̅^ε, obtained by suitably regularizing ω̅ from [Step1.1]Step 1.1, which is also unstable for some small ε>0. Similarly to [Step1.1]Step 1.1, now we need to solve the Rayleigh stability equation (<ref>) in the intervals B_ε(r_j) for j=1,2. We
rescale variables around r_j writing
 r=r_j+εα with α∈ I=(-1,1).
Next, we  make the asymptotic expansions for eigenfunctions and eigenvalues. Namely, we write 

    h^ε(r)
    =h(r_j)+ε g_j(α),
        
    z^ε
    =z+ε y,

for some profiles g=(g_1,g_2)∈ L^2(I)^2, and  a constant y∈, to be determined.
We also expand 

    A^ε
    g=v̅_θ^ε/rg+1/n∫_0^∞ K_n(r/s)(g∂_rw̅^ε)(s) s
    =(A+ε B)g,

where B is another linear operator in L^2(I)^2.
Therefore, the Rayleigh stability equation (<ref>) 
can be rewritten in the rescaled variables as

    (A^ε-z^ε)h^ε
    =
    (A-z)h
    +ε(A-z)g+ε (B-y)h + ε^2 (B-y)g=0.









Notice that the zero order term vanishes by (<ref>). Since ε>0, 
we want to find (g,y) satisfying 

    (A-z)g=(y-B)h+ε (y-B)g.

The first step consists of “inverting” the operator (A-z).
Unfortunately, since z,z^* are the eigenvalues of A, the kernel and the image of (A-z) are given by Ker(A-z)=span(h) and Im(A-z)=span(h^*). We bypass this obstacle by choosing y in such a way that the right hand side of (<ref>) becomes “parallel” to h^*. This allows writing (<ref>) in the form

    (g,y)=H+ε R,

where H only depends on the (h,z) solving (<ref>), while the remainder R depends also on (g,y).
Thus, there is  an explicit solution for ε=0, and   for small enough ε>0  a fixed point argument in L^2(I)^2× yields a solution as well.
Once ε>0 is fixed, Theorem <ref> holds by taking

    Ω̅=ω̅^ε.


From now on the smooth vortex Ω̅ is fixed and thus we will omit ε for the sake of simplicity. The use of capital letters will become clear in the following step.



 §.§ Step 2. Self-similar instability

Let us start by writing the Euler equation in self-similar coordinates

    τ=1/ablog t,
        
    X=x/(abt)^1/a,

in terms of two parameters 0<a,b≤ 1 to be determined.
Notice that the logarithmic time interval is  (instead of _+). In particular, the physical time t=0 corresponds to
logarithmic time τ=-∞.
It is straightforward to check that (ω,f) given by the change of variables

    ω(t,x)=1/ab tΩ(τ,X),
        
    f(t,x)=1/(ab t)^2F(τ,X),

is a solution to the Euler equation if and only if (Ω,F) solves the self-similar Euler equation

    ∂_τΩ
    +V·∇Ω
    -bSΩ
    =F,

where S=S_a is the linear operator defined by

    SΩ=(a+X·∇)Ω.

Notice that (<ref>) agrees with the Euler equation (<ref>) but for the extra term bSΩ, which vanishes (formally) as b→ 0.
The corresponding velocities and Eulerian forces and are linked by

    v(t,x)=(abt)^1/a-1V(τ,X),
        
    g(t,x)=(abt)^1/a-2G(τ,X),

that is, (v,V,g,G) are recovered from (ω,Ω,f,F) through the Biot-Savart law (<ref>)
 respectively.
 





















We have adopted the overall convention in <cit.>: Given some object related to the Euler equation, the corresponding object in self-similar coordinates will be denoted with the same letter in capital case. The only exception will be (<ref>).



In <cit.> the parameters are given by α=a and β=1/ab. The reason why we consider this change of variables is purely cosmetic. For instance, it makes more manageable the spectral analysis in the limit b→ 0.


In the setting of Theorem <ref>, we take the background solution as the time-dependent vortex

    ω_0(t,x)
    =1/abtΩ̅(X),
        
    v_0(t,x)
    =(abt)^1/a-1V̅(X),

where Ω̅ is the unstable vortex from Theorem <ref>, and V̅ the corresponding velocity  given by the Biot-Savart law  (<ref>).
Given 2<p<∞, we fix the parameter a in the regime

    0<a p< 2,

to guarantee that the integrability condition (<ref>) is satisfied. Indeed, we have the scaling

    ω_0(t)_L^p
    =(abt)^2/a p-1Ω̅_L^p,
        v_0(t)_L^2
    =(abt)^2/a-1V̅_L^2.


The force is defined ad hoc in such a way that (<ref>) becomes a solution. By the radial symmetry of the vortex (and defining the pressure properly) the quadratic term in the Euler equation vanishes, that is, V̅·∇Ω̅=0. Thus, the force is defined by

    ∂_t ω_0 = f,
        ∂_t v_0 = g,

or, equivalently,

    F(X)
    =-bSΩ̅,
        
    G(X)
    =b(a-S)V̅.

Observe that (Ω̅,V̅,F,G) are supported on {|X|≤ r_2+ε}, and that the integrability condition (<ref>) is also satisfied

    ∫_0^t
    f(s)_L^p s
    =(abt)^2/ap-1/2/ap-1F_L^p,
        ∫_0^t
    g(s)_L^2 s
    =(abt)^2/a-1/2/a-1G_L^2.
 


Analogously to [Step1]Step 1, the deviation Ω of a close but different solution
Ω̅+ϵΩ
to the self similar Euler equation 
must satisfy

    (∂_τ-L_b)Ω
    +ϵ (V·∇Ω)=0,

where L_b=L_b,Ω̅ is the linearization of the self-similar Euler equation (<ref>) around Ω̅

    L_b
    =L + bS,

for S defined in (<ref>). The unstable solutions in the limit ϵ→ 0 are governed by exponentially growing solutions to the linearized problem, which we denote by 

    Ω^lin(τ,X)
    =e^λτW(X).

Hence, we need to understand the corresponding eigenvalue problem

    L_bW=λ W,

this time for the linear operator  L_b=L_b,Ω̅.

 
We say that the vortex Ω̅ is self-similarly unstable if, for some n∈, there exists 0≠ W∈ U_n satisfying (<ref>) with λ∈_+. 






An obvious but crucial observation is that

    e^λτ
    =t^λ/ab→ 0,

as τ→ -∞ (or equivalently t→ 0), and therefore

    Ω^lin|_τ=-∞=0.


The eigenvalue problem (<ref>) for W∈ U_n is the self-similar Rayleigh stability equation

    (V̅_θ/R-b/inR∂_R)W_n +∂_RW̅/n∫_0^∞ K_n(R/S)
    W_n(S) S
    =zW_n,

in polar coordinates X=Re^iθ,
where we have replaced λ=ab-inz.
Unfortunately, the new term R∂_R W_n prevents from reproducing [Step1]Step 1 for b>0.
Remarkably, Vishik circumvented this obstacle by showing that the spectrum σ(L_b)∩_+ converges (in a suitable sense) to σ(L)∩_+ as b→ 0.
To this end, it is useful to decompose

    L_b=ab+T_b+K,

where

    T_bΩ   =-(V̅-bX)·∇Ω,
    
    KΩ   =-V·∇Ω̅,

with V=∇^⊥Δ^-1Ω. 
Notice that the first term only shifts the spectrum by ab. 
As we will explain in Section <ref>, the transport operator T_b is invertible in _+. Since K is compact, the Operator theory allows to conclude that σ(L_b)∩_+ consists of isolated eigenvalues. 
Since the dependence on b is continuous, there is hope to find an eigenvalue λ_b∈σ(L_b)∩_+ close to λ∈σ(L)∩_+. In Section <ref> we will review and streamline the proof of the following result from <cit.>:


The vortex Ω̅ from Theorem <ref> is self-similarly unstable for some b>0. Moreover, the corresponding eigenfunction satisfies W∈ C_c^2(^2). 






 §.§ Step 3. Nonlinear instability


The last step for proving Theorem <ref> requires controlling the nonlinear effects.
Coming back to [Step2]Step 2 (recall (<ref>)-(<ref>)) we decompose the deviation into

    Ω=Ω^lin+ϵΩ^cor,


where the correcting term Ω^cor must satisfy the equation

    (∂_τ - L_b)Ω^cor
    +(V^lin+ϵ V^cor)·∇(Ω^lin+ϵΩ^cor)_ℱ
    =0,

coupled with the initial condition

    Ω^cor|_τ=-∞=0.

Recall that Ω^lin
=(e^λτW), while V^lin, V^cor are the velocities associated to Ω^lin, Ω^cor respectively.
If we interpret ℱ as a forcing term, since the linear part decays as e^λτ and the contribution of the quadratic term is expected to be negligible for short times, one can naively expect to
gain a slightly faster exponential decay by exploiting the Duhamel formula. In Section <ref> we will review this strategy following <cit.> and
provide some simplifications, mainly due to the compact support of our vortex. We will prove the following result: 

  
There exists a solution Ω^cor to (<ref>)(<ref>) satisfying

    Ω^cor(τ)_L^p
    =o(e^λτ),

as τ→ -∞.




 §.§ Proof of Theorem <ref>

We conclude this section by showing how Theorems <ref> and <ref> allow proving Theorem <ref>.
Coming back to the Eulerian coordinates, 
Theorems <ref> and <ref> imply  that
the vorticity 

    ω_ϵ
    =ω_0
    +ϵω^lin
    +ϵ^2ω^cor,

satisfies that ω_ϵ|_t=0=0, because of (<ref>), (<ref>) and (<ref>), and it is a solution to the Euler equation. We just need to  show that for different ϵ's we have different solutions.  
We  recall that

    ω_0(t,x)
    =1/abtΩ̅(X),
        ω^lin(t,x)
    =1/abtΩ^lin(τ,X),
        ω^cor(t,x)
    =1/abtΩ^cor(τ,X),

where we have applied the convention that capital letters and τ stand for self-similar variables. 
Thus, by applying the scaling

    ω^lin(t)_L^p
    = (abt)^2/ap-1(e^λτW)_L^p,
        ω^cor(t)_L^p
    =(abt)^2/ap-1o(e^λτ),

and the reverse triangle inequality, we obtain

    (abt)^1-2/ape^-λτ(ω_ϵ-ω_ϵ̅)(t)|_L^p≥
    |ϵ-ϵ̅|
    ((e^iλτW)_L^p
    -o(1)).

If λ≠ 0, the right-hand side is positive in a sequence of times τ_k=τ_0-2π k/ z→ -∞ as k→∞. If λ=0, we can assume from the beginning that W is real-valued. Otherwise, by Remark <ref>, it would suffice to take its imaginary part instead.
This allows concluding that
ω_ϵ≠ω_ϵ̅ whenever ϵ≠ϵ̅. Finally, we will prove within Section <ref> that the velocities v_ϵ=∇^⊥Δ^-1ω_ϵ belong to L_t^∞ L^2.


Notice that we have only considered ϵ=0,1 in Theorem <ref>, while there are indeed infinitely many different solutions indexed by ϵ≥ 0. Although these solutions depend on ϵ, we have abbreviated Ω_ϵ^cor=Ω^cor to lighten the notation.



In principle, the solutions we have constructed so far blow up in infinite time due to the scaling (<ref>). However, this can be easily solved by taking f and g vanishing for t≥ 1.





§ PIECEWISE CONSTANT UNSTABLE VORTEX


In this section we prove Theorem <ref>.
We start by rigorously deriving the Rayleigh stability equation (<ref>). 
Firstly, we write the (unforced) Euler equation
in polar coordinates x=re^iθ

    ∂_tω
    +v_r∂_rω
    +1/rv_θ∂_θω
    =0,

where v_r=v· e_r and v_θ=v· e_θ are the radial and angular components of the velocity respectively,
in the standard basis e_r=e^iθ and e_θ=ie^iθ.

As we mentioned in Section <ref>, we restrict to the case of vortices

    ω̅(x)
    =w̅(r).

It is straightforward to see that the corresponding velocity  satisfies

    v̅(x)
    =v̅_θ(r)e_θ,

namely, div v̅=0, while
curl v̅=ω̅ is equivalent to

    rw̅=∂_r(rv̅_θ).



The linearization (<ref>) of the Euler equation (<ref>) around ω̅ reads as

    Tw=-1/rv̅_θ∂_θ w,
        
    Kw=-v_r∂_rw̅,

where v is recovered from w through the Biot-Savart law (<ref>).
Notice that only the radial component v_r contributes in (<ref>).
In the next lemma we compute v_r for vorticities in the class (<ref>).


For any n≥ 1 and w∈ U_n,

    v_r
    =ie^inθ∫_0^∞
    K_n(r/s)
    w_n(s) s,

where

    K_n(ρ)
    ={[ 1/2ρ^-1+n,       ρ<1,; 1/2ρ^-1-n,       ρ>1. ].



By writing the Biot-Savart law (<ref>) in polar coordinates (x=re^iθ, y=se^iϑ)

    v(x)^*
    =1/2π i∫_^2w(y)/x-y y
    =1/2π i∫_0^∞∫_0^2πw(se^iϑ)/re^iθ-se^iϑϑ
    s s,

we get

    v_r
    =(v^*e^iθ)
       =/2π∫_0^∞∫_0^2πw(se^iϑ)/r-se^i(ϑ-θ)ϑ
    s s
       =-1/2π∫_0^∞∫_0^2πsinϑ/|r-se^iϑ|^2w(se^i(θ-ϑ))ϑ s^2 s.

We remark that this is the expression for the real operator v_r (respectively K). Next, we consider v_r acting on complex-valued vorticities. 
Hence, for w(x)=w_n(r)e^inθ, we have

    v_r
    =-e^inθ/2π∫_0^∞∫_0^2πsin(ϑ)e^-inϑ/|r-se^iϑ|^2ϑ w_n(s) s^2 s=ie^inθ∫_0^∞
    K_n(r/s)
    w_n(s) s,

where

    K_n(ρ)
    =1/2π∫_0^2πsin(ϑ)sin(nϑ)/|ρ-e^iϑ|^2ϑ
    ={[ 1/2ρ^-1+n,       ρ<1,; 1/2ρ^-1-n,       ρ>1. ].

The last integral can be computed by means of the Residue Theorem (see Appendix <ref>).


Therefore, the eigenvalue problem Lw=λ w can be written as

    v̅_θ/rw_n +∂_rw̅/n∫_0^∞
    K_n(r/s)
    w_n(s) s
    =zw_n,

where λ=-inz.
Since w_n must vanish wherever ∂_rw̅=0 provided that z≠ 0, we can write

    w_n
    = h∂_rw̅,

and thus the equation (<ref>) reduces to solve

    v̅_θ(r)/rh(r) +1/n∫_0^∞
    K_n(r/s)
    (h∂_rw̅)(s) s
    =zh(r),
        
    r∈supp(∂_rw̅),

for some profile h≠ 0 and eigenvalue z with z>0, to be determined.



 §.§ Ansatz


Given some parameters 0<r_1<r_2<∞ and c>0 to be determined, we consider the piecewise constant vorticity profile (see Figure <ref>)

    w̅(r)
    =
    {[          c,   0<r≤ r_1,;         -1, r_1<r≤ r_2,;          0,      r>r_2, ].

with corresponding (angular) velocity profile (<ref>)

    v̅_θ(r)
    =
    {[            c r/2,         0<r≤ r_1,; r/2((r_2/r)^2-1),       r_1<r≤ r_2,;                0,            r>r_2, ].

where we have chosen c satisfying

    (1+c)ξ=1
      with  ξ=(r_1/r_2)^2.

In this case we have

    ∂_rw̅
    =-(1+c)δ_r_1 +δ_r_2.

Formally, it remains to solve the Rayleigh stability equation (<ref>) at r=r_1,r_2. These are two conditions for the vector h=(h(r_1),h(r_2)) that form the linear system

    A
    h
    =zh,

where we split A=D+C into

    D=
    [
    [ v̅_θ(r_1)/r_1             0;             0 v̅_θ(r_2)/r_2 ]],
        
    C=
    [
    [       -1+c/nK_n(1)    1/nK_n(r_1/r_2); -1+c/nK_n(r_2/r_1)          1/nK_n(1) ]].

Notice that the diagonal matrix D comes from the transport operator T, and C from the compact operator K.

In any case, there exists an eigenvector  h≠ 0 for an eigenvalue z if and only if (A-z)=0.  In the next lemma we compute explicitly this characteristic polynomial of A.

 Let A be given by (<ref>) and r_1,r_2,c related to ξ by (<ref>). Then, 

    (A-z)=
    z^2-n-1/n1-ξ/2ξz
    +1-ξ/4nξ
    -1-ξ^n/4n^2ξ.



The determinant of A is a quadratic polynomial in z

    (A-z)
       =(v̅_θ(r_1)/r_1-1+c/nK_n(1)-z)(v̅_θ(r_2)/r_2+1/nK_n(1)-z)
    +1+c/n^2K_n(r_1/r_2)K_n(r_2/r_1)
       =z^2-(v̅_θ(r_1)/r_1+v̅_θ(r_2)/r_2-c/nK_n(1))z
       +(v̅_θ(r_1)/r_1-1+c/nK_n(1))
    (v̅_θ(r_2)/r_2+1/nK_n(1))
    +1+c/n^2K_n(r_1/r_2)K_n(r_2/r_1).

By plugging (recall (<ref>) and (<ref>))

    v̅_θ(r_1)/r_1=c/2,
        v̅_θ(r_2)/r_2=0,
        
    K_n(1)=1/2,
        
    K_n(r_1/r_2)K_n(r_2/r_1)
    =ξ^n/4,

and (<ref>) into (<ref>), we get (<ref>).













For any n≥ 2 there exists 0<ξ<1 such that the roots z,z^* of the characteristic polynomial (<ref>) satisfy z> 0. For n=2 we can take ξ=1/2.


The discriminant Δ of the quadratic polynomial (<ref>) equals

    Δ
    =p_n(ξ)/ξ^2  with  
    p_n(ξ)
    =(n-1/n)^2(1-ξ)^2/4
    -ξ(1-ξ)/n
    +ξ(1-ξ^n)/n^2.

Hence, it remains to show that p_n(ξ)<0
for some 0<ξ<1.
By computing the first two derivatives

    p_n'(ξ)
       =-(n-1/n)^2(1-ξ)/2
    -(1-ξ)/n
    +(1-ξ^n)/n^2
    +ξ/n(1-ξ^n-1),
    
    p_n”(ξ)
       =1/2(n-1/n)^2
    +2/n(1-ξ^n-1)
    -n-1/nξ^n-1,

and noticing that

    p_n'(1)=0,
        
    p_n”(1)
    
    
    
    
    
    
    =1-n^2/2n^2<0,

we conclude the first statement. For n=2 we have




p_2(1/2)<0.




§ REGULARIZATION


Unfortunately, the unstable vortex ω̅ we constructed in Section <ref> is not regular enough: its eigenfunction w=e^inθh∂_rw̅ is a measure concentrated on r=r_1,r_2. In this section we show that there is a regularization  ω̅^ε that is also an unstable vortex for some small ε>0.

We take a standard mollifier η∈ C_c^∞(I) with I=(-1,1), ∫η=1, and define as usual

    η^ε(α)
    =1/εη(α/ε).

By mollifying v̅_θ (instead of w̅) a few computations are simplified because we get rid of some nonlocal terms.
Since we want to regularize the singularities without modifying the vortex outside B_ε({r_1,r_2}), 
we only mollify the indicator functions.
To this end, we write the velocity (<ref>) as

    v̅_θ
    =v̅_θ,11_[0,r_1)
    +v̅_θ,21_[r_1,r_2),

where we have abbreviated

    v̅_θ,1
    =c r/2,
        v̅_θ,2
    =r/2((r_2/r)^2-1).

Let 0<ε<1/3min{r_1,r_2-r_1}, to be determined.
It is straightforward to check that the profile

    v̅_θ^ε
    ={[ v̅_θ,1
    	+(v̅_θ,2-v̅_θ,1)
    	1_[r_1,r_2)*η^ε,                                 |r-r_1|<ε,;                   v̅_θ,2
    	1_[r_1,r_2)*η^ε,                                 |r-r_2|<ε,;                                      v̅_θ,                                 otherwise, ].

is smooth (and agrees with v̅_θ outside B_ε({r_1,r_2}) by definition).
The corresponding vorticity profile w̅^ε is then given by (<ref>)

    w̅^ε
    =1/r∂_r(rv̅_θ^ε),

which is smooth and agrees with w̅ outside B_ε({r_1,r_2}). Therefore, since

    ∂_rw̅^ε
    =0
      outside  
    B_ε({r_1,r_2}),

it remains to find an eigenvalue z^ε∈ with z^ε>0 and a profile h^ε satisfying the Rayleigh stability equation (<ref>)

    (v̅_θ^ε(r)/r-z^ε)h^ε(r) +1/n∫_0^∞
    K_n(r/s)
    (h^ε∂_rw̅^ε)(s) s
    =0,
        
    r∈ B_ε({r_1,r_2}).





 §.§ Rescaling


In this section we zoom into each interval B_ε(r_j) through the change of variables

    r=r_j+εα,

for α∈ I=(-1,1) and j=1,2. 
From now on we will denote

    c_1=-(1+c),
    	    
    	c_2=1,

to make the notation more compact.


It holds

    v̅_θ^ε(r)/r
    =v̅_θ(r_j)/r_j
    +εu_j(α)/r^2,

for r=r_j+εα, where

    u_j(α)
    =c_j/2α(2r_j+εα)(σ(α)-j+1)
      with  σ(α)
    =∫_-1^αη(β)β.



Firstly, notice that

    v̅_θ,1(r)/r=v̅_θ(r_1)/r_1
    =c/2,
        v̅_θ(r_2)/r_2=0.

Hence, by computing (recall (<ref>))

    v̅_θ,2-v̅_θ,1/r
    =-1+c/2r^2-r_1^2/r^2,
        v̅_θ,2/r
    =-1/2r^2-r_2^2/r^2,

and

    1_[r_1,r_2)*η^ε
    =1/ε∫_r_1^r_2η(r-s/ε) s
    ={[     σ(α), r=r_1+εα,;   1-σ(α), r=r_2+εα, ].

the lemma follows from the definitions (<ref>) and (<ref>).



It holds

    ε∂_rw̅^ε(r)
    =
    c_j(η̅
    +εζ_j)(α),

for r=r_j+εα, where

    η̅
    =η+∂_α(αη),
        ζ_j
    =-1/2∂_α(α^2η/r_j+εα).



By applying Lemma <ref>,
it is straightforward to check that

    ε∂_rw̅^ε
    =ε∂_r(1/r∂_r(rv̅_θ^ε))
    =∂_α(∂_α u_j/r_j+εα).

Then, by computing

    ∂_α u_j(α)/r_j+εα
    =c_j(σ(α)-j+1+αη(α)-ε/2α^2η(α)/r_j+εα),

we get,

    ∂_α(∂_α u_j/r_j+εα)
    =c_j(η+∂_α(αη)-ε/2∂_α(α^2η/r_j+εα)).

This concludes the proof.


We define h^ε in each interval B_ε(r_j) as

    h^ε(r)
    =h_j+ε g_j(α),

where h_j=h(r_j),
for some profiles g_j∈ L^2(I), to be determined. Typically, we will deal with j=1,2
and will speak of g=(g_1,g_2) ∈ L^2(I)^2 to deal with both equations simultaneously and make the notation more compact.  
Similarly, we define

    z^ε
    =z+ε y,

for some y∈, to be determined. 

We also make the asymptotic expansion of the kernel

    K_njk^ε
    =K_n(r_j+εα/r_k+εβ)
    =K_njk^0
    +ε J_njk.

After all this preparation, we can write the Rayleigh stability equation in the rescaled variables.


It holds

    (v̅_θ^ε(r)/r-z^ε)h^ε(r) +1/n∫_0^∞
    K_n(r/s)
    (h^ε∂_rw̅^ε)(s) s
    =ε E_j(α),

for r=r_j+αε. 
We split the error E=(E_1,E_2) into

    E=(A-z)g+(B-y)h+ε (B-y)g,

where 

    (Ag)_j
       =v̅_θ(r_j)/r_jg_j
    +1/n∑_k=1,2c_kK_njk^0∫_I
    g_kη̅β,
    
    (Bg)_j
       =
    u_j/r^2g_j
    +1/n∑_k=1,2c_k∫_I
    J_njk
    g_kη̅β
    +1/n∑_k=1,2c_k∫_I
    K_njk^ε
    g_k ζ_kβ.



On the one hand, by Lemma <ref>,

    (v̅_θ^ε/r-z^ε)h^ε   =(v̅_θ(r_j)/r_j-z)h_j
    +ε(u_j/r^2-y)h_j
    +ε(v̅_θ(r_j)/r_j-z)g_j
    +ε^2 (u_j/r^2-y)g_j.

On the other hand, by Lemma <ref>,

    1/n∫_0^∞
    K_n(r/s)
    (h^ε∂_rw̅^ε)(s) s
       =1/n∑_k=1,2 c_kh_k
    K_n(r_j/r_k)
       +ε/n∑_k=1,2 c_kh_k∫_I
    1/ε[K_n(r_j+εα/r_k+εβ)
    -K_n(r_j/r_k)]
    η̅(β)β
       +ε/n∑_k=1,2 c_kh_k∫_I
    K_n(r_j+εα/r_k+εβ)
    ζ_k(β)β
       +ε/n∑_k=1,2c_k∫_I
    K_n(r_j+εα/r_k+εβ)
    (g_kη̅)(β)β
       +ε^2/n∑_k=1,2c_k∫_I
    K_n(r_j+εα/r_k+εβ)
    (g_k ζ_k)(β)β,

where we have applied

    ∫η̅
    =∫η=1.

Finally, notice that the first terms in (<ref>) and (<ref>) cancel out because Ah=zh by Section <ref>.




 §.§ Inverting the linear operator


Notice that we can split A=D+C into

    (Dg)_j=v̅_θ(r_j)/r_jg_j,
        
    (Cg)_j=1/n∑_k=1,2c_kK_njk^0
    ∫_I g_kη̅β.

Observe that when g is constant we get back the matrix A from (<ref>). We omitted this detail in the introduction for the sake of simplicity. Notice that D is a real-valued diagonal operator, and therefore (D-z) is invertible, and that C is constant-valued, that is, C:L^2(I)^2→^2. The next lemma takes advantage of this structure to reformulate the Rayleigh stability equation (<ref>)=0 in order to find the corresponding positive eigenvalue.


The Rayleigh stability equation

    (A-z)g=(y-B)h+ε (y-B)g,

is equivalent to

    g   =f
    +γ h^* + δ h,
    
    y   =⟨ Cf,h^*⊥⟩/⟨ h,h^*⊥⟩,

where f=f(g,y)

    f=(D-z)^-1(-Bh+ε (y-B)g),

and γ,δ∈, with γ=γ(f)

    γ
    =-1/2iz_2|h|^2(⟨ h,h^*⟩/⟨ h,h^*⊥⟩⟨ Cf,h^*⊥⟩
    -⟨ Cf,h^*⟩).

Since δ is arbitrary, we will take δ=0 for simplicity. 


We start by making the change of variables

    g=f
    +μ,

in terms of some μ, to be determined. It turns out that μ must be a constant vector satisfying

    (A-z)μ
    =yh-Cf.

Notice that A in (<ref>) is acting on constant vectors, and it is therefore given by (<ref>).
Since Ker(A-z)=span(h), Im(A-z)=span(h^*), and {h,h^*} is a basis of ^2,
we take

    μ=γ h^*+δ h,

in terms of some γ,δ∈, to be determined. Notice that

    (A-z)μ
    =γ(A-z)h^*
    =γ(A-(z^*+2iz_2))h^*
    =-2i z_2γ h^*.

Hence, 

multiplying (<ref>) by h^*⊥=(-h_2^*,h_1^*), we get the compatibility condition for y

    0
    =y⟨ h,h^*⊥⟩
    -⟨ Cf,h^*⊥⟩.

Notice that ⟨ h,h^*⊥⟩≠ 0 since {h,h^*} is a basis of ^2. On the other hand, multiplying (<ref>) by h^*, and using (<ref>),
we can determine γ from the following equality

    -2iz_2γ
    |h|^2
    =y⟨ h,h^*⟩
    -⟨ Cf,h^*⟩.

The proof of the lemma is concluded. 




 §.§ Fixed point argument


For ε=0 the equation (<ref>) for (g,y) is explicit and thus we can find a solution (g^0,y^0). For small ε>0 we will apply a fixed point argument. 
It can be shown by a bootstrapping argument that the eigenfunction g^ε is indeed smooth. However, since this additional information is not necessary at this point, we omit the proof.

 
For every M>(g^0,y^0)_L^2(I)^2× there exists ε_0>0 satisfying that: for any 0≤ε≤ε_0 there exists (g^ε,y^ε)∈ L^2(I)^2× solving (<ref>) with 
(g^ε,y^ε)_L^2(I)^2×≤ M. In particular, z^ε>0 for ε< z/M.


First we check that the linear operators B and C














are bounded in L^2(I)^2. The operator C can be easily bounded by

    Cg_L^2≤|I|^1/2/n∑_j=1,2∑_k=1,2|c_k||K_njk^0|
    g_k_L^2η̅_L^2.

Similarly, the operator B can be bounded by

    Bg_L^2≤∑_j=1,2(
    u_j/r^2_L^∞g_j_L^2
    +|I|^1/2/n∑_k=1,2|c_k|
    (
    J_njk_L^∞η̅_L^∞
    +K_njk^ε_L^∞ζ_k_L^∞)g_k_L^2).

From the expression (<ref>) and (<ref>) it follows that

    K_njk^ε_L^∞≤1/2,
        J_njk_L^∞≤1/2(n+1)
    r_j+r_k/(r_k-ε)^2.




Let us denote by F the map on the right hand side of (<ref>), that is, we rewrite this equation compactly as

    (g,y)=F(g,y).

On the one hand, since all the operators involved in F are bounded and (g^0,y^0)_L^2(I)^2×<M, there exists ε_0>0 such that F maps the ball B_M of L^2(I)^2× into itself.
On the other hand, given a pair (g,y) and (g̅,y̅) in B_M,  the  corresponding pair
f and f̅ given by (<ref>) satisfy

    f-f̅
    =ε(D-z)^-1((y-B)g-(y̅-B)g̅),

and thus

    f-f̅_L^2≤ε(D-z)^-1_L^∞(|y-y̅|g_L^2(I)^2
    +|y̅|g-g̅_L^2
    +B(g-g̅)_L^2(I)^2).



The boundedness of the operators B and C and the constant M in the statement of the proposition imply that we can choose ε_0>0 such that F becomes a contraction on B_M. Then, we can apply the classical Banach fixed point theorem to find our required solution. 








§ SELF-SIMILAR INSTABILITY


In this section we prove Theorem <ref>, and provide growth bounds of the semigroup generated by L_b.
We start by decomposing

    L_b
    =
    ab
    +T_b+K,

where T_b is the transport operator

    T_bΩ
    =-V̅_b·∇Ω  with  V̅_b
    =V̅-b X,

and K is the compact operator

    KΩ=-V·∇Ω̅  with  
    V=∇^⊥Δ^-1Ω.



As these operators arise from the self-similar Euler equation, we will keep the convention of using uppercase letters for maintaining coherence in notation. It is important to observe that the results applicable to b=0 correspond to the (original) Euler equation.


From now on we consider n≥ 2 fixed.
We define these operators on the subspace L_n^2 of vorticities Ω∈ L^2 which has zero mean and are n-fold symmetric 

    Ω(X)=Ω(e^2π i/nX).
 
By writing the Fourier expansion of Ω in polar coordinates X=Re^iθ,

    Ω(X)=∑_k∈Ω_k(R)e^ikθ,
        Ω_k(R)
    =1/2π∫_0^2πΩ(Re^iθ)e^-ikθθ,

it follows that
the n-fold symmetry (<ref>) is equivalent to the vanishing of the indices k that are not multiples of n, and the zero mean condition to

    ∫_0^∞Ω_0(R)R R=0.

Therefore, we can decompose L_n^2 into the orthogonal direct sum

    L_n^2=⊕_j∈U_jn,

of the invariant subspaces given in (<ref>), while U_0 is given by the condition (<ref>).
More precisely, by the Plancherel identity

    Ω_L^2^2
    =2π∑_j∈∫_0^∞ |Ω_jn(R)|^2R R,

we consider sums of elements Ω_jn for which (<ref>) is finite.
We remark that the operators under consideration are closed and densely defined. In fact, the domain of K is D(K)=L_n^2, and the domain of L_b and T_b equals

    D(L_b)=D(T_b)
    ={Ω∈ L_n^2 : div(V̅_bΩ)∈ L_n^2}.



The reason for considering the direct sum (<ref>) is that the nonlinear term in the Euler equation lacks invariance in U_n, but it is invariant in the entire space L_n^2 (see Proposition <ref>). 
Specifically, in Section <ref> we will need the growth bound of the semigroup generated by L_b acting on the full space L_n^2. 


Before embarking on the proof of Theorem <ref>, we recall some classical results in Operator theory. Then, we will analyze T_b and K separately, and later L_b. After that, we will compute the growth bound of the semigroup generated by L_b, and provide regularity properties of the eigenfunction.



 §.§ Preliminaries


In this section we recall some classical results in Operator theory that will be helpful during the analysis. In general, we will consider a linear operator A:D(A)⊂ H→ H acting on some Hilbert space H, where D(A) is the domain of A. For a fixed H, we will denote ℒ and 𝒦 by the space of bounded and compact operators on H respectively. In the next sections we will consider H=L_n^2.



  §.§.§ Spectral theory

The spectrum of A is defined as  

    σ(A)={λ∈ :  (A-λ) is not invertible}.

Let us suppose that A is a  bounded operator. 
Then, A is called a  Fredholm Operator if both the kernel Ker(A) and the cokernel H/Im(A) are finite dimensional. In this case, the index of A is the integer

    Ind(A)
    =(Ker(A))
    -(H/Im(A)).

We recall a classical result in Spectral theory: the stability of the index of Fredholm operators with respect to compact perturbations
(see e.g. <cit.>).




	Let A∈ℒ be a Fredholm operator and K∈𝒦. Then, A+K∈ℒ is a Fredholm operator with Ind(A+K)=Ind(A).











  §.§.§ Semigroup theory

We recall several classical results in Semigroup theory. The first one gives a characterization of strongly continuous semigroups (see e.g. <cit.>).
 
Given w∈ and a linear operator A, the following are equivalent:

    
  * A generates a strongly continuous semigroup satisfying e^sA_ℒ≤ e^ws for all s≥ 0.
    
  * A is closed, densely defined, and for any λ∈ with λ>w, it holds
    
    (A-λ)^-1_ℒ≤1/λ - w.




Next, we recall that, given a strongly continuous semigroup e^sA generated by an operator A,
its growth bound is defined as

    ω_0(A)
    =inf{w∈ : there exists C_w≥ 1 such that e^s A_ℒ≤ C_w e^w s for all s≥ 0}.

The second result from semigroup theory that we need (see e.g. <cit.>) relates (<ref>) with the essential bound 

    ω_ess(A)
    =inf_s>01/sloge^s A_ℒ/𝒦,

where ℒ/𝒦 is the quotient space (the so-called Calkin algebra),
and the spectral bound

    s(A)
    =sup{λ : λ∈σ(A)}.



Given an operator A generating a strongly continuous semigroup, it holds

    ω_0(A)
    =max{ω_ess(A),s(A)}.

Moreover,
σ(A)∩{λ≥ w}
is finite for any w>ω_ess(A).


The third result in semigroup theory that we need is the stability of the essential bound with
respect to compact perturbations (see e.g. <cit.>).


Given an operator A generating a strongly continuous semigroup, and K∈𝒦, it holds

    ω_ess(A+K)
    =ω_ess(A).





 §.§ The transport operator T_b





	The operator T_b generates a strongly continuous
 semigroup
	
    e^sT_bΩ
    	=Ω∘X̅_b(s,·)^-1,

	where X̅_b is the flow map

    ∂_sX̅_b
    	=V̅_b(X̅_b),
    	    X̅_b|_s=0=id.

	Moreover,

    e^sT_b_ℒ
    	=e^-bs.



Notice that the flow map is defined through a Lipschitz vector field

    V̅_b_Lip≤V̅_Lip
    +b.

Hence, the first statement follows from the Cauchy-Lipschitz theory applied to the transport equation.
For the second statement, by solving the ODE

    ∂_sJ_X̅_b=div(V̅_b)J_X̅_b  with  div(V̅_b)=-2b,

we deduce that the Jacobian of the flow map equals

    J_X̅_b=e^-2bs.

Therefore,

    ∫_^2|e^sT_bΩ|^2 X
    =e^-2bs∫_^2|Ω|^2 Y,

where X=X̅_b(s,Y).



The resolvent map

    (T_b-λ)^-1:
    {λ>-b}   →ℒ
    
    (b,λ)    ↦(Ω↦-∫_0^∞
    e^s(T_b-λ)Ω s)

is well defined with

    (T_b-λ)^-1_ℒ≤1/λ+b.

For any Ω∈ L_n^2, the map
(b,λ)↦ (T_b-λ)^-1Ω is continuous.


The first statement follows directly from Lemma <ref> and Proposition <ref>.
In fact, a simple integration by parts shows that (<ref>) is the inverse of (T_b-λ).
Alternatively, (<ref>) is the Laplace transform of the semigroup e^sT_b. 









For the second statement, given Ω∈ L_n^2∩ C_c^∞ and s>0, the map (b,λ)↦ e^s(T_b-λ)Ω is continuous since the flow map (<ref>) is defined through a continuous in b uniformly Lipschitz vector field. The continuity (indeed analiticity) in λ is clear from the definition.
Hence, (<ref>) allows to apply the dominated convergence theorem in (<ref>). Finally, by applying that L_n^2∩ C_c^∞ is strongly dense and (<ref>), 
we conclude the proof.




 §.§ The compact operator K


As pointed in <cit.>, while it is not feasible to extend the Biot-Savart operator throughout L^2, it can be achieved within the subspace L_n^2, thanks to the following proposition.  By a slight abuse of notation, we will continue to denote the Biot-Savart operator as V=∇^⊥Δ^-1Ω.
By combining (<ref>) with the Rellich-Kondrachov Theorem, and the fact that Ω̅ is compactly supported, we deduce the compactness of K as a corollary. The second estimate (<ref>) will imply that the velocity of the eigenfunction belongs to L^2.


There exists C>0 such that

    1/RV_L^2(B_R)+DV_L^2≤ CΩ_L^2,

for any R>0 and Ω∈ L_n^2, where V=∇^⊥Δ^-1Ω. 
If supp(Ω)⊂ B_R̅ for some R̅>0, then also

    V_L^2≤ CR̅Ω_L^2.



Let Ω∈ L_n^2∩ C_c. 





Firstly, by applying the n-fold symmetry (<ref>), we deduce that

    V(X)
    =(1/2π i∫_^2Ω(Y)/X-Y Y)^*
    =(1/2π i∫_^2Ω(Y)/X-e^-2π i k/nY Y)^*
    =e^-2π i k/n V(e^2π i k/nX).

Therefore, by integrating V on the ball B_R, we get

    ∫_B_RV(x) x
    =e^-2π i k/n∫_B_RV(x) x,

from which we deduce that

    ∫_B_RV(x) x
    =(1/n∑_k=0^n-1e^-2π i k/n)
    ∫_B_RV(x) x
    = 0.

Therefore, by applying the Poincaré inequality, we get

    V_L^2(B_R)≤ CRDV_L^2(B_R).

Secondly, by applying the Plancherel identity on the Biot-Savart law (<ref>), we obtain


    DV_L^2=Ω_L^2.
 
These inequalities allow to extend the Biot-Savart operator in L_n^2 by density. For the second statement, since we already have (<ref>) for R=2R̅, it is enough to estimate the L^2-norm of V outside B_2R̅. By applying the zero-mean condition of Ω, we get

    V_L^2(^2∖ B_2R̅)   =1/2π(∫_^2∖ B_2R̅|∫_B_R̅Ω(y)(1/X-Y-1/X) Y|^2 X)^1/2
       ≤C/R̅∫_B_R̅|Ω (Y)||Y| Y
    ≤ CR̅Ω_L^2,

where we have applied |X-Y|≥1/2|X| in the first inequality, and the Hölder inequality in the second one. The constant C>0 changes from line to line, but it keeps universal.



The operator K is compact.




 §.§ The operator L_b



For any λ∈ with λ>b(a-1), the operator (L_b-λ) is Fredholm with index zero.
Therefore, σ(L_b)∩{λ> b(a-1)} consists of eigenvalues with finite multiplicity.


We split

    L_b-λ
    =(T_b-(λ - ab))+K.

Since λ>b(a-1), the first term is invertible by Lemma <ref>. In particular, it is a Fredholm operator with index zero.
Since K is compact, the operator (<ref>) is also Fredholm with index zero (in particular positive) by Proposition <ref>. Thus the second claim follows.



 The operator L_b generates a strongly continuous semigroup.


By applying Lemma <ref> and that K is compact, we deduce that the map

    C:
    	{λ>b(a-1)}   →𝒦
    
    	(b,λ)    ↦
    	(T_b-(λ - ab))^-1∘ K,

is well defined and continuous. 
In fact, 
the continuity of (<ref>) in the operator norm follows from combining the fact that the resolvent map (<ref>) is continuous for any fixed Ω, and that K can be approximated by finite rank operators. 
The map C allows decomposing (<ref>) into

    L_b-λ
    =(T_b-(λ - ab))∘(I+C(b,λ)).

Let λ∈ with λ+b(1-a)≥ 2K_ℒ. By applying (<ref>), we deduce that

    C(b,λ)_ℒ≤K_ℒ/λ +b(1-a)≤1/2,

and thus the Neumann series

    (I+C(b,λ))^-1
    =∑^∞_n=1 (-C(b,λ))^n,

converges in the operator norm. Hence, I+C(b,λ) is invertible with the bound

    (I+C(b,λ))^-1_ℒ≤ 1.

Finally, by applying again (<ref>), combined with (<ref>) and (<ref>), we deduce that

    (L_b-λ)^-1_ℒ≤1/λ + b(1-a)≤1/λ + b(1-a)-2K_ℒ,

for any λ + b(1-a)≥ 2K_ℒ. Then, the claim follows by applying Proposition <ref>.



The essential bound
satisfies 

    ω_ess(L_b)≤ (a-1)b≤ 0.

Therefore, σ(L_b)∩{λ>w} is finite for any w> (a-1)b.


Since the identity commutes with any operator, we can decompose 

    e^sL_b
    =e^sabe^s(T_b+K).

Thus,

    ω_ess(L_b)
    ≤ ab+ω_ess(T_b+K).

Then, by applying 
(<ref>), that K is compact, and Proposition <ref>, we get that 

    ω_ess(T_b+K)
    =ω_ess(T_b)
    = -b.

The second part of the statement follows from Proposition <ref>.




 §.§ Proof of Theorem <ref>


Firstly, we claim that there exists b>0 satisfying

    σ(L_b)∩{λ>3b}≠∅.

Since L=L_0, we know from Theorem <ref> that (<ref>) holds for b=0. More precisely, there exists an eigenvalue λ_0∈σ(L)∩_+ and an eigenfunction 0≠ W_0∈Ker(L-λ_0).
We will prove by contradiction that necessarily (<ref>) is true for some 0<b≤λ_0/4.

Let us suppose that (<ref>) is false. Thus, (L_b-λ) is invertible for all λ∈_+ and 0<b≤λ_0/4. We will first prove that in fact (L_b-λ)^-1 is continuous as a function of (b, λ).

By applying Lemma <ref>  to b=0, we can take a ↺-oriented circle Γ:→_+∖σ(L) of radius strictly less than λ_0/4 surrounding λ_0. By applying Lemma <ref> coupled with (<ref>) and (<ref>), our hypothesis implies that I+C maps continuously the compact set [0,λ_0/4]×Γ() into 

    {A∈ℒ : A invertible}
    =⋃_N∈{A∈ℒ : A^-1_ℒ< N}.

Since the last union forms an open cover, we deduce that there exists N∈ such that

    (I+C(b,λ))^-1_ℒ≤ N,

uniformly in (b,λ)∈ [0,λ_0/4]×Γ(), 
and thus, by applying (<ref>) and (<ref>), also

    (L_b-λ)^-1_ℒ≤N/λ + b(1-a).

By applying these bounds and the resolvent identity

    (I+C(b,λ))^-1-(I+C(b',λ'))^-1
    =(I+C(b,λ))^-1∘(C(b,λ)-C(b',λ'))∘(I+C(b',λ'))^-1,

it follows that 
the map (b,λ)↦ (I+C(b,λ))^-1 is continuous from [0,λ_0/4]×Γ() into ℒ. As a consequence, the same can be deduce for the map
(b,λ)↦ (L_b-λ)^-1.

Once continuity is obtained, we can consider the Riesz projection

    P_b,Γ=-1/2π i∫_Γ (L_b-λ)^-1λ.

and take limits under the integral sign by the dominated convergence theorem. Therefore, we have

    P_b,Γ(W_0)
    → P_0,Γ(W_0),

as b→ 0. However, P_b,Γ(W_0)=0 for any 0<b≤λ_0/4 by hypothesis, while P_0,Γ(W_0)=W_0≠ 0, which is a contradiction. 

Therefore, there exists λ_b∈σ(L_b)∩{λ>3b} for some 0<b≤λ_0/4. 
By Lemma <ref>, there exists 0≠ W_b∈Ker(L_b-λ_b).


Since L_b is invariant in each U_jn, the functions W_b,jn(R)e^ijn θ are also eigenfunctions. Furthermore, since Ker(L_b-λ_b) is finite dimensional, W_b,jn is null
for all but a finite number of j's. For j=0, since L_b W_b,0=λ_b W_b,0 reads as

    b(a+R∂_R)W_b,0=λ_b W_b,0,

necessarily
W_b,0=0. Therefore, there exists 
0≠ W_b,jn(r)e^ijnθ∈Ker(L_b-λ_b)∩ U_jn
for some j≠ 0.




 §.§ Growth bound

In the [sec:ProofthmLb]Proof of Theorem <ref> we have seen the existence of an eigenvalue λ∈σ(L_b)∩{λ>3b}. 
In this section we select the eigenvalue with largest real part, and we compute the growth bound of L_b. In the section about the nonlinear instability we will need the estimate for the growth of the semigroup norm. 


There exists λ∈σ(L_b)∩{λ>3b} such that ω_0(L_b)=s(L_b)=λ.
Therefore, for any δ>0 there exists C_δ≥ 1 such that

    e^s L_b_ℒ≤ C_δ e^(λ+δ)s,

for all s≥ 0.

 By Section <ref> we know that there exists λ_b∈σ(L_b)∩{λ>3b}. 
Hence, by combining Proposition  <ref> and Lemma <ref>, we deduce the statement.





 §.§ The eigenfunction


In this section we prove that the eigenfunction associated to the eigenvalue λ appearing in Proposition <ref> is smooth and compactly supported. Here we see, once again,  the advantages of writing the Rayleigh stability equation in vorticity form and dealing with vortices with compact support.


Let W∈Ker(L_b-λ)∩ U_jn with j≠ 0. Then, 

    W∈ C^∞(^2∖{0})
    ∩ C_c^γ(^2),

with
γ =(λ/b-a)≥ 2. 


The self-similar Rayleigh stability equation (<ref>) can be rewritten as

    ((λ-ab)+ijnV̅_θ/R-bR∂_R)W_jn +i∂_RW̅∫_0^∞ K_jn(S)
    W_jn(R/S) S/S^2
    =0.

We remark that (<ref>) is well defined since W∈ D(L_b) by construction.
On the interval (0,r_1-ε), since V̅_θ=cR/2 and ∂_RW̅=0, we deduce that

    W_n(R)
    =C_0
    R^2(λ-ab)+ijn c/2b,

for some constant C_0. On the interval (r_2+ε,∞),  
since V̅_θ=∂_RW̅=0, we deduce that

    W_n(R)
    =C_1
    R^λ-ab/b.

In this case, since W_n∈ L^2 and λ>3b, necessarily C_1=0. As a consequence, we have
supp(W_n)⊂[0,r_2+ε]. 
We have checked that W_n is smooth outside the interval B_ε([r_1,r_2]). Let us check that it is smooth inside the interval I_ε=B_2ε([r_1,r_2]).
Notice that the last integrand in (<ref>) is supported on S≥R/r_2+ε.
Hence, it follows from (<ref>) that W_n∈ H^1(I_ε). By bootstrapping, the same formula allows to prove that W_n∈ H^k(I_ε) for any k≥ 1.



It is straightforward to check that, by substituting the constant in front of b in equation (<ref>) and also adjusting Proposition <ref>, we can achieve γ≥ k for any given k∈. 
Since k=2 is sufficient to prove Theorem <ref>, we opt to keep it that way for the sake of simplicity.




§ NONLINEAR INSTABILITY


In this section we prove Theorem <ref>. 
To this end, for any k∈, we consider the unique solution Ω_k^cor to (<ref>) coupled with the initial condition

    Ω_k^cor|_τ=-k=0.

The existence and uniqueness of this solution is guaranteed by the Yudovich theory. More precisely, in Eulerian coordinates we have that

    ω_ϵ,k=ω_0
    +ϵω^lin
    +ϵ^2ω^cor_k,

is a solution to the Euler equation (<ref>) with the smooth initial condition

    ω_ϵ,k|_t=t_k=(ω_0+ϵω^lin)(t_k)
    =1/abt_k(Ω̅+ϵΩ^lin(-k)),

where t_k=e^-abk.

 
On the one hand, recall that Ω̅ and V̅ are smooth,  compactly supported, and radially symmetric. In particular, V̅∈ L^2. On the other hand, recall that Ω^lin=(e^λτW) with W∈ L_n^2∩  C_c^2 by Proposition <ref>. In particular, V^lin∈ L^2 by Lemma <ref>.
Since the Euler equation is invariant under rotations and the initial data (<ref>) are n-fold symmetric, the uniqueness implies that necessarily ω_ϵ,k (and thus also ω_k^cor) are n-fold symmetric as well.


Thus, our task consists in showing that Theorem <ref> holds for 
Ω^cor_k uniformly in k.
As we will see, the use of the Duhamel formula allows gaining extra exponential decay, but at the cost of having to control the bound under a stronger norm.
Following <cit.>, we introduce the subspace Y of vorticities Ω∈ L_n^2 satisfying

    ‖Ω‖_Y:=‖Ω‖_L^2+‖|X|∇Ω‖_L^2+‖∇Ω‖_L^4<∞.

In <cit.> it is proven that  Y satisfies certain properties that allow closing the energy estimates. Due to the compact support of our vortices, some of them can be omitted, namely the control of the decay. In the following proposition we present a simpler version with the minimum necessary bounds, and provide a shorter proof for the sake of completeness.

There exists C>0 such that

    Ω_L^∞
    +
    1/|X||V|+|DV|_L^∞≤ CΩ_Y,

for any Ω∈ Y, where V=∇^⊥Δ^-1Ω. Moreover, V·∇Ω∈ L_n^2.


We start by proving that 

    Ω_L^∞≤ CΩ_Y.

Given some fixed  X∈^2, let us denote by B_X the unit ball centered at X, and by (Ω)_B_X the average value of Ω in B_X. By applying the Poincaré and Hölder inequalities, we deduce that

    Ω_L^4(B_X)   ≤Ω-(Ω)_B_X_L^4(B_X)
    +|B|^1/4|(Ω)_B_X|
       ≤ C∇Ω_L^4(B_X)
    +|B_X|^-1/4Ω_L^2(B_X)≤ CΩ_Y,

where C is a constant that changes from line to line, but it is independent of X. Hence, by applying the Morrey inequality we get

    Ω_L^∞(B_X)≤ CΩ_W^1,4(B_X)≤ CΩ_Y.

Since this estimate holds in every ball B_X, we conclude (<ref>). As a corollary, by applying the log-convexity of the L^p-norms, we deduce that

    Ω_L^p≤ CΩ_Y,

for any 2≤ p≤∞. In particular, for p=4 this implies that Y↪ W^1,4.
Therefore, by applying first the Morrey inequality and the Calderon-Zygmund theory, we get

    DV_L^∞≤ CDV_W^1,4≤ CΩ_W^1,4≤ CΩ_Y.

In particular, V is continuous. Therefore, letting R→ 0 in (<ref>), we observe that V(0)=0.
Consequently, by integrating DV on the segment with endpoints 0 and X we get

    |V(X)|≤ C |X|Ω_Y.

For the last statement, we have

    V·∇Ω_L^2≤1/|X|V_L^∞|X|∇Ω_L^2≤Ω_Y^2.

It remains to check that V·∇Ω=div(VΩ) is n-fold symmetric. By applying (<ref>) and (<ref>), we get

    div(VΩ)(X)
    =div(e^-2π i/n(VΩ)(e^2π i/n X))
    =div(VΩ)(e^2π i/n X),

for any X∈^2.


Since ω_0, ω^lin and the forcing term f are smooth and compactly supported for any t≥ t_k, it is classical that ω_ϵ,k (and thus also ω_k^cor) enjoy the same properties.








Due to (<ref>), given some fixed 0<δ_0≤1/2λ,  we can define -k<τ_k≤∞ to be the largest time such that

    ‖Ω^cor_k(τ)‖_Y≤ e^(λ+δ_0)τ,

for any -k≤τ≤τ_k.
Our task is to show that the times τ_k satisfying (<ref>) do not diverge to -∞ as k→∞. If τ_k remained positive, we were done. Hence, let us assume from now on that τ_k≤ 0 for all k sufficiently large. 
Thus, in particular,

    ‖Ω^cor_k(τ_k)‖_Y= e^(λ+δ_0)τ_k.

Firstly, we will  improve the bound (<ref>) by using the following lemma. Its proof is similar to that of <cit.> and it is the content of our Sections <ref> and <ref>.

There exists C>0 independent of k such that 

    ‖Ω^ cor_k(τ)‖_Y≤ C e^1/2(3λ+δ_0)τ,

for all -k≤τ≤τ_k.




 §.§ Proof of Theorem <ref>


In this section we show how Lemma <ref> allows proving Theorem <ref>.
Notice that (<ref>) together with (<ref>) implies C e^1/2(λ-δ_0) τ_k≥ 1. Thus, we can take 

    τ̅
    =inf_kτ_k≥ -2log C/λ-δ_0>-∞.

In Eulerian coordinates, the classical solutions ω_ϵ,k satisfy, for all t≥ t_k, the bound

    ω_ϵ,k(t)_L^p   ≤ω_ϵ,k(t_k)_L^p
    +∫_t_k^tf(s)_L^p s,

and similarly the corresponding velocities v_ϵ,k satisfy

    v_ϵ,k(t)_L^2≤v_ϵ,k(t_k)_L^2
    +∫_t_k^tg(s)_L^2 s.

By the scaling identities (<ref>), (<ref>) and Remark <ref>, the sequence of vorticities and velocities are uniformly bounded in L_t^∞ (L^1∩ L^p) and L_t^∞ L^2, respectively. In this case, it is classical (see e.g. <cit.>) that this sequence converges (by taking a subsequence if necessary) to a weak solution of the Euler equation (<ref>).
Finally, by applying the sequential weak lower semicontinuity of the L^p-norm, the embedding L^p↪ L^2∩ L^∞↪ Y (recall Proposition <ref>), and (<ref>), we get

    Ω^cor(τ)_L^p≤lim inf_k→∞Ω_k^cor(τ)_L^p≤ e^(λ+δ_0)τ,

for all τ≤τ̅.



 §.§ Baseline L^2 estimate

In this section we deal with the L^2  part of the Y-norm  for the proof of Lemma <ref>.

There exists C>0 such that

    ‖Ω^ cor_k(τ)‖_L^2≤ C e^2λτ,

for all -k≤τ≤τ_k.


Firstly, recall that
Ω_k^cor satisfies equation (<ref>)
with forcing 

    ℱ(s)=(V^lin+ϵ V^cor)·∇(Ω^lin+ϵΩ^cor)(s)∈ L_n^2,

by Proposition <ref> and Remark <ref>.
Hence, by exploiting the Duhamel formula

    Ω_k^cor(τ)
    =-∫_-k^τe^(τ-s)L_bℱ(s) s,

together with Proposition <ref>,
we estimate,
for some fixed 0<δ≤λ/2 and C=C_δ≥ 1,

    ‖Ω^ cor_k(τ)‖_L^2≤ C∫_-k^τ e^(λ+δ)(τ-s)‖ℱ(s)‖_L^2 s.

On the one hand, Ω^lin=(e^λτW). On the other hand, Ω_k^cor satisfies the estimate (<ref>). Thus, by applying (<ref>), we obtain the bound

    ‖ℱ(s)‖_L^2≤ C(Ω^lin+ϵΩ_k^cor)(s)_Y^2
    ≤ Ce^2λ s,

for all -k≤ s≤τ_k. Thus, (<ref>) yields

    ‖Ω^ cor_k(τ)‖_L^2≤ C  e^(λ+δ)τ∫_-k^τ e^(λ-δ)sτ≤ C e^2λτ,

for all -k≤τ≤τ_k.




 §.§ Energy estimates on derivatives


With (<ref>) proven we now proceed to show that

    ‖ |X|DΩ^ cor_k(τ_k)‖_L^2   ≤ C e^2λτ_k,
    ‖ DΩ^ cor_k(τ_k)‖_L^4   ≤ C e^1/2(3λ+δ_0)τ,

for all -k≤τ≤τ_k.
These estimates appear in Lemmas <ref> and <ref>.
Recalling  (<ref>), we rewrite (<ref>) as

    ∂_τΩ^cor_k - abΩ^cor_k+(V̅-bX+𝒱)·∇Ω^cor_k
    +𝒢
    =0,











where we have abbreviated

    𝒱   :=ϵ V^lin+ϵ^2 V^cor_k,
    𝒢   :=V^lin·∇Ω^lin
    +ϵ  V^cor_k·∇Ω^lin+V^cor_k·∇Ω̅.









Throughout the energy estimates, we will need to control  the norms appearing in (<ref>)(<ref>) for the terms involving 𝒱 and 𝒢. These bounds appear in the next two lemmas.


There exists C>0 such that

    |X|(1/|X||𝒱|+|D𝒱|)|DΩ^cor_k|(τ)_L^2   ≤ Ce^(2λ+δ_0)τ,
    (1/|X||𝒱|+|D𝒱|)|DΩ^cor_k|(τ)_L^4   ≤ Ce^(2λ+δ_0)τ,

for all -k≤τ≤τ_k.


It follows by applying Proposition <ref>.
Recall that Ω^lin is smooth, compactly supported, and proportional to e^λ t, and that Ω_k^cor satisfies the bound (<ref>).



There exists C>0 such that

    ‖ |X|D𝒢(τ)‖_L^2   ≤ Ce^2λτ,
    ‖ D𝒢(τ)‖_L^4   ≤ Ce^12(3λ+δ_0)τ,

for all -k≤τ≤τ_k.


We have 

    ∂_j𝒢
       = ∂_jV^lin·∇Ω^lin
    +ϵ∂_jV^cor_k·∇Ω^lin+∂_jV^cor_k·∇Ω̅
       +   V^lin·∇∂_jΩ^lin
    +ϵ  V^cor_k·∇∂_jΩ^lin+V^cor_k·∇∂_jΩ̅.

We proceed to bound the worst behaved terms. These are the terms involving both V_k^cor and Ω̅.
Recall that Ω̅ (and Ω^lin) are smooth and supported on a ball of radius R̅.
On the one hand, by applying Lemma <ref> coupled with the baseline L^2 estimate (Lemma <ref>), we get

    1/R̅V_k^cor_L^2(B_R̅)+DV_k^cor_L^2≤ CΩ_k^cor_L^2≤ Ce^2λτ.

Therefore, we deduce that

    |X|DV_k^corDΩ̅_L^2≤R̅DV_k^cor_L^2DΩ̅_L^∞≤ Ce^2λτ,
    |X|V_k^corD^2Ω̅_L^2≤R̅V_k^cor_L^2(B_R̅)D^2Ω̅_L^∞≤ Ce^2λτ,

and, by additionally applying Proposition <ref> coupled with (<ref>) and the baseline estimate , we get

    DV_k^corDΩ̅_L^4≤DV_k^cor_L^2^1/2DV_k^cor_L^∞^1/2DΩ̅_L^∞≤ C e^12(3λ+δ_0)τ,
    V_k^corD^2Ω̅_L^4≤V_k^cor_L^2(B_R̅)^1/2V_k^cor_L^∞(B_R̅)^1/2D^2Ω̅_L^∞≤ C e^12(3λ+δ_0)τ.

The remaining terms are bounded similarly and have the same (or faster) decay.


Notice that until now we have neglected the terms V̅ and X.  In fact, these terms are very problematic since they do not give any extra decay. Following <cit.>, we bypass this problem by estimating the derivatives with respect to the polar coordinates 

    D
    =e_R∂_R+e_θ1/R∂_θ.

More precisely, 
by noticing that

    V̅·∇Ω^cor_k=V̅_θ(R)/R∂_θΩ^cor_k,
        
    X·∇Ω^cor_k=R∂_RΩ^cor_k,

it becomes convenient to perform first energy estimates with respect to the angular derivative. This allows to gain an extra decay on ∂_θΩ^cor_k, and then close the energy estimates for  ∂_RΩ^cor_k. 




  §.§.§ Estimates on the angular derivative


 
There exists C>0 such that

    ‖∂_θΩ^cor_k(τ)‖_L^2   ≤ Ce^2λτ,
    ‖1/R∂_θΩ^cor_k(τ)‖_L^4   ≤ Ce^12(3λ + δ_0)τ,

for all -k≤τ≤τ_k.


For (<ref>), by differentiating (<ref>) with respect to θ, we get

    ∂_τ∂_θΩ^cor_k - ab∂_θΩ^cor_k+(V̅-bX+𝒱)·∇∂_θΩ^cor_k   
    
    =
    -∂_θ𝒱_R∂_RΩ_k^cor
    -∂_θ𝒱_θ∂_θΩ_k^cor/R
    -∂_θ𝒢   =:ℋ_1,

where we have applied (<ref>)

    ∂_θ(V̅·∇Ω^cor_k)
    =V̅·∇∂_θΩ^cor_k,
        ∂_θ(X·∇Ω^cor_k)
    =X·∇∂_θΩ^cor_k,

and also

    ∂_θ(𝒱·∇Ω_k^cor)
    
    
    =𝒱·∇∂_θΩ_k^cor
    +∂_θ𝒱_R∂_RΩ_k^cor
    +∂_θ𝒱_θ∂_θΩ_k^cor/R.

Multiplying (<ref>) by ∂_θΩ^cor_k and integrating by parts yields

    1/2∂_τ‖∂_θΩ^cor_k‖_L^2^2=b(a-1)‖∂_θΩ^cor_k‖_L^2^2+∫_^2ℋ_1∂_θΩ^cor_k X
    ≤‖ℋ_1‖_L^2‖∂_θΩ^cor_k‖_L^2,

where we have applied that divX=2 and a<1. Thus,

    ‖∂_θΩ^cor_k(τ)‖_L^2≤∫_-k^τ‖ℋ_1(s)‖_L^2 s.

Then, (<ref>) follows by applying Lemmas <ref> and <ref> since

    ℋ_1(s)_L^2≤ C e^2λ s,

for all -k≤ s≤τ_k.
For (<ref>), multiplying (<ref>) by 1/R yields

    ∂_τ∂_θΩ^cor_k/R - b(a+ 1)∂_θΩ^cor_k/R+(V̅-bX+𝒱)·∇∂_θΩ^cor_k/R   
    
    =
    1/R𝒱_R∂_θΩ^cor_k/R
    +1/Rℋ_1   =:ℋ_2,

where 𝒱_R=𝒱· e_R, and we have applied that V̅=V̅_θ e_θ, X=Re_R and
∇(1/R)
=-1/R^2e_R.
After multiplying (<ref>) by (1/R∂_θΩ^cor_k)^3 and integrating by parts, we get

    1/4∂_τ‖1/R∂_θΩ^cor_k‖_L^4^4
       =b(a+1/2)‖1/R∂_θΩ^cor_k‖_L^4^4+∫_^2ℋ_2(1/R∂_θΩ^cor_k)^3 X
       ≤ b(a+1/2)‖1/R∂_θΩ^cor_k‖_L^4^4+‖ℋ_2‖_L^4‖1/R∂_θΩ^cor_k‖_L^4^3.

Therefore,

    ‖1/R∂_θΩ^cor_k(τ)‖_L^4≤∫_-k^τe^b(a+1/2)(τ-s)‖ℋ_2(s)‖_L^4 s.

Then (<ref>) follows by applying Lemmas <ref> and <ref> since

    ℋ_2(s)_L^4≤ Ce^12(3λ+δ_0)s,

for all -k≤ s≤τ_k.
Notice that we need 
b(a+1/2)≪1/2(3λ+δ_0),
but this is satisfied since Proposition <ref> guarantees λ>3b.




  §.§.§ Estimates on the radial derivative

 
There exists C>0 such that

    ‖ R∂_RΩ^cor_k(τ)‖_L^2   ≤ Ce^2λτ,
    ‖∂_RΩ^cor_k(τ)‖_L^4   ≤ Ce^12(3λ+δ_0)τ,

for all  -k≤τ≤τ_k.


For (<ref>), by differentiating (<ref>) with respect to R, we get

    ∂_τ∂_R Ω^cor_k - b(a+1)∂_RΩ^cor_k+(V̅-bX+𝒱)·∇∂_RΩ^cor_k   
    
    =
    -∂_R𝒱_R∂_RΩ_k^cor
    -(∂_R𝒱_θ-𝒱_θ/R)∂_θΩ^cor_k/R
    -(∂_RV̅_θ-V̅_θ/R)∂_θΩ^cor_k/R
    -∂_R𝒢   =:ℋ_3.

where we have applied (<ref>)

    ∂_R(V̅·∇Ω^cor_k)
       =V̅·∇∂_RΩ^cor_k
    +(∂_RV̅_θ-V̅_θ/R)∂_θΩ^cor_k/R
    ,
    ∂_R(X·∇Ω^cor_k)
       =X·∇∂_RΩ^cor_k
    +∂_RΩ^cor_k,

and also

    ∂_R(
    𝒱·∇Ω_k^cor)
    
    
       =𝒱·∇∂_θΩ_k^cor
    +∂_R𝒱_R∂_RΩ_k^cor
    +(∂_R𝒱_θ-𝒱_θ/R)∂_θΩ^cor_k/R.

After multiplying (<ref>) by R we get

    ∂_τ (R∂_R Ω^cor_k) - a b R∂_RΩ^cor_k+(V̅-bX+𝒱)·∇(R∂_RΩ^cor_k)   
    
    =𝒱_R∂_R Ω^cor_k
    +Rℋ_3
       =:ℋ_4,

where we have applied that V̅_R=0, X=Re^iθ and
∇ R=e_R.
Multiplying (<ref>) by R∂_RΩ^cor_k and integrating by parts yields

    1/2∂_τ‖ R∂_RΩ^cor_k‖_L^2^2=b(a-1)‖∂_RΩ^cor_k‖_L^2^2+∫_^2ℋ_4 R∂_RΩ^cor_k X
    ≤‖ℋ_4‖_L^2‖ R∂_RΩ^cor_k‖_L^2,

and thus,

    ‖ R∂_RΩ^cor_k(τ)‖_L^2≤∫_-k^τℋ_4(s)_L^2 s.

For (<ref>), by multiplying (<ref>) by (∂_R Ω^cor_k)^3 and integrating by parts we get

    1/4∂_τ∂_RΩ^cor_k_L^4^4
       = b(a+1/2)∂_RΩ^cor_k_L^4^4
    +∫_^2ℋ_3
    (∂_RΩ^cor_k)^3 X
       ≤
     b(a+1/2)∂_RΩ^cor_k_L^4^4
    +ℋ_3_L^4∂_RΩ^cor_k_L^4^3,

and thus

    ‖∂_RΩ^cor_k(τ)‖_L^4≤∫_-k^τe^b(a+1/2)(τ-s)ℋ_3(s)_L^4 s.

Similarly to Lemma <ref>, we conclude the proof by applying Lemmas <ref>  and <ref>, but now coupled with Lemma <ref>.
This was the reason for estimating the angular derivative before.









§ PROOF OF FORMULA (<REF>)


We start by writing (z=e^iϑ)

    K_n(ρ)
    =1/2π∫_0^2πsin(ϑ)sin(nϑ)/1+ρ^2-2ρcos(ϑ)ϑ
    =1/8π i ρ∫_
    f(z)
     z,

where f is the meromorphic function

    f(z)
    =(z^2-1)(z^2n-1)/z^n+1(z-ρ)(z-ρ^-1).

Hence, by the Residue Theorem we have

    K_n(ρ)
    =1/4ρ∑_z=0,ρ,ρ^-1Ind(,z)
    Res(f,z).

We have

    Ind(,0)=1,
        Ind(,ρ)=1_(0,1)(ρ).

It is easy to compute
the residue for the simple poles

    Res(f,ρ)
       =(ρ^2-1)(ρ^2n-1)/ρ^n+1(ρ-ρ^-1)
    =ρ^n-ρ^-n,
    Res(f,ρ^-1)
       =(ρ^-2-1)(ρ^-2n-1)/ρ^-(n+1)(ρ^-1-ρ)
    =ρ^-n-ρ^n.

For the multiple pole z=0 we write the Maclaurin series of f

    f(z)
    =(z^2-1)(z^2n-1)/z^n+1g(z)
    =(z^n+1-z^n-1-1/z^n-1+1/z^n+1)g(z),

where

    g(z)=1/(z-ρ)(z-ρ^-1)
    =1/ρ-ρ^-1(
    1/z-ρ-1/z-ρ^-1)
    =1/ρ-ρ^-1∑_j=0^∞
    (ρ^j+1-ρ^-(j+1))z^j.

Hence, for any n≥ 1, we deduce that

    Res(f,0)
    
    
    =ρ^n+ρ^-n.

By plugging everything together we get (<ref>).



 §.§ Acknowledgments

The authors thank all participants of the reading seminar on the uniqueness or lack thereof for the Navier-Stokes and Euler equations, which took place in Madrid during the spring of 2023.
D.F. and F.M. acknowledge the hospitality and financial support of the Institute for Advanced Study in Princeton during various periods in 2021-2022, as well as the discussions held there with Camillo De Lellis and his group on topics related to Vishik's construction. 

A.C.  acknowledges financial support from Grant PID2020-114703GB-I00 funded by MCIN /AEI/ 10.13039/501100011033, Grants RED2022-
134784-T and RED2018-102650-T funded by MCIN/ AEI/ 10.13039/501100011033 and by a 2023
Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The BBVA Foundation
accepts no responsibility for the opinions, statements, and contents included in the project and/or
the results thereof, which are entirely the responsibility of the authors.
A.C. and D.F.  acknowledge financial support from the Severo Ochoa Programme
for Centres of Excellence Grant CEX2019-000904-S funded by MCIN/ AEI/ 10.13039/501100011033. D.F. acknowledges the financial support of QUAMAP and the ERC Advanced Grant 834728. D.F., F.M. and M.S. acknowledge support from PI2021-124-195NB-C32. D.F. and F.M. acknowledge support from CM through the Line of Excellence for University Teaching Staff between CM and UAM.
F.M. acknowledges support from the 
Max Planck Institute for Mathematics in the Sciences. 
M.S. acknowledges support from PID2022-136589NB-I00 founded by MCIN/AEI/10.13039/501100011033/ and by FEDER (Una manera de hacer Europa) and from “Conselleria d'Innovació, Universitats, Ciència i Societat Digital”: project AICO/2021/223 and programme “Subvenciones para la contratación de personal investigador en fase postdoctoral” (APOSTD 2022),  Ref. CIAPOS/2021/28.





abbrv





	Ángel Castro

	Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM

		28049 Madrid, Spain

	E-mail address: angel_castro@icmat.es 





	Daniel Faraco

	Departamento de Matemáticas, Universidad Autónoma de Madrid, Instituto de
Ciencias Matemáticas, CSIC-UAM-UC3M-UCM

		28049 Madrid, Spain

	E-mail address: 
daniel.faraco@uam.es 





	Francisco Mengual

	Max Planck Institute for Mathematics in the Sciences

		04103 Leipzig, Germany

	E-mail address: fmengual@mis.mpg.de





	Marcos Solera

	Departament d’Anàlisi Matemàtica, Universitat de València

		46100 Burjassot, Spain

	E-mail address: marcos.solera@uv.es




