Nonuniqueness of Solutions to the Euler Equations with Vorticity in a Lorentz Space

For the two dimensional Euler equations, a classical result by Yudovich states that solutions are unique in the class of bounded vorticity; it is a celebrated open problem whether this uniqueness result can be extended in other integrability spaces. We prove in this note that such uniqueness theorem fails in the class of vector fields u with uniformly bounded kinetic energy and vorticity in the Lorentz space L1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1, \infty }$$\end{document}.


Introduction
Let us consider the 2-dimensional Euler equation where u : [0, 1] × T 2 → R 2 is the velocity of a fluid and p : [0, 1] × T 2 → R the pressure.This system can be equivalently rewritten as the two dimensional Euler system in vorticity formulation, which is a transport equation for the vorticity ω = curl(u), i.e.
In the latter formulation it is clear that L p norms of the vorticity are formally conserved for any p ∈ [1, ∞].
For p > 1, this was used in [11] to prove existence of distributional solutions starting from an initial datum with vorticity in L p .A similar existence result is much more involved for p = 1, and it was obtained by Delort [10] (see also [11,12]), improving the existence theory up to measure initial vorticities in H −1 (this latter condition guarantees finiteness of the energy) whose positive (or negative) part is absolutely continuous.As regards uniqueness, the classical result of Yudovich [15,16] (see also the proof in [17]) states that, given an initial datum ω 0 ∈ L ∞ , there exists a unique bounded solution to (2) starting from ω 0 .However, the classical problem raised by Yudovich about the sharpness of his result is still open.Let u 0 be an initial datum in L 2 with curl u 0 in some function space X.Is the solution of the Euler equation in vorticity formulation unique in the class L ∞ (X)?
The main result of this paper provides a negative answer when X is the Lorentz space L 1,∞ .
Recently, there have been formidable attempts to disprove this conjecture for X = L p , none of which has by now fully solved it.Vishik [22,23] proposed a complex line of approach to this problem, which however has the price of showing nonuniqueness only with an additional degree of freedom, namely a forcing term in the right-hand side of the equation (2) in the integrability space L 1 (L p ).The nonuniqueness suggested 1 MSC classification: 35F50 (35A02 35Q35).Keywords: Euler Equation, vorticity formulation, convex integration, uniqueness.
by this work is of symmetry breaking typeand, in contrast with the ideas of this paper, his nonuniqueness stems from the linear part of the equation, by carefully choosing an initial datum that sees the instability directions of a linearized operator.
A second attempt has been pursued by Bressan and Shen [2], based on numerical experiments which share the symmetry breaking type of nonuniqueness of Vishik.Their work is a first step in the direction of a computer assisted proof.
Our approach is instead of different nature and stems from the convex integration technique.The latter was introduced by De Lellis and Székelyhidi [9] in the context of nonlinear PDEs, inspired by the work of Nash on isometric embeddings [20], which found striking applications in recent years to different PDEs (see for instance [5-7, 14, 18, 19] and the references quoted therein).As such, our proof would probably be less constructive with respect to the strategies of [22,23] and [2], where an initial datum for which nonuniqueness is expected is described fairly explicitly as well as the mechanism for the creation of two different singularities.Conversely, the latter approaches see the drawbacks described above and are by no means "generic" in the initial data, whereas it is known (see for instance [8,21]) that convex integration methods yield not only the lack of uniqueness/smoothness for certain specific initial data, but also that solutions are typical (in the Baire category sense).
1.1.Strategy of proof.The guiding thread of this construction is an iterative procedure, where one starts from a solution (u 0 , p 0 , R 0 ) of the Euler equations with an error term in the right-hand side, namely and iteratively corrects this error by adding a fastly oscillating perturbation to the approximate solution.
The nonlinear interaction of this perturbation with itself generates a resonance which allows for the cancellation of the previous error; the other terms are mainly seen as new error terms, with smaller size with respect to the previous error.More precisely, we define the new solution (u 1 , p 1 , R 1 ) by setting where λ ≫ 1 is a higher frequency with respect to the typical frequencies in u 0 , w is called building block of the construction and enjoys suitable integrability properties, a is a slowly varying coefficient.The cancellation of error happens because the low frequency term in a 2 w λ ⊗ w λ satisfies This forces us to require that On the contrary, we wish to control the quantity Du 1 X and for this end we need Dw λ X arbitrarily small.This imposes us a restriction on the space X since the Sobolev inequality in Lorentz spaces (see [1]) states that giving that ∇w λ L 1,2 = λ ∇w L 1,2 λ w L 2 ∼ λ ≫ 1 , when applied with p = 1 and q = 2.In particular, with the current method of proof (and in particular with the current way to cancel the error in the iteration), X = L 1 or X = L 1,2 are not allowed; only X = L 1,q for q > 2 could be obtained.To avoid technicalities, we present the proof with X = L 1,∞ .
The main novelty in the proof of Theorem 1.1 regards the construction of a new family of building blocks.They are designed as a bundle of almost solutions to Euler, suitably rescaled and periodized in order to saturate the L 1,∞ norm.To this aim we take advantage of intermittent jets, introduced in [4], and we bundle them in a similar spirit to the atomic decomposition of Lorentz functions.A challenge is to keep different building blocks disjoint in space-time, since we work in two dimensions and since each component of the bundle has its own characteristic speed.We refer the reader to Section 4 for the precise construction and more explanations on our choice of building blocks.
Remark 1.2.The proof of Theorem 1.1 is flexible enough, due to the exponential convergence of the iterative sequence, to give ω ∈ L 1,q for some q ≫ 1.A technical refinement of the current proof, based on Remark 4.4, would give q > 4.
Acknowledgments.EB was supported by the Giorgio and Elena Petronio Fellowship at the Institute for Advanced Study.MC was supported by the SNSF Grant 182565.The author wish to thank Camillo De Lellis for interesting discussions on the theme of the paper.

Iteration and Euler-Reynolds system
We consider the system of equations (3) in [0, 1] × T, where R is a traceless symmetric tensor.
As already remarked, our solution to (1) is obtained by passing to the limit solutions of (3) with suitable constraints on u and R. The latter are built by means of an iterative procedure based on the following.Proposition 2.1.There exists M > 0 such that the following holds.For any smooth solution (u 0 , p 0 , R 0 ) of (3), there exists another smooth solution (u 1 , p 1 , R 1 ) of (3) such that Proof of Theorem 1.1 given Proposition 2.1.Fix λ > 0. We start the iteration scheme with Applying iteratively Proposition 2.1 with t 0 = 1/2 we build a sequence {(u n , p n , R n ) : n ∈ N} of smooth solutions to (1) such that, for any n ≥ 0, it holds and , where u satisfies the assumptions of Theorem 1.1.To prove that Du ∈ C 0 (L 1,∞ ), a bit of extra care is needed since only the weak triangle inequality f ∞ holds true.However, the latter is enough for our purposes The remaining part of this note is devoted to the proof of Proposition 2.1.In Section 4 we introduce the building blocks of our construction, in Section 5 we use them to define the perturbation u 1 − u 0 , finally in Section 6, we introduce the new error term R 1 and show that it can be made arbitrarily small.

Preliminary lemmas
3.1.Lorentz spaces.For every measurable function f : (see e.g.[13]) and we define the Lorentz space L r,q with r ∈ [1, ∞), q ∈ [1, ∞], as the space of those functions f such that f L r,q < ∞.Note that, in spite of the notation, • L r,q is in general not a norm but for (r, q) = (1, ∞) the topological vector space L r,q is locally convex and there exists a norm ||| • ||| r,q which is equivalent to • L r,q in the sense that the inequality C −1 |||f ||| r,q ≤ f L r,q ≤ C|||f ||| r,q holds.3.2.Improved Hölder inequality.We recall the following improved Hölder inequality, stated as in [18,Lemma 2.6] (see also [3,Lemma 3.7]).If λ ∈ N and f, g : T 2 → R are smooth functions, then we have ( When T 2 g = 0, then Here Sym 2 denotes the space of symmetric matrices in and that DR 0 is a Calderon-Zygmund operator, in particular it holds ) Notice that ( 7) and ( 8) allow showing that where v λ (x) := v(λx) for some λ ∈ N. The latter is immediate for p in the case p = 1 and p = ∞ we need to take advantage of the Sobolev embedding theorem: To prove (10) we use ( 9) and ( 6): Remark 3.2.The operator R can be also defined on scalar functions f : and arguing as in Lemma (3.1) we can easily show that div Proof.Set T (A) := R(∇a • div A).By duality, it suffices to show that , where T * and R + 0 denote the adjoint of T and R 0 , respectively.To this aim we employ the Sobolev embedding and the fact that DT * R * 0 (B) maps L p into L p for any p ∈ (1, ∞):

Building blocks
In this section we introduce the building blocks of our construction.They will be employed in Section 5 to define the principal term of u 1 − u 0 in Proposition 2.1.
(iv) the following estimates hold i is the principal term, it has zero mean, high frequency λ ≥ ε −1 , is controlled in the relevant norms (cf.(iv)), and satisfies the fundamental property (iii): the quadratic interaction W p i ⊗ W p i produces the lower order term , is used to cancel the error R 0 out.To achieve the crucial bound DW p i L 1,∞ we design the principal term as where K, n 0 ≫ 1 are big parameters and ξ i is one of the four directions appearing in the statement of Proposition 4.1.In a first stage, we build W p i (x, t) for a fixed parameter i, ignoring the issue that, for different parameters, such functions will not have disjoint support as requested in Proposition 4.1 (v); only in Section 4.6 we make sure to suitably time-translate them, making substantial use of their special structure, to guarantee that Proposition 4.1 (v) holds .The vector fields W k (x, t), k = n 0 + 1, . . ., n 0 + K, are the 2-dimensional counterpart of the intermittent jets introduced in [4].They have L 2 norm equal to 1, and are supported on disjoint balls of radius 2 −k r, for some r ≪ 1, which move in direction e i with speed µ2 k , where µ ≫ 1.The fast time translation is used to make W k "almost divergence free" and "almost solutions to the Euler equation".In more rigorous terms, it means that there exist vector fields W p k , Q k , that are smaller than W k satisfying div (W k + W p k ) = 0 and . The vector fields W p i and Q i are defined bundling together W p k and Q k as we did in (12).Another important property we need is that W i ⊗W j = 0 when i = j.It is ensured by (iv) in Proposition 4.1, which builds upon a delicate combinatorial lemma presented in section 4.6.
We finally explain the role of the matrix A i in our construction.Let us begin by noticing that the principal term W p i has big time derivative, being fast translating in time.Hence, the term ∂ t W p i cannot be treated as an error.To overcome this difficulty we impose an extra structure on W p i and W c i .We construct them in order to have the identity ∂ t (W p i + W c i ) = div (A i ), for some symmetric matrix A i which has small L 1 -norm.The latter can be added to the new error term R 1 .

General notation. Given a velocity field
Let us fix r ⊥ ≪ r ≪ 1 and k ∈ N. We adopt the following convention: given any ρ : R → R supported in (−1, 1) we write With a slight abuse of notation we keep denoting by ρ k r ⊥ , ρ k r : T → R their periodized version.

Time correction. Let us now set
and observe that The time corrector is defined as The proof of Lemma 4.2 is a simple computation, so we omit it.It implies the following, summing on k and remininding that then terms in the sum in (14) have disjoint support, (in particular, this says that the principal part is much smaller than the corrector), and
We claim the following: suppose that for a certain t > 0 and k ∈ {n 0 , . . ., n 0 +K} we have suppW k for every h ∈ {n 0 , . . ., n 0 + K}.The previous claim excludes the simultaneous presence at any t > 0 of the support of W k (ξ 1 ) (•, t) and the support of W h (ξ 2 ) (•, t + t 0 ) in B R (0), thereby concluding the proof of the lemma.We now prove the claim.Let us fix a time t such that suppW k ) is moving at constant speed µ2 k along the tube on the torus, there exists t such that |t − t| ≤ Rµ −1 2 −k and suppW k (ξ 1 ) (•, t) = suppW k (ξ 1 ) (x, 0).At time t we have information about the position of suppW h (ξ 2 ) (•, t + t 0 ); more precisely, we have that because the ratio between the (constant) velocity of suppW k (ξ 1 ) (•, t) and the velocity of suppW k (ξ 2 ) (•, t) is of the form 2 j for some j ∈ {−K, ..., K}.
In the union in the right-hand side of ( 23), thanks to the upper bound on t 0 , the choice n = 0 identifies the ball of the (finite) union at minimal distance from the origin for every k.By the lower bound on t 0 and the fact that the minimal velocity is µ2 n 0 , we get that this distance is greater than 2 n 0 −7K .At time t the distance between suppW h (ξ 2 ) (•, t + t 0 ) and B R (0) is therefore bigger than This concludes the proof of the claim.
,K,n 0 and A i+1 := A ξ,K,n 0 satisfy (v) in Lemma 4.1.We refer the reader to Lemma 4.5 for the construction of A ξ,K,n 0 .Properties (i) and (ii) in Lemma 4.1 are now immediate from (15), (16) and Lemma 4.5.We are left with the proof of (iii) and (iv) in Lemma 4.1.To do so we have to choose appropriately the parameters λ, µ, K, r ⊥ and r .Let δ < 1/2 to be chosen later in terms of ε > 0, we set leaving r ⊥ ≪ r ≪ 1 free.From Lemma 4.3, Lemma 4.5, ( 17), ( 18) and ( 19) we deduce The conclusions (iii) and (iv) in Lemma 4.1 follow by choosing first δ small enough so that Cδ ≤ ε, and after r ⊥ ≪ r ≪ 1 so that r ⊥ r ≤ ε and
For ε > 0 to be chosen later, we consider the functions W p i , W c i , Q i , A i from Proposition 4.1.We define the new velocity field as the sum of the previous one, a principal perturbation, a divergence corrector and a temporal corrector 1 , where We refer the reader to Remark 3.2 for the definition of R.
From now on, in order to simplify our notation, for any function space X and any map f which depends on t and x, we will write f X meaning f L ∞ (X) .
5.1.Estimate on u 1 − u 0 L 2 and on u 1 − u 0 L 1 .By the triangular inequality, and we estimate the right-hand side separately as where in the second line we used the improved Holder inequality ( 5) and (iii) in Proposition 4.1.From Remark 3.2 we deduce Finally we employ (iv) in Proposition 4.1 to get we estimate the right-hand side separately as where we employed (iv) in Proposition 4.1.Using that DR is a Calderon-Zygmund operator we deduce

New error
We define R 1 in such a way that which, by subtracting the equation for u 0 , is equivalent to We are going to define 1 , where the various addends are defined in the following paragraphs, and show that ) .The proof of Proposition 2.1 will follow by choosing ε small enough.
thanks to (25) it holds Using that for some pressure term P , it is immediate to verify the identity Since R and R 0 send L 1 to L 1 (cf.Lemma 3.1 and Remark 3.2), we have that From (iv) in Proposition 4.1 we get .
By employing (11) we bound 6.3.Quadratic error terms.Let us set and show that (26) holds.In view of ( 27), ( 24) and (28) it amounts to check that div (R The latter easily follows by noticing that, as a consequence of (ii) in Proposition 4.1, one has Let us finally prove that R (q) 1 L 1 ≤ εC(t 0 , R 0 C 2 ).We begin by observing that 1 L 2 , u 1 −u 0 L 2 and Lemma 3.1 we deduce