Lagrangian solutions to the 2D Euler system with L1 vorticity and infinite energy

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Lagrangian solutions to the 2D Euler system with L1 vorticity and infinite energy

Anna Bohun, François Bouchut, Gianluca Crippa

To cite this version:

Anna Bohun, François Bouchut, Gianluca Crippa. Lagrangian solutions to the 2D Euler system with L1 vorticity and infinite energy. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2016, 132, pp.160-172. �10.1016/j.na.2015.11.004�. �hal-01184975�

WITH L¹ VORTICITY AND INFINITE ENERGY

ANNA BOHUN, FRANC¸ OIS BOUCHUT, AND GIANLUCA CRIPPA

Abstract. We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under the only assumption ofL¹ weak convergence of the initial vorticity. The exis- tence of Lagrangian solutions to the Euler system follows for arbitraryL¹ vorticity. Relations with previously known notions of solutions are established.

Keywords: Euler system, L1 vorticity, Lagrangian solutions, symmetrized solutions, infinite energy

1. Introduction

The incompressible Euler equations for a two-dimensional inviscid fluid are given by

$

'

&

'

%

Btv div^pv^bv^q ∇p0, v^p0,^qv^0px^q,

divv0,

(1.1) wherev^pt, x^qis the velocity of particles at positionxand timet, andp^pt, x^qthe scalar pressure, that sustains the incompressibility constraint divv 0. The two-dimensional incompressible Euler equations may be rewritten as a transport equation for the scalar vorticity ω defined by

ωcurlv^B₁v₂^B₂v₁, (1.2)

which is advected by the velocity v. This gives the vorticity formulation

#

Btω div^pωv^q0,

ω^p0,^qω^0px^q, (1.3)

withω⁰ curlv⁰. The coupling (1.2) can alternatively be written via the Biot-Savart convolu- tion law

v^pt, x^q 1 2π

»

R²

pxy^qK

|xy^|2 ω^pt, y^qdy K

xω, (1.4)

where we denote by ^px1, x2^qKpx2, x1^q and by K^px^q 1

2π x^K

|x^|2 1 2π

x₂

|x^|2, x₁

|x^|2

(1.5) the Biot-Savart kernel.

In this paper we deal with the existence and stability of infinite kinetic energy solutions associated to initial vorticities lying inL^1pR^2q. In this context, because of the lack of (even local) kinetic energy bound, the velocity formulation (1.1) cannot be given the usual distributional meaning (see Definition 3.1). Though, a symmetrized velocity formulation can be used, see Definition 3.2.

For vorticities ω^PL^8pp0, T^q;L^1pR^2qq, the decomposition

vK1ω K2ω, (1.6)

1

where K1 K1_B1p0^{q P}L^1pR^2q and K2 K1_B1p0^{qc P}L^8pR^2q, gives immediately with Young’s inequality that v ^PL^8pp0, T^q;L^1pR^2qq L^8pp0, T^q;L^8pR^2qq. Nevertheless, as for the velocity formulation, no direct distributional formulation is available for the vorticity equation (1.3), since the factors in the product ωv are not summable enough to define a locally integrable product. The symmetrized formulation can however be used again. More generally, one can consider three alternate formulations of weak solutions for the vorticity equation, defined as follows.

(1) Renormalized solutions [11], defined by the requirement that β^pω^q is a distributional solution to the transport equation (1.3) for a suitable class of functionsβ:

#

Bt^pβ^pω^qq div^pβ^pω^qv^q0,

β^pω^qp0,^qβ^pω^0q, (1.7)

(2) Symmetrized vorticity solutions [10, 22, 20], defined by exploiting the antisymmetry of the Biot-Savart kernelK, so that multiplying (1.3) by a test functionφand integrating gives the formulation

»T 0

»

R²

Btφ^pt, x^qω^pt, x^qdxdt

»T 0

»

R²

»

R²

Hφ^pt, x, y^qω^pt, x^qω^pt, y^qdxdydt

»

R²

φ^p0, x^qω^0px^qdx0, (1.8) whereH_φis the bounded function

H_φ^pt, x, y^q 1

2K^pxy^q ∇φ^pt, x^q∇φ^pt, y^q

, (1.9)

(3) Lagrangian solutions, i.e. solutions ω transported by a suitable flow associated to the velocityv, to be precisely defined in the sequel.

The notions (1) and (2) of solutions have been considered previously, but no study has been made concerning (3) in our low regularity context ω ^PL¹. In this paper we prove that initial vorticities in L¹ give rise to well-defined weak solutions that are transported by flows. The key point of our strategy relies on a priori error estimates (under bounds that are natural in our setting) for a class of flows which are measure preserving, called regular Lagrangian flows.

These estimates were developed in [8]. The novelty of this approach for the Euler equations, in contrast with [10, 22, 18, 15], is that it entirely relies on the Lagrangian formulation, and therefore proves existence of solutions which are naturally associated to flows. In this setting we also allow for velocities with locally infinite kinetic energy.

The usual strategy for proving existence of solutions to (1.1) is by smoothing the initial data, and using estimates that enable passing to the limit in the weak formulation. For initial velocities belonging to H^s, s ^¡2, well-posedness of classical solutions is due to Wolibner [24].

Existence and uniqueness of solutions to (1.1) is known for vorticities inL^1XL⁸, and was first proved by Yudovich [25]. For compactly supported initial vorticities in L^p, with 1   p   ⁸, existence was first proved by DiPerna and Majda [12]. The proof relies on suitable Sobolev embeddings, that guarantee strong convergence inL²_loc on the approximate velocities.

On the other hand, while a uniform L¹ bound on the vorticities is still sufficient to guar- antee the L¹_loc convergence of the smoothed velocities, it is generally insufficient for the strong convergence in L²_loc, see for instance Example 11.2.1 in [19]: the approximate velocities may concentrate. However, concentrations may occur for sequences whose limit still satisfies (3.1), in spite of the lack of strongL²_loc convergence: this is referred to as concentration-cancellation and has been studied in [14, 19]. This happens for instance if the vorticity is a measure with distinguished sign [10]. The key point in proving that concentration-cancellations occur is to prove distributional convergence of the antisymmetric quantitiesv¹_nv_n² andv_n¹v_n². ForL¹ vor- ticities with compact support, without necessarily distinguished sign, and initial velocities with locally finite kinetic energy, the propagation of the equi-integrability guarantees concentration- cancellations [22]. However, these solutions were not proved to be Lagrangian, and only weak L¹ convergence was obtained onωeven for strongly convergent initial data. This is nevertheless

sufficient to pass to the limit in the symmetrized formulation (1.8).

A stability estimate for flows associated to velocity fields with gradient given by the sin- gular integral of an L¹ function was derived in [8], building on previous results in [9]. This regularity of the field is weaker than the one classically used, namely W^1,1 or BV [11, 1]. Our assumptions in the context of Euler equations fall under the theory in [8]. From this theory it follows that Lagrangian flows associated to velocities whose curl are equi-integrable are strongly precompact, and thus stable under approximation, so that the limit flow solves the ODE with the limit velocity. We shall therefore conclude that vorticities in L¹ are strongly stable under approximation, in the sense that if ω_n⁰ converges strongly in L¹ to ω⁰, then the solution ω_n of the corresponding vorticity formulation converges strongly in L¹ to a Lagrangian solution ω.

Additionally, even for weakly convergent initial vorticities, the flow always converges strongly.

The main results of this paper were announced in [7].

A classical difficulty in proving strong compactness is related to time oscillations. Indeed, when dealing with velocity formulations, the strong compactness in space follows from the L¹ bound on the vorticity, but the compactness in time relies on bounds on ^B_tv_n in L⁸_t ^pD_x¹q in order for Aubin-Lions’ lemma to apply. Without the assumptionv ^PL²_loc, we do not have such regularity in time ofvand we cannot apply Aubin-Lions’ lemma. We thus propose a refinement of the stability estimates in [8] so that weak time convergence of the velocities is still sufficient for the stability of regular Lagrangian flows. We nevertheless prove a posteriori the strong compactness ofv in time and space.

Main notations. We setB_R:B_R^p0^q. We denote byL^0pR^dqthe space of all measurable real valued functions on R^d, defined a.e. with respect to the Lebesgue measure, endowed with the convergence in measure. We denote byL⁰_loc^pR^dqthe same space, endowed with local convergence in measure (see definition below). The space logL^pR^dq contains all functionsu :R^dÑ Rsuch that

³

R^dlog^p1 ^|u^px^q|qdx   ⁸, with logL_loc^pR^dq defined accordingly. We refer to BpE, F^q as the space of bounded functions between sets E and F. We also introduce the following seminorm:

Definition 1.1. Letu be a measurable function on Ω^€R^d. For 1^¤p ⁸, we set

|||u^|||p_Mp

pΩ^qsup

λ^¡0

!

λ^pL^d tx^PΩ : ^|u^px^q|¡λ^u

)

and define the weak Lebesgue space M^ppΩ^q as the space consisting of all such measurable functions u: Ω^ÑRwith^|||u^|||_M^ppΩ^q 8. For p⁸, we setM^8pΩ^qL^8pΩ^q.

Definition 1.2. We say that a sequence of measurable functions un :R^dÑR converges locally in measure in R^dto a measurable function u:R^dÑRif for every γ^¡0 and everyr ^¡0 there holds

L^Nptx^PBr:^|un^px^qu^px^q|¡γ^uqÑ0, n^Ñ8. 2. Regularity of the velocity field

We summarize in the present section some integrability and regularity estimates for the vector fieldv given by (1.4), whenω ^PL^1pR^2q.

(I) The Biot Savart kernelK belongs toL¹_loc^pR^2q and has distributional derivatives given by the following singular kernels. Fori, j 1,2, we have

BjK^ipx^qB_j 1 2π

x2

|x^|2, x1

|x^|2

i

. (2.1)

The Fourier transform of (2.1) is bounded and is given by

z

BjK^ipξ^qξ_j

ξ2

|ξ^|2, ξ1

|ξ^|2

i

PL^8pR^2q. (2.2)

It is well-known that the operators S_jⁱ of convolution with such kernels (2.1) defined by S_jⁱu ^BjKⁱ u extend to bounded operators on L^2pR^2q, and have bounded extensions

on L^ppR^2q for 1   p   ⁸. For p 1 and u ^P L^1pR^2q, Sⁱ_ju is a tempered distribution S_jⁱu^PS¹pR^2q defined via the formula

xS_jⁱu, ϕ^yxu,S˜_jⁱϕ^y ϕ^PSpR^2q, (2.3) where ˜S_jⁱ is the singular integral operator associated to the kernel^pB_jK^iqpx^q. Thus for v given by (1.4) withω^PL^1pR^2q and fori, j 1,2, we have

p∇v^q_ij ^B_jvⁱ S_jⁱω ^PS¹pR^2q. (2.4) (II) From (2.4) it follows that divv0 in D¹, and curlvω inD¹.

(III) We have from (1.6) that vorticities bounded in L^8pp0, T^q;L^1pR^2qq are associated to ve- locities bounded inL^8pp0, T^q;L^1pR^2qq L^8pp0, T^q;L^8pR^2qq.Moreover, the weak Hardy- Littlewood-Sobolev inequality (see Lemma 4.5.7 in [16]) gives that

}v^}_L⁸pp0,T^q;M^2pR^{2qq ¤}c^}ω^}_L⁸pp0,T^q;L^1pR^2qq, (2.5) which implies in particular the embeddingv^pt, x^qPL^p_loc^pr0, T^sR^2qfor any 1^¤p 2.

3. Weak solutions

Several weak formulations can be considered. If the velocity has locally finite kinetic energy, v^PL²_loc^pR^2q, the usual weak formulation of (1.1) is available:

Definition 3.1 (Weak velocity formulation). We say that v ^P L^8pp0, T^q;L²_loc^pR^2qq is a weak solution of the Euler velocity formulation with initial datum v^{0 P} L²_loc^pR^2q if for all φ^pt, x^{q P} C_c^1pr0, T^qR²,R^2q with divφ0, there holds

»T

0

»

R²

Btφv ∇φ: v^bv

dxdt

»

R²

φ^p0, x^qv^0px^qdx0, (3.1) and v is divergence free in distributional sense.

3.1. Symmetrized velocity solutions. In order to deal with solutions with locally infinite kinetic energy we can propose a weaker formulation than the one in Definition 3.1. It is in the same spirit as the symmetrized vorticity formulation (1.8). Using the identity div^pv^bv^q v∇vω v^K ∇^|v|2

2 , that is valid when divv0, we can formally rewrite (1.1) as

Btv ωv^K ∇p¹0, (3.2)

wherep¹p ^|v₂^|2. This modified pressurep¹ can be eliminated by taking suitable test functions as in (3.1). With this form (3.2) we can observe that only the quantitiesv₁v₂ andv₁²v₂² need to be in L¹, since we can write ωv^K div^pv^bv^p|v^|2{2^qId^q, and the entries of the matrix v^bv^p|v^|2{2^qIdare just these two scalarsv1v2 andv₁²v²₂. However, without such assumptions, we observe that the term ωv^K has a priori no pointwise meaning when ω only belongs to L^p for some p   4^{3, since in such a case ω and v would not have conjugate summabilities.

Nevertheless, with the only assumptionω^PL¹, that yields v^PM² (butv^RL²_loc in general), we can give a meaning in distribution sense to this term by exploiting the symmetrization technique analog to that in [10, 22], that uses the antisymmetry property K^px^qK^px^q.

Letφ^PC_c^1pr0, T^qR²,R^2q. Then using the Biot-Savart law we can write

»T 0

»

R²

pωv^Kqpt, x^qφ^pt, x^qdxdt

»T 0

»

R²

»

R²

ω^pt, x^qω^pt, y^qK^pxy^qKφ^pt, x^qdxdydt

»T

0

»

R²

»

R²

ω^pt, y^qω^pt, x^qK^pxy^qKφ^pt, y^qdxdydt

1 2

»T

0

»

R²

»

R²

ω^pt, x^qω^pt, y^qK^pxy^qK φ^pt, x^qφ^pt, y^q

dxdydt

»T 0

»

R²

»

R²

ω^pt, x^qω^pt, y^qH¯_φ^pt, x, y^qdxdydt,

(3.3)

where ¯H_φ^pt, x, y^q is the function on^r0, T^qR²R² given by H¯_φ^pt, x, y^q 1

2K^pxy^qK φ^pt, x^qφ^pt, y^q

. (3.4)

For φ ^P C_c^1pr0, T^qR²,R^2q we have that ¯H_φ is a bounded function, continuous outside the diagonal, that tends to zero at infinity. Indeed we have

|H¯_φ^pt, x, y^q|¤ 1

4πLip^pφ^pt,^qq. (3.5)

Thus for vorticities belonging to L^8pp0, T^q;L^1pR^2qq, the last integral in (3.3) is well-defined.

This motivates the next definition of weak solutions.

Definition 3.2 (Symmetrized velocity formulation). Let^pω⁰, v^0qPL^1pR^2qM^2pR^2q, withω⁰ curlv⁰. We say that the couple^pω, v^qis a symmetrized velocity solution of (1.1) in ^r0, T^q with initial datum^pω⁰, v^0q, if

(1) ω^PL^8pp0, T^q;L^1pR^2qq,

(2) the velocity field v is given by the convolution in (1.4),

(3) for all test functions φ^PC_c^1pr0, T^qR²,R^2q with divφ0, we have

»T

0

»

R²

Btφv dxdt

»T

0

»

R²

»

R²

H¯_φ^pt, x, y^qω^pt, x^qω^pt, y^qdxdydt

»

R²

φ^p0, x^qv^0px^qdx0, (3.6) where ¯H_φis the function on ^r0, T^qR²R² given by (3.4).

3.2. Three formulations of the vorticity equation. According to the introduction, we now define three notions of solution to the vorticity formulation (1.3) when the vorticity is only L¹ summable. Since we do not assume v^{0 P} L²_loc^pR^2q, we deal with velocities that belong to M^2pR^2q, a consequence of the Hardy-Littlewood inequality (2.5).

Definition 3.3 (Renormalized solutions). Let^pω⁰, v^0qPL^1pR^2qM^2pR^2qwithω⁰ curlv⁰. We say the couple^pω, v^q is a renormalized solution to (1.3) with initial data^pω⁰, v^0q, if

(1) ω^PL^8pp0, T^q;L^1pR^2qq,

(2) the velocity field v is given by the convolution in (1.4),

(3) for every nonlinearity β^PC^1pR^q withβ bounded, we have that

#

Bt^pβ^pω^qq div^pβ^pω^qv^q0,

β^pω^qp0,^qβ^pω^0q (3.7)

hold in the sense of distributions.

For smooth solutions this is equivalent to the classical notion of solution (as can be seen by multiplying the equation byβ^1pω^q and applying the chain rule.) This formulation derives from the classical DiPerna-Lions [11] framework for transport equations.

Definition 3.4 (Symmetrized vorticity formulation). As mentioned in the introduction, the sym- metrization technique for the term div^pωv^q provides a second formulation of the vorticity equa- tion. Letφ^PC_c^2pr0, T^qR^2q. Computations as in (3.3) give

»T 0

»

R²

div^pωv^qpt, x^qφ^pt, x^qdxdt

»T 0

»

R²

»

R²

H_φ^pt, x, y^qω^pt, x^qω^pt, y^qdxdydt, (3.8) with

H_φ^pt, x, y^q 1

2K^pxy^q ∇φ^pt, x^q∇φ^pt, y^q

. (3.9)

We say that^pω, v^q is a symmetrized vorticity solution to (1.3) if (1), (2) above are satisfied and if for all test functionsφ^PC_c^2pr0, T^qR^2q there holds

»T

0

»

R²

Btφ^pt, x^qω^pt, x^qdxdt

»T

0

»

R²

»

R²

H_φ^pt, x, y^qω^pt, x^qω^pt, y^qdxdydt

»

R²

φ^p0, x^qω^0px^qdx0.

(3.10) Proposition 3.5. We have the following equivalence of notions of solutions to the Euler system.

(1) Symmetrized velocity solutions (Definition 3.2) are symmetrized vorticity solutions (Def- inition 3.4), and conversely.

(2) If ^pω, v^q is such that v^PL^8pp0, T^q;L²_loc^pR^2qq, then it is a symmetrized velocity solution if and only if it is a weak velocity solution (Definition 3.1).

Proof. For (1), taking a test function of the form ∇^Kφ in (3.6) we see that a solution to the symmetrized velocity formulation is also a solution to the symmetrized vorticity formulation, indeed one has ¯H∇^KφH_φ. The converse is also true since all functions ¯φ^PC_c^2pr0, T^qR²,R^2q with div ¯φ0 can be written ¯φ∇^Kφfor someφ^PC_c^2pr0, T^qR^2q. For ¯φonly C¹ one just approximates it by aC² function. It follows that Definitions 3.2 and 3.4 are indeed equivalent.

Finally, the statement (2) follows from the next lemma.

Lemma 3.6. Letω ^PL^1pR^2q, definevKω withK the Biot-Savart kernel (1.5), and assume that v^PL²_loc^pR^2q. Then for all φ^PC_c^1pR²,R^2q withdivφ0, we have

»

R²

»

R²

H¯φ^px, y^qω^px^qω^py^qdxdy

»

R²

∇φ^px^q: v^px^qbv^px^q

dx (3.11)

where H¯_φ is given by (3.4).

Proof. For smoothω andv, the formula is just the weak form of the already mentioned identity div^pv^bv^qω v^K ∇^|v|2

2 , taking into account the computation (3.3). The general case follows easily by smoothingω andv by a regularizing kernel and passing to the limit.

3.3. Lagrangian solutions. We describe now a third class of weak solutions which are trans- ported by a measure-preserving flow in an “almost everywhere” sense. When the velocity is not globally bounded, the associated flow X is not locally integrable in R², thus the ODE defining the flow has to be taken in the renormalized sense.

Let us recall the following general definition on R^N. Assume that a vector field b^pt, x^q :

p0, T^qR^{N Ñ}R^N can be decomposed as b^ps, x^q 1 ^|x^|

˜b1^ps, x^q ˜b2^ps, x^q, (R1) with ˜b1 ^PL^1pp0, T^q;L^1pR^Nqq, ˜b2 ^PL^1pp0, T^q;L^8pR^Nqq. (3.12)

Definition 3.7. If b is a vector field satisfying (R1), then for fixed t ^Pr0, T^q, a map X^ps, t, x^q satisfying

ps, x^qÞÑX^ps, t, x^{q P}C^prt, T^ss;L⁰_loc^pR^Nx^qqXBprt, T^ss; logLloc^pR^Nx^qq (3.13) is a regular Lagrangian flow (in the renormalized sense) relative tobstarting attif we have the following:

(i) The equation

Bs β^pX^ps, t, x^qq

β^1pX^ps, t, x^qqb^ps, X^ps, t, x^qq (3.14) holds in D¹ppt, T^qR^Nq, for every function β ^P C^1pR^N;R^q that satisfies ^|β^pz^q|¤ C^p1 log^p1 ^|z^|qqand ^|β^1pz^q|¤C^{p1 ^|z^|qfor all z^PR^N, for some constant C,

(ii) X^pt, t, x^qx forL^N-a.e x^PR^N,

(iii) There exists a compressibility constantL^¥0 such that

³

R^Nϕ^pX^ps, t, x^qqdx^¤L

³

R^Nϕ^px^qdx for all measurableϕ:R^{N Ñr}0,^8q.

By now this is the usual definition of flows for weakly differentiable vector fields satisfying the general growth condition (R1).

We next consider the condition that the components of∇bcan be written as singular integrals of L¹ functions,

BjbⁱS_jⁱgⁱ_j inD¹pp0, T^qR^Nq, (R2) whereS_jⁱ are singular integral operators in R^N, andg_j^{i P}L^1pp0, T^qR^Nq.

Finally we consider the conditions

b^PL^p_loc^pr0, T^sR^Nq for somep^¡1, (R3) and

divb^PL^1pp0, T^q;L^8pR^Nqq. (R4) According to [8], under the assumptions (R1), (R2), (R3), (R4), a regular Lagrangian flow X as in Definition 3.7, except that now s^{P r}0, T^s instead of s ^Prt, T^s (the forward-backward flow defined in Corollary 6.6 in [8]) exist and is unique and stable.

With this notion of flow, we can define in accordance with [8] our class of Lagrangian solutions

pω, v^q to the Euler equations by the relation ω^pt, x^qω⁰

X^ps0, t, x^q , for all t^Pr0, T^s. (3.15) Definition 3.8 (Lagrangian solution). Let ^pω⁰, v^{0q P} L^1pR^2qM^2pR^2q with ω⁰ curlv⁰. We say the couple^pω, v^q is a Lagrangian solution to (1.3) in^r0, T^swith initial data ^pω⁰, v^0q, if

(1) ω^PC^pr0, T^s;L^1pR^2qq,

(2) the velocity field v is given by the convolution in (1.4),

(3) for all t^Pr0, T^s,ω is given by the formula in (3.15), whereX is the regular Lagrangian flow associated tov.

Note that according to the properties stated in Section 2, the vector fieldbv satisfies the properties (R1), (R2), (R3), (R4), with g_jⁱ ω, justifying the existence of X. We remark that according to [8], Lagrangian solutions in the sense of Definition 3.8 are also renormalized solutions in the sense of Definition 3.3.

4. Strong stability of Lagrangian flows

We now recall from [8] the following stability result. It gives a quantitative estimate in measure on the flows difference, in terms of the L¹ norm of the difference of the vector fields.

Theorem 4.1 (Fundamental estimate for flows). Let b and ¯b be two vector fields, b satisfying assumptions (R1)-(R3)and¯bsatisfying only (R1). Fix t^Pr0, T^q and letX and X¯ be regular

Lagrangian flows starting at time t associated to b and¯b respectively, with compressibility con- stants L and L. Then for every¯ γ ^¡ 0 and r ^¡0, and for every η ^¡0 there exist λ ^¡0 and C_γ,r,η ^¡0 such that

L^NpB_r^Xt|X^ps,^qX¯^ps,^q|¡γ^uq¤C_γ,r,η^}b¯b^}_L1pp0,T^qBλ^q η for alls^Prt, T^s. (4.1) The constant λandC_γ,r,η depend on γ, r, η and on the bounds on the operator norms ofS_jⁱ, the norms involved in the estimates from(R1) and(R3), the compressibility constants L andL¯ of X and X, and the equi-integrability of¯ g_jⁱ from assumption (R2) for b.

In previous literature (see again [19]), strong L¹_loc convergence of smoothed velocities was guaranteed for initial datav⁰ belonging toL²_loc^pR^2q. In order to allow for solutions with infinite kinetic energy, we bypass this assumption and use the weaker M² estimate arising in (2.5). As seen in the estimate (4.1), we need the strong convergence in time and space of the vector field.

We have a priori only compactness in space, and we shall therefore use a general argument to deal with only weak convergence in time. We shall show a posteriori that given equi-integrable vorticity data, the associated velocities are indeed strongly compact in time and space. An alternative way to get compactness is also explained in Remark 6.4.

We now expand in the following Proposition 4.4 a remark from [9] proving thatweak conver- gence is sufficient for stability, and adapt the proof from the setting of Sobolev regularity to the regularity given by(R2). We begin with two lemmas, the first arising from standard analysis.

Lemma 4.2. Let K be the Biot-Savart kernel (1.5), and denote by τ_hK^px^qK^px h^q. Then for any1 p 2 and all h^PR² one has

}τ_hKK^}_L^ppR^2q¤c_p^|h^|α (4.2) withα2^{p1^¡0. In particular, the linear mappingL^1pR^2qÑL¹_loc^pR^2qdefined byg^ÞÑKg is a compact operator.

Proof. See Lemma 8.1 in [6] for the proof of the first inequality. Next, denoteT gKg, and take an exponent1 p 2. Whenever ^}g^}_L1pR^2q¤1,T g is bounded inL¹_loc and one has

}τ_h^pT g^qT g^}_Lp

pR^{2q }p}τ_hKK^qg^}_Lp

pR^2q

¤}τ_hKK^}_Lp

pR^2q

¤cp^|h^|α.

(4.3) Thusτ_h^pT g^qT g is uniformly small inL¹_loc ash^Ñ0. Applying the Riesz-Fr´echet-Kolmogorov

criterion gives the result.

The second lemma states that given classical flows associated to Lipschitz vector fields, weak convergence of the vector fields suffices for the associated flows to converge uniformly.

Lemma 4.3. Let b_n be a sequence of vector fields uniformly bounded in L^8pp0, T^q R^Nq with ∇_xb_n uniformly bounded in L^8pp0, T^q R^Nq. Assume that there exists a vector field b^PL^8pp0, T^qR^Nq with∇_xb^PL^8pp0, T^qR^Nq, such that b_n^áb in L^8pp0, T^qR^Nqw. Let X_n^ps, t, x^qand X^ps, t, x^q be Lagrangian flows (in the DiPerna-Lions sense) associated tob_n and b. Then X_n^ps, t, x^qÑX^ps, t, x^q in L^8pp0, T^q2;L⁸_loc^pR^Nqq.

Proof. It follows from the uniform bounds on b_n and ∇_xb_n that ∇_xX_n is uniformly bounded, with

|∇_xX_n^ps, t, x^q|¤exp T^}∇_xb_n^}_L⁸pp0,T^qR^Nq

. (4.4)

Then, the ODE forX_n implies that^B_sX_n is uniformly bounded. Also, the transport equation

BtX_n b_n^pt, x^q∇_xX_n0 (4.5)

implies also that^BtX_n is bounded. Thus up to modifyingX_n on a Lebesgue negligible set,X_n is uniformly bounded in Lip^pr0, T^s2R^Nq. By Arzel`a-Ascoli’s theorem there existsY^ps, t, x^qP Lip^pr0, T^s2R^Nq such that up to a subsequence Xn^ps, t, x^{q Ñ} Y^ps, t, x^q locally uniformly in