Strong convergence of the vorticity and conservation of the energy for the α-Euler equations

In this paper, we study the convergence of solutions of the α-Euler equations to solutions of the Euler equations on the two-dimensional torus. In particular, given an initial vorticity ω 0 in Lxp for p∈(1,∞) , we prove strong convergence in Lt∞Lxp of the vorticities q α , solutions of the α-Euler equations, towards a Lagrangian and energy-conserving solution of the Euler equations. Furthermore, if we consider solutions with bounded initial vorticity, we prove a quantitative rate of convergence of q α to ω in Lp , for p∈(1,∞) .


Introduction
In this paper, we consider the incompressible α-Euler equations on the two-dimensional torus, which, given α > 0, read as and our primary objective is the rigorous study of the limit as α → 0 of solutions of (1).Formally, if we substitute α = 0 in (1), we obtain that v α = u α , and then, employing the identity and defining π α := p + |u α | 2 /2, we obtain the two-dimensional incompressible Euler equations:    The α-Euler equations are part of a larger class of approximation schemes called Large Eddies Simulations (LES), which have been first introduced in [25] by Smagorinsky and later generalized by Leonard [18].LES approximations are relevant for numerical simulations of fluids in turbulent regime.Indeed, due to the high number of scales needed in turbulent dynamics, Direct Numerical Simulation (DNS) of fluid equations are computationally very demanding.The idea of LES models is based on the fact that the whole range of flow scales may not be necessary in order to have an accurate approximation.Therefore, a filter cutting the small scales is applied to the velocity.Since the operation of filtering does not commute with the nonlinear convective term, one needs to estimate the commutator between the filter and the convective term and this procedure gives rise to an approximation.We refer to [1] and [16] for more details on the derivation and the motivations of LES models.In particular, we refer to [14,22] for the derivation of the α-Euler equations.
Concerning the system (1), the filter is given by the so-called Helmholtz filter, which is exactly the second equation in (1), namely The action of the filter (3) can be written in Fourier variables as The denominator of the right hand side of (4) diverges as the frequency grows, cutting the high frequencies as a consequence.
The analysis of the limit as α → 0 for solutions of the α-Euler equations has been widely studied in literature.In particular, we mention the important result in [20], where the authors analyse the convergence in L 2 for the velocity fields in two dimensions for smooth solutions of the (1) equations on a bounded domain with Dirichlet boundary conditions towards smooth solutions of the Euler equations.The importance relies on the fact that in the limit of α → 0 no boundary layers are created for the velocity.The result in [20] has been extended in [2] where the convergence of the velocity for the problem posed on bounded domains with Dirichlet boundary conditions has been proved for less regular solutions.We also recall the paper [3], where the convergence as α → 0 is studied for initial vorticity in the space of positive Radon measures.Finally, we refer to [17,24] where the relationship between the α-Euler equations and the vortex blob method (another commonly used numerical approximation of the Euler equations) has been investigated.
An important feature of the system (1) is that the vorticity structure of the 2D Euler equations is preserved in the approximation.Indeed, thanks to the presence of the term j v α j ∇u α j in the momentum equations of (1), if we consider q α = curl v α , we obtain the following vorticity formulation of (1)          ∂ t q α + u α • ∇q α = 0, on (0, T ) × T 2 q(0, •) = q α 0 , on T 2 div u α = div v α = 0, on (0, T ) × T 2 curl v α = q α , on (0, T ) × T 2 . ( The vorticity formulation (5) is particularly important for the purpose of this paper, since we are primarily interested in the analysis of the convergence of the vorticity q α towards the vorticity of the 2D Euler equations in strong norms.This is reminiscent of the analogous results for the vanishing viscosity limit.In two-dimensional turbulence, the transport-structure for the vorticity and the appearance of an inverse cascade (from small to large scales) entail that vanishing viscosity solutions enjoy better properties compared to general weak solutions.In this paper, we show that the same principle holds for solutions that are the limit of the α-Euler equations.
In our first result, we show the strong convergence in L p of the vorticity to a solution of the Euler equations which conserves the energy and is Lagrangian, namely the vorticity is transported by the flow of the associated velocity, see Definition 2.3.We refer to Section 2 for the relevant notations and definitions.In particular, k is the Biot-Savart kernel on the torus (cf.(18)).
Theorem 1.1.Let T > 0 arbitrary and finite, p ∈ (1, ∞) and ω 0 ∈ L p (T 2 ) with T 2 ω 0 = 0. Let {q α 0 } be a sequence of functions uniformly bounded with respect to α in L p (T 2 ) with T 2 q α 0 = 0 such that q α 0 → ω 0 strongly in L p (T 2 ).Let (u α , q α ) be the solution of the α-Euler equations with initial datum q α 0 .Then, up to subsequences, there holds, and and (u, ω) is a Lagrangian solution of the Euler equations.Moreover, for any δ > 0, there exists K(δ, ω 0 ) such that for α small enough it holds Finally, the solution u conserves the kinetic energy, namely The novelty of our approach consists in the application of techniques related to the Lagrangian perspective, introduced in the non-smooth settings in [11] and employed for a vanishing viscosity scheme in [8].We first prove the convergence of the velocity adapting the proof of [2] to this setting.Then, by using Lagrangian techniques, we show strong convergence of the vorticity providing a certain quantification of the convergence.Nonetheless, the rate of convergence is not fully quantitative since it depends (logarithmically) on the rate of convergence of the velocities.
Weak solutions of the Euler equations with vorticity in L p with p ≥ 3/2 conserve the kinetic energy, but this is not known for p < 3/2 (cf.[6]).In Theorem 1.1, we also prove that limit solutions are energy conserving, for every p ∈ (1, ∞], as previously done for the vanishing viscosity in [6] and for the vortex blob in [7].
The second main result of this paper concerns the study of the rate of convergence in the case of solutions belonging to the Yudovich class.In particular, the next theorem shows that if we consider bounded initial vorticity, we can obtain a rate of convergence independent on Let (u α , q α ) be the solution to the α-Euler equations with initial datum q α 0 and let (u, ω) be its limit, which is the unique Yudovich solution to the Euler equations.Then, there exist two constants C 1 and C 2 (depending on it holds Moreover, there exists a value α 0 = α 0 (T, M, ω 0 ) and a continuous function ψ ω0,p,M : R + → R + vanishing at zero, such that for every α ≤ α 0 there holds where One can check that thanks to (8), it is always possible to find α small enough such that (9) holds for a fixed time T and vice versa a positive time T such that (9) holds for α ∈ (0, α].Finally, assuming additional regularity on the initial datum, it is possible provide an explicit expression for the function ψ ω0,p,M . Corollary 1.3.Under the same assumptions of Theorem 1.2, let s ∈ (0, 1), if ω 0 belongs to the Besov space B s p,∞ (T 2 ), then the function ψ ω0,p,M (•) is controlled, yielding where C depends on M .
Theorem 1.2 is the analogous of the results obtained in [8,9,23] for the vanishing viscosity limit and Corollary 1.3 corresponds to [9], Corollary 2, for the vanishing viscosity limit.In particular, the interest in considering initial vorticity in B s p,∞ relies on the fact that some classes of vortex patch are in those spaces.Indeed, in [10] it has been proved that if χ Ω is the characteristic function of Ω ⊂ R 2 whose boundary ∂Ω has box-counting dimension dim F (Ω) < 2, then Finally, we mention that in the case of the α-Euler equations, when considering a vortex patch of boundary C 1,γ , in [19] the authors obtained a rate of convergence of order α 3 4 according to our notation.The proof of [19] is built upon the observations that the C 1,γ regularity of the boundary of a vortex patch is preserved in time and the vortex patches under this assumption belong to the space L ∞ (0, T ; B

Organization of the paper
In Section 2, we introduce the notations used throughout the paper and we recall some standard results on the Euler equations and the known results on the well-posedness of the α-Euler equations.Section 3 is devoted to the proof of Theorem 1.1.In particular, the proof is split between Proposition 3.2 where we prove strong convergence of the velocity and Proposition 3.7 where we prove strong convergence of the vorticity.Finally, in Section 4 we prove Theorem 1.2 and Corollary 1.3 where we estimate the rate of convergence under the additional hypothesis of bounded initial datum.
Acknowledgments This research has been partially supported by the SNF Project 212573 FLUTURA -Fluids, Turbulence, Advection.The first author acknowledges the hospitality of the University of Basel, where a large part of this work was carried out.The work of the third author is partially supported by INdAM-GNAMPA and by the project PRIN 2020 "Nonlinear evolution PDEs, fluid dynamics and transport equations: theoretical foundations and applications".
2 Notations and preliminary results

Notations
Throughout this work, we always consider as a domain the two-dimensional flat torus T 2 = R 2 /Z 2 and we denote the Lebesgue measure on it by L 2 .The distance on the torus is defined as and it is immediate to check that d(x, y) ≤ |x − y|.
We use the standard definitions of functional spaces L p = L p (T 2 ), H s = H s (T 2 ) and W s,p = W s,p (T 2 ), in which we avoid writing explicitly the dependence on T 2 for the norms.To simplify the notation, we use L p t X x for the Bochner spaces L p (0, T ; X(T 2 )).The space C ∞ c ([0, T ) × T 2 ) denotes the space of smooth functions with compact support in time and periodic in space.Finally, we introduce an ad-hoc norm for the solution of the α-Euler equations, namely the α-norm defined by In this exposition, we use multiple times a generic constant C.This constant does not depend on α unless the dependence is specified and even if the constant C appears more than one time in the same computation, its value may change from one line to the next.

The α-Euler equations
For the sake of completeness, we derive the vorticity formulation of (1).Let us define q α := curl v α and ω α := curl u α .
We recall that in dimension two the curl is a scalar and we employ the standard identities Taking the curl of (1), we get We compute the two curls and obtain Therefore, by ( 14)-( 15), simplifying the opposite terms, identity (13) becomes Employing the incompressibility constraint, we obtain that q α = curl v α satisfies where v α is related to the vorticity q α as div The system (17) yields the existence of a stream function The solution of this Poisson equation is given in terms of the Green function on the torus, under the condition T 2 q α dx = 0, which is preserved in time by the equation at least formally.The Green function on the torus reads and the corresponding Biot-Savart kernel, which can be used to represent the solution, reads This relation implies that the Calderon-Zygmund estimates hold, see [21], namely where C p ∼ p for p > 2.
We state the well-posedeness of the α-Euler equations whose proof can be adapted from [2,3] to the two-dimensional torus.
) and there exists a unique solution u α ∈ L ∞ (0, T ; W 3,p (T 2 )) of (1).Moreover, the solution u α conserves the α-norm, namely The conservation of the α-norm can be shown formally by considering (1) and testing it against u α , integrating over the torus, which yields Exploiting div u α = 0 and integrating by part, removing the trivially zero terms, we obtain d dt where the two terms simplified by swapping the indices in the sum.

The two-dimensional Euler equations
Let T > 0 be arbitrary and finite, the two-dimensional Euler equations on the torus are which in vorticity formulation reads where We introduce the Lagrangian description of (22).
for any given t ∈ [0, T ].By the theory of characteristics, if u is smooth, we know that the unique solution of the two-dimensional Euler equations with initial datum ω 0 satisfies In order to extend the definition to the non-smooth case, we need to introduce the following definition.

Definition 2.2 (Regular Lagrangian flow
) is called a Lagrangian flow for the vector field u ∈ L 1 (0, T ; L 1 (T 2 )) if • the map s → X t,s (x) is an absolutely continuous solution of (24) for almost every x ∈ T 2 and any t ∈ [0, T ); • the map x → X t,s (x) is measure preserving with respect to the Lebesgue measure on the torus for any s, t ∈ [0, T ).
The definition of Lagrangian solution in the non-smooth setting is the following.

Definition 2.3 (Lagrangian solution to the Euler equations
x is called Lagrangian solution to the two-dimensional Euler equations if there exists a regular Lagrangian flow in the sense of Definition 2.2 and the couple (u, ω) satisfies (25) for almost every (t, x) ∈ (0, T ) × T 2 .

Quantitative strong convergence of the vorticity
In this section, we present the proof of Theorem 1.1 which is split between Proposition 3.2 and Proposition 3.7.We recall and adapt some lemmas introduced in [2].In the first proposition, we prove the convergence in velocity and we show it implies conservation of energy.In the second proposition, we introduce the proof of convergence in vorticity through Lagrangian techniques.

Preliminaries
In this paragraph, we show some bounds which are needed to prove the convergences in Theorem 1.1.We know that the solution u α is regular enough so that the L p norms of q α are preserved in time, thanks to Theorem 2.1.Hence, using standard elliptic estimate and the Calderon-Zygmund inequality (19), we get Moreover, the elliptic equation v α = u α − α ∆ u α yields the additional regularity This elliptic estimate can be adapted on the torus from [15], Theorem 8.10.The inequality ( 27) is not uniform with respect to α; nevertheless, it is used in Lemma 3.1 to produce an improvement of ( 26) to control the L 2 norm of the gradient and the Laplacian of u α , even for p < 2. The proof is an adaptation of Proposition 3.1 in [2].This bound refines the proof for the convergence in velocities with respect to [19].Moreover, the energy conservation for the limit solution is proven as a direct consequence of this estimate.

Strong convergence of the velocity and energy conservation
At this point, we have all the tools to show the strong convergence of the velocity.
Proposition 3.2.Let T > 0 arbitrary and finite, let q α 0 and ω 0 under the assumptions of Theorem 1.1 and let the couple (u α , q α ) be the corresponding global solution to the α-Euler equations.Then, there exists a couple (u, ω) such that, up to subsequences, it holds and u is a distributional solution of the Euler equations with initial datum u 0 = k * ω 0 .Finally, u conserves the kinetic energy, namely Proof.
Step 1: Weak convergences.We know that the sequence q α is bounded uniformly in L ∞ (0, T ; L p (T 2 )) by Theorem 2.1.By a standard compactness argument, we have that up to (non relabelled) subsequences By (26), the corresponding velocities u α are bounded uniformly in α with respect to the L ∞ t W 1,p xnorm, hence, up to a (sub)subsequence, we have which implies Step 2: Strong convergence of the velocity.We rewrite (1) as thanks to standard tensor identities and π α = p + |u α | 2 /2.Now, we want to control the time derivative of the velocity and use Aubin-Lions Lemma, in order to show the strong convergence of u α .Let ϕ ∈ C ∞ (T 2 ), then there exists ). Multiplying (35) by ϕ α and integrating over T 2 , after some integrations by parts, we infer On the left hand side of (36), we get Hence, we need to bound every term on the right hand side of (36) to control {∂ t u α }.The control on the first term in the right hand side of (36) is straightforward For the other terms, we employ the bound on the α-norm given by (20).In particular, for the second term of the right hand side of (36), we obtain (39) Finally, the third and fourth term are respectively bounded in the following way Lastly, we need to estimate the pressure term.Let us consider (35) and take its divergence.
Owing to the incompressibility constraint, we get Here, we consider the pressure as the sum of two contributions For the term π α 1 , we notice that it can be bounded analogously to (38), which is The control on the term π α 2 is exactly equivalent to the ones in (39)-( 40)-(41), indeed Now, the estimates for the non-linear terms (38)-( 39)-( 40)-( 41) and the ones for the pressure (42)-( 43) complete the control of the right hand side of (36).Indeed, employing the elliptic estimate ϕ α H 4 ≤ C ϕ H 4 , we infer Thus, ∂ t u α is uniformly bounded with respect to α in L ∞ t H −4 x .The immersion of L 2 (T 2 ) is continuous in H −4 (T 2 ) and by (33) the velocity converges weakly- * in L ∞ (0, T ; W 1,p (T 2 )) with W 1,p (T 2 ) compactly embedded in L 2 (T 2 ).Hence, we use Aubin-Lions lemma to infer that up to a new (sub)subsequence Step 3: Equation for the velocity.We want to show that the limit u a solution to the Euler equation with initial datum u 0 .First, we recover strong convergence of u α 0 to u 0 in L 2 (T 2 ).We recall that the initial data are defined as Owing to (19) and q α 0 → ω 0 in L p (T 2 ) by hypothesis, we have that Then, we consider Lemma 3.1 and if p ≥ 2, we deduce Whereas, if p ∈ (1, 2), we obtain By ( 47)-( 48), we infer from (46) that Now, we are left to show that the limit u is a distributional solution to the velocity formulation of the Euler equations.Thus, we need to pass to the limit into (35).The term ∂ t (u α −α ∆ u α ) → ∂ t u in the sense of distribution thanks to (34).Indeed, for any Let p ∈ (1, 2), considering the right hand side of (35), thanks to Lemma 3.1, we obtain Analogously for p ∈ [2, ∞), we use (26) to deduce After integration by parts, for any p ∈ (1, ∞) inequalities (51)-(50) imply the following convergences in the sense of distribution Hence, in the right hand side of (35), we are left to pass to the limit for div(u α ⊗ u α ).However, this is implied by strong convergence of the velocity in (45).
Step 4: Energy conservation.By Theorem 2.1, we that the α-norm of the solution is conserved, namely and we want to pass to the limit as α → 0. Considering Lemma 3.1, we obtain Moreover, by (26) we get We proceed in analogous way to control ∇u α 0 and we infer α ∇u α 0 2 Employing ( 53)-( 54) and ( 55), we pass to the limit into (52) to obtain the thesis where we have used the strong convergence of the velocity in L 2 (T 2 ) expressed by ( 45)-(49).

Strong convergence of the vorticities
We proceed to prove that the limit solution of the α-Euler equations is a Lagrangian solution of the Euler equations.This proof is analogous to the one in [12] for the vanishing viscosity scheme.Let us introduce the transport equation as with div b = 0. We define in the following way a renormalized solution.
Definition 3.3.A measurable function ρ is a renormalized solution of (57), if it solves in the sense of distribution We first recall the following lemma, given by Theorem II.6, [13].
Lemma 3.4.Let b be a vector field such that )) be a renormalized solution of the transport equation according to Definition 3.3.Let ξ ∈ L ∞ (0, T ; L q (T 2 )), where q = p−1 p , be a renormalized solution of the following backward transport problem where χ ∈ L 1 (0, T ; L q (T 2 )) and ξ T ∈ L q (T 2 ).Then, it holds With the introduction of this setting, we are able to prove that the limit is a Lagrangian solution of the Euler equations.Proposition 3.5.Let (u, ω) be the limit of (u α , q α ) according to Proposition 3.2.Then, (u, ω) is a Lagrangian solution of the Euler equations according to Definition 2.3.

Proof.
Step 1: Consistent limit.We begin the proof by showing that u = k * ω.The starting point is the equation Considering the left hand side, we obtain where we exploited the smoothness of ϕ and (45).Instead, for the right hand side we have Here, being ϕ ∈ C ∞ c ((0, T ) × T 2 ), by Young inequality we have Employing (32), taking the limit as α → 0 + and switching back the convolution, we infer ).This implies u = k * ω for almost every (t, x) ∈ (0, T ) × T 2 as wanted.
Step 2: Lagrangian solution.We show that (u, ω) is a Lagrangian solution.We only need to prove it for p ∈ (1, 2), because for p ≥ 2 it follows directly from uniqueness of the solution of the transport equation (see [13], Theorem II.3).Let us consider (16), let χ ∈ C ∞ ((0, T ) × T 2 ) and let us define the backward transport problem for u α of the form and the limit backward problem Thanks to the stability theorem in DiPerna-Lions, [13] II.4, it holds that where ξ α solves (60).We recall that q α ⇀ * ω in L ∞ (0, T ; L p (T 2 )) and q α 0 → ω 0 in L p (T 2 ), therefore passing to the limit in (62), we obtain where ξ is the unique solution in C([0, T ]; L q (T 2 )) of (61).Using Lemma 3.4 on the limit backward problem, we infer where ω L is the unique renormalized solution, thus Lagrangian of the transport equation ( 57) with velocity field u and initial datum u 0 (cf.[13] Theorem II.3).Subtracting ( 63) and ( 64) we get We notice that for the α-Euler equations, we can introduce the flow map X α using the classical theory of characteristic (cf.( 24)-( 25)), since the velocity field u α Lipschitz due to the embedding W 3,p ֒→ W 1,∞ .Knowing that the limit solution is Lagrangian for any p ∈ (1, ∞), we want to study the convergence of the flows as it has been done in [11].Lemma 3.6.Let T > 0 arbitrary and finite and let (u α 0 , q α 0 ) be under the assumptions of Theorem 1.1 and let (u α , q α ) be the sequence of unique solutions of α-Euler equations according to Theorem 2.1.Let (u, ω) be the limit of (u α , q α ) obtained in Proposition 3.2 and let X α t,s and X t,s be the corresponding Lagrangian flows according to (24) and Definition 2.2.Then, it holds where the constant C depends on ∇u L 1 t L p x and on T .
x .Let us define the quantity Let us consider x → log 1 + x δ , increasing in [0, ∞).We use Chebyshev inequality to infer for every ε > 0. We have used that on the torus it d(X α t,s (x), X t,s (x)) ≤ |X α t,s (x)−X t,s (x)|.We split the integral over the torus in two complementary sets as We use (67) to infer The inequality (68) holds true for any ε > 0, thus we can choose ε = √ δ.We recall that δ goes to zero as α → 0, by definition.Therefore, we can take a value α for which δ < 1, which yields Substituting into (68) with (66), we infer By Definition 2.2 and (66), we know g δ (t) = 0 and where we have summed ±u α (τ, X α t,τ ) in the numerator and we have used the triangular inequality.The first term in the right hand side of (69) is controlled as thanks to (69).We are left to bound the second term in the right hand side of (69).Let M be the maximal function operator defined on L 1 functions as We recall that for any f ∈ L p with p ∈ (1, ∞], it holds Moreover, for any f ∈ W 1,1 , there exists a set N ⊂ T 2 such that L 2 (N ) = 0 and This allows us to deduce thanks to the measure preserving property of the flows X α t,s and X t,s .Substituting (70) and ( 72) into (69), we obtain x , we have the thesis.Lemma 3.6 allows to prove a result on the strong convergence of the vorticities.Proposition 3.7.Let T > 0 arbitrary and finite and let (u α 0 , q α 0 ) be under the assumptions of Theorem 1.1.Let the couple (u α , q α ) be the corresponding global solution to the α-Euler equations.Let (u, ω) be the limit of (u α , q α ) obtained in Proposition 3.2, then for any δ > 0, there exists K(δ, ω 0 ) such that for α small enough it holds which implies q α → ω strongly in C([0, T ]; L p (T 2 )). (74) Proof.We recall that Lipschitz function are dense in L p (T 2 ), for every p < ∞.Thus, we take a sequence of Lipschitz function {ω j 0 } j∈N , such that ω j 0 → ω as j → ∞ in L p (T 2 ).We notice .
Knowing that ω j 0 are Lipschitz and using the estimate on the flows (65), we deduce If we send first α → 0 and then j → ∞, by employing the strong convergence of u in L ∞ t L 2 x , thus in L 1 t L 1 x , we get that q α → ω strongly in C t L p x .Given δ > 0, we can take the sequence {ω j 0 } j∈N such that δ ≥ 2 ω j 0 (t) − ω 0 (t) L p , for any j ∈ N, and infer (73) from (75).The convergence (74) follows from u α − u L 1 t L 1 x → 0 as α → 0. This result completes the proof of Theorem 1.1.

Rate of convergence for bounded vorticity
In this section, we analyse rate of convergence for the initial vorticity in L ∞ (T 2 ).The main tool we employ is a standard generalization of Gronwall inequality, known as Osgood's lemma.We recall here the statement as it has been proven in [5].

Rate of convergence for the velocities
As a first step to prove Theorem 1.2, we use a technique introduced by Chemin in [4] to control the convergence of the velocities, employing Lemma 4.1.Proposition 4.2.Let q α 0 and ω 0 be under the assumptions of Theorem 1.2.Let u α 0 := (I −α ∆) −1 k * q α 0 and u 0 := k * ω 0 , then Let (u α , q α ) be the solution to the α-Euler equations with initial datum q α 0 and let (u, ω) be its limit, which is the unique Yudovich solution to the Euler equations.Then, there exist two constants C 1 and C 2 (depending on M := ω 0 L ∞ ) such that, if α ∈ (0, α] and T > 0 satisfy Proof.We observe that considering the assumptions of Theorem 1.2, we have that Theorem 1.1 holds true for every p < ∞.Therefore, we know that q α converges to ω in C([0, T ]; L p (T 2 )), for every p < ∞ and it is uniformly bounded in L ∞ ((0, T ) × T 2 ).Hence, we deduce that ω ∈ L ∞ ((0, T )×T 2 ), which implies that u is the unique Yudovich solution to the Euler equations (cf.[26]).Now, set Let us take the difference between the Euler equations and the α-Euler equations and test it against y α .Summing the terms Using the definitions of v α and y α , some computations, we infer Owing to the incompressibility constrains, we cancel out some terms and we obtain We first control the last term in the right hand side of (77) with a technique analogous to the one used in [4].Using Hölder inequality and ( 19), we get We observe that y α = −k * (ω − q α ).By Calderon-Zygmund inequality, for q > 2, we have The right hand side of (78) is controlled as We are left to control the first two terms in the right hand side of (77).By Lemma 3.1, we deduce √ α ∆ u α L 2 ≤ C q α 0 L 2 .Exploiting this fact, we control the first two terms in the right hand side of (77) with some applications of Hölder inequality and we deduce where we have used (79).Finally, we substitute back into (77) the estimates (78)-( 81) and (80) and by definition (76), we obtain the ordinary differential inequality Thanks to (47)-( 48)-(49), we deduce (8), namely Thus, let α be such that γ α 0 ≤ 1/2, for every α ≤ α.Hereafter until the end of the proof, we consider α ∈ (0, α] and an application of triangular inequality yields Then, there exists a value α 0 = α 0 M, ω 0 ) and a continuous function ψ ω0,p,M : R + → R + vanishing at zero, such that for every α ≤ α 0 holds sup t∈(0,T ) q α (t) − ω(t) L p ≤ CM 1− 1 p max ψ ω0,p,M (K(α, t)), K(α, t) , where C depends on M . Proof.
Integrating over the torus, we bound the first term owing to the measure preserving property of X α t,τ and using Young inequality.The second term is bounded thanks to (71), hence we infer Step 2: Quantitative estimate for the vorticities.Since ω 0 ∈ L ∞ (T 2 ) and therefore in L 1 (T 2 ), there exists a modulus of continuity in L 1 , that is a continuous function ψ ω0 (•) vanishing at zero, such that ω 0 (• + h) − ω 0 (•) L 1 (T 2 ) ≤ ψ ω0 (|h|), for any h ∈ T 2 .
Exploiting the fact that X α t,s is measure preserving and using inequality (89), we infer Finally, we take ε = K(α, t)e − CT Proof of Corollary 1.3.We prove a control on the modulus of continuity under the additional hypothesis ω 0 ∈ B s p,∞ (T 2 ).A well known characterization of the Besov spaces B s p,∞ (T 2 ), for s ∈ (0, 1), is given by Hence, the characterization (91) implies thus, we can take ψ ω0 (|h|) ≃ |h| s into (10) and therefore (11) follows.

4
and interpolate L p between L 1 and L ∞ to obtain the thesis.Combining Proposition 4.2 with Proposition 4.3 completes the proof of Theorem 1.2.Lastly, the proof of Corollary 1.3 follows directly from the definition of Besov spaces.