Research article

Yudovich type solution for the two dimensional Euler-Boussinesq system with critical dissipation and general source term

  • Received: 19 March 2023 Revised: 14 May 2023 Accepted: 19 May 2023 Published: 01 June 2023
  • MSC : 35Q35, 35B65, 76D03

  • The present article investigates the two-dimensional Euler-Boussinesq system with critical fractional dissipation and general source term. First, we show that this system admits a global solution of Yudovich type, and as a consequence, we treat the regular vortex patch issue.

    Citation: Oussama Melkemi, Mohammed S. Abdo, Wafa Shammakh, Hadeel Z. Alzumi. Yudovich type solution for the two dimensional Euler-Boussinesq system with critical dissipation and general source term[J]. AIMS Mathematics, 2023, 8(8): 18566-18580. doi: 10.3934/math.2023944

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  • The present article investigates the two-dimensional Euler-Boussinesq system with critical fractional dissipation and general source term. First, we show that this system admits a global solution of Yudovich type, and as a consequence, we treat the regular vortex patch issue.



    The generalized 2d Euler-Boussinesq system with fractional dissipation is given by the following typical system:

    {τv+vv+b=H(ϱ)if (τ,x)R+×R2,τϱ+vϱ+κ|D|αϱ=0if (τ,x)R+×R2,divv=0,(v,ϱ)|τ=0=(v0,ϱ0), (1.1)

    where v=(v1;v2)T is the velocity of the fluid, ϱ represents the density/temperature and the pressure b is a scalar function. The action of the buoyancy forces are represented by the function H(ϱ)=(H1(ϱ),H2(ϱ))T. The coefficient κ0 stands for the thermal diffusivity and the operator |D|α denotes the fractional Zygmund operator and it is defined formally as follows,

    |D|αf(x)F1(||αF(f))(x),

    where F denotes the Fourier transform and F1 is its inverse. In [5], Brenier introduced and studied the optimal transport and the magnetic relaxation concepts related to the Boussinesq system with general source term. The system (1.1) is a natural generalization for the 2d classical Boussinesq system, in fact, the system (1.1) coincides with the classical Boussinesq system when H(s)=(0,s)T. Now, we list some results regarding the classical Boussinesq system i.e. (H(ϱ)=ϱe2). The existence of local solution of the inviscid system (with κ=0) was proved by many authors in a different spaces, see for instance [7,8]. For the case κ>0 and α=2 Chae proved in [6] the global existence for initial datum lies in Hs, with s>2. More recently, Hmidi and Keraani [17] extended the result of [6] from the subcritical regularities to the critical regularities. The study of the classical Boussinesq system in the critical case (α=1) attracted the attention of many authors, where, Hmidi et al. [18] established a new technique to prove the global existence of solutions, this technique uses deeply the structure of the equation and they introduced the "coupled function" ΓwRϱ with w stands for the vorticity and R=1|D|1 is the Riesz transform. Thereafter, Ye [30] gave another approach to studying the classical Boussinesq system in the critical regime. The method used by [30] is based on a prestigious result obtained by Ye and Xue [28], (see Theorem 1.6 below). Concerning the existence/uniqueness problem for the Boussinesq system with general source term (1.1), Sulaiman showed in [25] that the system (1.1) is globally well-posed whenever α(1,2). Recently, Wu and Zheng established in [26] that the system (1.1), with the presence of the vertical dissipation in the velocity and density equations, admits a unique global solution. For more results on this subject we may refer to the papers [1,20,22,27].

    The notion of vorticity (w×v) plays an interesting role in the study of the fluid dynamics. By performing the operator × to the Eq (1.1)1, we get the following new system:

    {τw+vw=×(H(ϱ))if (τ,x)R+×R2,τϱ+vϱ+κ|D|αϱ=0if (τ,x)R+×R2,divv=0,(w,ϱ)|τ=0=(w0,ϱ0). (1.2)

    We say that initial data is of Yudovich type if the initial vorticity w0 is bounded, where Yudovich in his famous paper [31] proved the global existence and uniqueness for Euler equation. Recently, Danchin and Paicu [12] proved that the system (1.1) admits unique solution of Yudovich type when α=2. Recently, Xu and Xue [29] extended the results of [12] in the supercritical/critical regime α(0,1], but in the supercritical case, they have imposed a smallness condition on the initial density ϱ0. A particular sub-class of Yudovich type data is the vortex patch, The initial vorticity w0 is said to be a regular vortex patch, if the initial vorticity w0 is the characteristic function of a regular domain D0. In [9] Chemin showed that, if the initial boundary D0 is C1+η for η(0,1), then Dτ is still C1+η, where DτΦ(τ,D0) is the flow image of the initial domain and Φ is the flow map defined as follows

    {τΦ(τ,x)=v(τ,Φ(τ,x))if (τ,x)R+×R2,Φ|τ=0(x)=x.

    The approach of Chemin consists to control the L norm of the gradient of the velocity with respect to the co-normal regularity of the vorticity by introducing the logarithmic estimate (see also Serfati [24] for another proof). In this context, for the case κ>0, and α=2, Hmidi and Zerguine [19] proved that for a smooth initial boundary then the velocity remains Lipschitz for any positive time. More recently, the reference [32] investigates the vortex patch issue for the classical Euler-Boussinesq system with critical dissipation. Hassainia and Hmidi [14] established the local existence for the regular/singular patch for the inviscid Euler-Boussinesq system, this result was improved later by Hmidi et al. in [16]. Also, the first author and Zerguine [21] established a similar results to [14] for the system (1.1)κ=0. For more information and details we refer the reader to [4,11,13,15,23] and the references therein. Inspired by the method used by [30] and motivated by the above works, we investigate the generalized Boussinesq system (1.1) with critical dissipation corresponding to the case α=1. The first theorem of this paper deals with the Yudovich type solutions.

    Theorem 1.1. Let α=1 and HC1(R;R2), such that H(0)=0. We assume that the initial datum (v0,ϱ0)L2(R2)×L(R2)H1(R2), and w0L3(R2)L(R2). Then the system (1.1)α=1 has a global unique output (w,ϱ) such that

    wL,loc(R+,L3(R2)L(R2)),ϱL,loc(R+,L(R2)H1(R2)),|D|3/2ϱL2,loc(R+,L2(R2)).

    Remark 1.2. In what follows, κ is a non-negative parameter (κ>0), and for the sake of simplicity, we assume that κ=1.

    Remark 1.3. Sulaiman [25] assumed that the function H is C5(R;R2) function, this assumption is rather strong. In the reference [26], Wu and Zheng supposed that the function H belongs to C2(R;R2). Compared with these results, in this theorem, we have weakened the assumption imposed on the general source term H(ϱ), where we have assumed that HC1(R;R2), which is quite natural because we need to deal with the vorticity equation.

    The second result of this manuscript is devoted to studying the regular vortex patch issue, and we have the following theorem:

    Theorem 1.4. Let HC3(R;R2), such that H(0)=0 and η(0,1) and D0 be a bounded smooth domain of R2 with D0C1+η(R2). Suppose the initial vorticity is of the form w0=1D0 such that w0Cη(X0) and ϱ0L(R2)H1(R2). Then the system (1.1)α=1 has a unique global output

    (v,ρ)L,loc(R+;˙W1,(R2))×L,loc(R+;L(R2)H1(R2)).

    Moreover, the boundary of the advected domain DτCη+1(R2), where Dτ=Φ(τ,D0).

    Remark 1.5. This result can be seen as a generalization of the results obtained in [25,32]:

    Zerguine [32] studied the vortex patch problem for the classical Euler-Boussinesq system (1.1) with critical dissipation i.e. (H(ϱ)=ϱe2 and α=1), where he have used the technique introduced by [18], that dealing with the coupled function Γ=wRϱ. Unfortunately, this method can not be adopted in our case, because the presence of the general source term H(ϱ) spoils the principle of the coupled function used in [18].

    Sulaiman [25] established the global well posedness for generalized Euler-Boussinesq system (1.1) with subcritical dissipation α(1,2). This paper treats the critical dissipation α=1.

    The proof of the above theorems heavily lies on the prestigious theorem proved by [28], which is given as follows:

    Theorem 1.6. (Differentiability of the drift-diffusion equation) Consider the advection fractional-diffusion equation with critical fractional dissipation:

    {τϱ+vϱ+κ|D|ϱ=0if (τ,x)R+×R2,divv=0. (1.3)

    Let ϱ0L2(R2). Suppose that vLloc(R+;Cδ(R2)) for some positive number δ. Then there exists some constant η belongs to (0,δ) such that ϱL,loc(R+;C1+η(R2)).

    Based on this theorem, we prove that v is a δ-Hölder function for some positive constant δ(0,13), thus we obtain a bound for wLτL. Then we get Yudovich's type solution. The main tool of the proof of Theorem 1.4 is to control the Lipschitz norm of the velocity by using the logarithmic estimate to do so we shall control the co-normal regularity Xτw in Hölder space of negative index Cη1, (see Definitions B.7 and B.8 in Appendix B for the directional derivative Xτ), in the way, we need to deal with the term H(ϱ)C1+η, to get rid of the nonlinear function H, we use the paralinearization theorem (see Theorem A.5), which leads us to assume that H is a C3 function, and by using results from Theorem 1.1, we bound the co-normal derivative of the vorticity. The rest of this paper is arranged as follows: The next section is devoted to prove Theorem 1.1. In Section 3, we investigate the vortex patch issue for the system (1.1). We finish this paper with two appendices, the first one gives some basic useful tools and definitions and the second one is devoted to giving some notions and results regarding the vortex patch.

    Notations: (1) We denote by C any positive constant that changes from line to line, we shall use the notation XY instead of the notation C0>0 such that XC0Y and C0 is a positive constant depending on the initial data.

    (2) For every p[1,],.Lp denotes the Lp norms.

    (3) The space Cη denotes the Hölder space. For ηR+N the Hölder space Cη coincides with Bη,.

    (4) j=xj and ij=2xixj, where x=(x1,x2) and dx=dx1dx2.

    (5) s on the involved norms of the initial data and the time.

    In this part, we perform only the classical A priori estimates. For the existence of solutions can be done by the classical arguments.

    Proposition 2.1. Let (v,ϱ) be a smooth solution for (1.1) such that divv0=0. Then we have:

    (i) ϱ(τ)Lηϱ0Lη,η[2,],

    (ii) H(ϱ)LC,

    (iii) v(τ)2L2+ϱ(τ)2L2+τ0|D|1/2ϱ(s)2L2dsμ0(τ).

    Proof. For (ⅰ) see [10]. For (ⅱ), we use the fact that H is a C1 function, so that

    Hi(ϱ)Lsup|x|ϱ(τ)L|Hi(x)|.

    According to (ⅰ) we have ϱ(τ)Lϱ0L. Thus

    Hi(ϱ)Lsup|x|ϱ0LHi(x)C. (2.1)

    Regarding (ⅲ), We start first by establishing the L2-estimate of the density, in fact we take the L2 inner product of (1.1)2 with ϱ, and integrating with respect to time, we obtain

    ϱ(τ)2L2+τ0|D|1/2ϱ(s)2L2dsμ0(τ). (2.2)

    In the next step, we take the L2 scalar-product of (1.1)1 with v and keep in mind divv=0, we obtain

    12ddτv(τ)2L2=R2H(ϱ)vdxH(ϱ)L2v(τ)L2.

    Since H(0)=0 we use Taylor's formula at order 1, we get

    H(ϱ)=ϱ10H(ϱ+rϱ)dr.

    From (ⅱ), we get

    H(ϱ)L2ϱL2.

    In view of (2.2) we obtain the desired inequality (ⅲ).

    Proposition 2.2. Let (v,ϱ) be a smooth solution for (1.1) such that divv0=0. Then we have:

    ϱ(τ)2L2+w(τ)3L3+τ0|D|3/2ϱ(s)2L2dsμ0(t), (2.3)
    w(τ)L+ϱ(τ)C1+ημ0(τ), (2.4)

    with η(0,δ) and δ(0,13).

    Proof. Testing the Eq (1.1) with Δϱ and integrating with respect to space variable, we find

    12ddτϱ(τ)2L2+|D|3/2ϱ(s)2=|R2(vϱ)Δϱdy|.

    From the incompressibility condition (divv=0), we obtain

    |R2(vϱ)Δϱdy|=|R2DkvjDjϱDkϱdy|.

    From Hölder inequality, we get

    |R2DkvjDjϱDkϱdy|vL3ϱ2L3,

    thus

    |R2DkvjDjϱDkϱdy|vL3ϱ2θ˙B1,ϱ2(1θ)˙B1/22,2wL3ϱ2θ˙B0,ϱ2(1θ)˙B3/22,2wL3ϱ2θL|D|3/2ϱ2(1θ)L2.

    From the inequality (ⅱ) in Proposition 2.1 and Young's inequality, we find out

    |R2DkvjDjϱDkϱdy|Cw3L3ϱ02L+14|D|3/2ϱ2L2. (2.5)

    Hence

    12ddτϱ(τ)2L2+||D|3/2ϱ(τ)2L2Cw3L3ϱ02L+14|D|3/2ϱ2L2. (2.6)

    The classical L3-estimates of the vorticity equation gives:

    13ddτw(τ)3L3HLϱL3w2L3.

    Combining (ⅱ)-Proposition 2.1, Young's inequality and interpolation theorem, we conclude

    13ddτw(τ)3L3ϱ2θ˙B1,ϱ2(1θ)˙B1/22,2w2L2Cw3L3ϱ01/2L+14|D|3/2ϱ2L2.

    Gathering the last inequality with (2.6) and using Gronwall's inequality, we obtain (2.3).

    At this stage, we prove that the velocity v belongs to Cδ, with δ(0,13). In fact, since CδBδ,, then according to Bernstein inequality, we find

    v(τ)CδΔ1v(τ)L2+supjN{2jδΔjvL}v(τ)L2+supjN{2j(δ+23)ΔjvL3}.

    From the Biot-Savart law, we have vΔ1w, then according to the fact that δ13<0, Propositions 2.1 and 2.2, we infer that

    v(τ)Cδv(τ)L2+supjN{2j(δ13)ΔjwL3}v(τ)L2+w(τ)L3μ0(τ).

    Hence by using the differentiability of the drift-diffusion equation (Theorem 1.6), we conclude that there exists δ(0,13) such that ϱL,loc([0,);C1+η(R2)). In particular ϱL,loc([0,);L(R2)). Therefore, we find

    w(τ)Lμ0(τ).

    The proof of Proposition 2.2 is now achieved.

    In this part, we explore the regular vortex patch issue for the nonlinear Euler-Boussinesq system with general source term.

    Proof of Theorem 1.4. Our first step is to control the Lipschitz norm of the velocity, we start by estimating the quantity Xτ,λwCη1, for this aim we will use the commutation beetwen the operator Xτ,λ with transport operator, then

    (τ+v)Xτ,λw=Xτ,λ(×(H(ϱ))).

    Using Lemma 5.10, we obtain

    Xτ,λwCη1eCV(τ)(X0,λw0Cη1+Cτ0eCV(τ)Xτ,λ(×(H(ϱ)))Cη1ds, (3.1)

    where V(τ)=τ0v(t)Ldt. Since Xτ,λ(×H(ϱ))div(×(H(ϱ))Xτ,λ)×(H(ϱ))divXτ,λ. Then

    Xτ,λ(×(H(ϱ)))Cη1div(×(H(ϱ))Xτ,λ)Cη1+(×(H(ϱ)))divXτ,λCη1.

    Since Cη is an algebra, then the embedding LCη1 and the paralinearization theorem (Theorem A.5), yellowgive

    Xτ,λ(×(H(ϱ)))Cη1×(H(ϱ))Xτ,λCη+(×(H(ϱ)))divXτ,λCη1ϱC1+η˜Xτ,λCη,

    see the identity (B.1) for the definition of the norm ˜Cη. Substituting the last estimate into (3.1), we deduce that

    Xτ,λwCη1eCV(τ)(X0,λw0Cη1+Cτ0eCV(s)ϱ(τ)C1+η˜Xτ,λCηds). (3.2)

    On the other hand, according to Lemma B.10, we find out

    Xτ,λCηeCV(τ)(X0,λCη+Cτ0eCV(s)Xτ,λv(s)Cηds).

    We make the following emphasis:

    Xτ,λvCηXτ,λwCη1+divXτ,λCηwL+Xτ,λCηvL,

    and for more information, see, e.g. [9,14]. Therefore

    eCV(τ)Xτ,λCηX0,λCη+Cτ0eCV(s)(Xs,λw(s)Cη1+divXs,λCηw(s)L)ds+τ0eCV(s)Xs,λCηv(s)Lds.

    From Gronwall's inequality, we obtain

    Xτ,λCηeCV(τ)(X0,λCη+Cτ0eCV(s)(Xs,λw(s)Cη1+divXs,λCηw(s)L)ds). (3.3)

    Putting together (B.4) and (3.3), we get

    ˜Xτ,λCηeCV(τ)(˜X0,λCη+divX0,λCηwL1tL+Cτ0eCV(s)Xs,λw(s)Cη1ds).

    From Proposition 2.2, we find

    eCV(τ)˜Xτ,λCη˜X0,λCη+CτdivX0,λCη+Cτ0eCV(s)Xs,λw(s)Cη1ds1+τ+τ0eCV(s)Xs,λw(s)Cη1ds. (3.4)

    Inserting (3.2) into (3.4), we obtain

    eCV(s)˜Xτ,λCη1+τ+τ0eCV(S)˜Xs,λCηϱ(s)C1+ηds.

    Gronwall's inequality ensures that

    ˜Xτ,λCηeCτeCV(τ)eϱL1τC1+η. (3.5)

    However, we also have

    ϱL1τC1+ητϱLτC1+η.

    According to inequality (2.4), we infer that

    ˜Xτ,λCηeC0τeCV(τ). (3.6)

    Consequently, (3.2) becomes

    Xτ,λwCη1eC0τeCV(τ). (3.7)

    Gathering together (3.6) and (3.7), we find

    Xτ,λwCη1+˜Xτ,λCηeC0τeCV(τ). (3.8)

    According to (B.3), we obtain

    Xτ,λwCη(Xτ)eC0τeCV(τ). (3.9)

    Combining the logarithmic estimate Theorem B.11 with the monotonicity of mapping yylog(e+ay) and (3.9), we find out

    v(τ)LC(wL3+wLlog(e+wCη(Xτ)))1+wL3+wL+t+τ0v(s)Lds. (3.10)

    According to Proposition 2.2, then (3.10) takes the following form

    v(τ)Lμ0(τ)+τ0v(s)Lds.

    Using Gronwall's lemma, we obtain

    v(τ)Lμ0(τ).

    Therefore

    wCη(Xt)μ0(τ). (3.11)

    To estimate the flow Φ in Cη(Xτ) we use the identity XτΦ(τ)Xτ,λΦ(τ) and XτΦ(τ)Xτ,λΦ(τ), we obtain

    Xτ,λΦ(τ)CηXτ,λCηΦ(τ)L.

    On the other hand we have the classical estimate Φ(τ)LCeCV(τ) (see for instance [9]). Therefore, we obtain

    Xτ,λΦ(τ)CηXτ,λCηeCV(τ). (3.12)

    Putting (3.6) into (3.12), we deduce that

    XτΦ(τ)Cημ0(τ).

    Gathering the last estimate with (3.11), we conclude that

    XτΦ(τ)Cη+wCη(Xt)μ0(τ).

    Boundary regularity. Since the boundary D0 is a curve of class C1+η, there exists a real function h0 and a neighborhood V0 such that

    (i) h0C1+η,h0(x)0, on V0.

    (ii) D0=h10({0})V0.

    Let ς be a smooth function satisfying

    suppςV0,ς(x)=1,xV1

    with V0 is a small nighbrohood of V1. Now we construct an admissible family as follows, we set

    X0,0(x)=(2h0(x)1h0(x)),X0,1(x)=(1ς(x))(01).

    Set X0=(X0,λ)λ{0,1}, we have  X0,λ  belongs to  Cη  as well as its divergence, we can check that X0,λ is admissible in the sense of Definition B.7, see for instance [19]. Since w0=1D0, we obtain

    h0w0=0.

    On the other hand 1ς0 on V1 then X0,1w0=0 in view of Theorem 1.1, the system (1.1) has a unique local solution

    (v,ϱ)Lloc(R+;W1,(R2))×Lloc(R+;L(R2)H1(R2)).

    Next, we move to prove the regularity of the transported initial domain Dτ. We parametrize the boundary D0 as follows:

    Let y0D0,γ0Cη(R+,R2) solves the following equation:

    {σγ0(s)=X0,0(γ0(s))γ0(0)=x0

    for every t0,

    γ(τ,s)Φ(τ,γ0(s)).

    Differentiating with respect to the variable s, we get

    sγ(τ,s)=(X0,0Φ)(τ,γ0(s)).

    Thus using that X0,0ΦL([0,T0],Cη(R2)), we infer that γ(τ)L([0,T0],C1+η(R2)). This completes the regularity persistence of the boundary Dτ.

    In this work, we have shown that the two-dimensional Boussinesq system with critical dissipation and general source term admits a unique global solution of the Yudovich type. Furthermore, the regular vortex patch issue has been successfully addressed.

    In this appendix, we give some useful tools and notions. We begin with the definition of the dyadic blocks.

    Definition A.1. There exist two radial positive functions (ϕ,ψ)D(R2)×D(R2{0}) such that

    xR2,ϕ(x)+j0ψ(2jx)=1.

    For every ϖS(Rd), the homogenous dyadic blocks are defined as follows: Let jZ,

    ˙Δjϖψ(2j)ϖ,

    and the nonhomogenous dyadic blocks are defined as follows:

    Δjϖ{0if j2ϕ()ϖif j=1,ψ(2j)ϖif jN.

    The following result is the famous Bernstein's inequality, for the proof see [3].

    Proposition A.2. Let 1pq. Suppose ϖLp then for every γNd, there exists two constants C1,C2 such that

    suppˆϖ{|ξ|A02q}γϖLqC12q(|γ|+d(1p1q)ϖLp,
    suppˆϖ{A12q|ξ|A22q}ϖLpC22qksup|γ|=kkϖLp,

    where A0,A1 and A2 are non-negative constant.

    Now we recall the homogeneous and nonhomogeneous Besov spaces in terms of the Littlewood-Paley operators.

    Definition A.3. The homogeneous Besov space ˙Bsp,r(Rd) is the set of all tempered distributions vS(Rd) with

    ϖ˙Bsp,r(2js˙ΔjϖLp)r(Z)<,

    for (s,p)R×[1,+] and 1r<. Besides, if r=

    ϖ˙Bsp,supjZ(2js˙ΔjϖLp)<.

    Definition A.4. The nonhomogeneous Besov space Bsp,r(Rd) is the set of all tempered distributions ϖS(Rd) with

    ϖBsp,r(2jsΔjϖLp)r(Z)<,

    for (s,p)R×[1,+] and 1r<. Besides, if r=

    ϖBsp,supjN{1}(2jsΔjϖLp)<.

    The next Theorem is the paralinearization theorem, see [2].

    Theorem A.5. Let HC[s]+2, with H(0)=0 and s[0,]. Assume that fBsp,rL, with (p,r)[1,+]2, then HfBsp,r and satisfying

    HfBsp,rC(s)sup|y|CfLH[s]+2(x)LfBsp,r.

    We end this part with an interpolation theorem, for more details see [3].

    Theorem A.6. Let 1a<b<, and α positive number. A constant exists, such that

    ϖLbCϖ1θ˙Bα,ϖθ˙Bβa,a,β=α(ba1);θ=ab.

    In this part, we provide some definitions and notations concerning the admissible family of vector fields and the anisotropic Hölder space. These quantities constitute the main ingredients concerning the vortex patch issue.

    Definition B.7. Let η (0, 1). Let X=(Xλ)(λΛ) be a family of vector fields. We assert that this family is admissible of class Cη if and only if:

    Regularity: Xλ, div XλCη.

    Non-degeneracy:

    J(X)infxRdsupλΛ|Xλ(x)|>0.

    We set

    ˜XληXλCη+div XλCη1, (B.1)

    and

    Mη(X)supλΛ˜XληJ(X).

    The action of the family Xλ on bounded real-valued functions u in the weak meaning as follows:

    Xλudiv(uXλ)udivXλ.

    Now, for all τ[0,T] the transported Xτ=(Xτ,λ) of an initial family X0=(X0,λ) by the flow Φ, is defined by

    Xτ,λ(x)(X0,λΦ(t))(Φ1(τ,x)). (B.2)

    The next definition deals with the concept of anisotropic Hölder space, denoted by Cη+k(X). We refer the reader to [9] for more details.

    Definition B.8 Let η(0,1) and kN. We assume that a family of vector fields X=(Xλ)λ be as in Definition B.7. Then, the space Cη+k(Xλ) is defined as the space of functions vWk, such that

    |α|kαvL+supλΛXλvCη+k1<,

    and we set

    vCη+k(X)Mη(X)|α|kαvL+supλΛXλvCη+k1J(X).

    Following that, we list a few family properties. For more information, we reference [9].

    Corollary B.9. There exists a constant C such that for any smooth solution (v,ϱ) of (1.1) on [0,T] and any time dependent family of vector field Xτ transported by the flow of v, we have for all τ[0,T],

    J(Xτ)J(X0)eCV(τ) (B.3)
    divXτ,λCηdivX0,λCηeCV(τ). (B.4)

    The next lemma is a useful result used in our proof, for the proof see [9].

    Lemma B.10. Let v be a smooth divergence-free vector field, and s]1,1[. Consider (ϖ,f) a couple of functions belonging to L,loc(R,Cs)×L1,loc(R,Cs) and such that

    {τϖ+vϖ=f.ϖ|τ=0=ϖ0.

    Then, we have

    ϖ(τ)Csϖ0CseCV(τ)+τ0f(r)CseCV(r)dr. (B.5)

    Where the constant C depends only on s and

    V(τ)τ0v(r)Ldr.

    Next, we state the main tool used in the proof of Theorem B.11, which is the logarithmic estimate introduced by Chemin [9].

    Theorem B.11. Let a(1,),η(0,1) and X be a family of vector fields as in Definition B.7. Consider wCη(X)La. Assume v be a divergence-free vector field with vorticity w. Then there exists C such that

    v(τ)LC(a,η)(w(τ)La+w(t)Llog(e+w(τ)Cη(X)w(t)L)).

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors are very grateful to the referees for the valuable comments and remarks which provide valuable insights and helped to polish the content of the paper and improve its quality.

    No conflicts of interest are related to this work.



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