Global well-posedness and asymptotics for a geophysical fluid system

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1 Global well-posedness and asymptotics for a geophysical fluid system Frédéric Charve Centre de Mathématiques de l École polytechnique, UMR 764 du CNRS, 9118 Palaiseau Cedex, France. charve@math.polytechnique.fr Résumé: On s intéresse au comportement des solutions fortes du système primitif lorsque le nombre de Rossby tend vers zéro. On montre que, sous des hypothèses de régularité suffisantes, lorsque le nombre de Rossby est assez petit, la solution du système primitif est unique et globale, et qu elle converge fortement vers la solution du système quasigéostrophique, elle aussi unique et globale, ceci sans aucune hypothèse de petitesse des données initiales. Classification AMS: 76U5, 76D5, 35Q35 Mots clés: Système primitif, système quasigéostrophique, anisotropie, inégalités de Strichartz Abstract: We focus on the behaviour of strong solutions of the primitive system when the Rossby number goes to zero. We show that, under strong enough regularity assumptions, when the Rossby number is small enough, the solution of the primitive system is unique and global, and that it converges strongly to the solution of the quasigeostrophic system wich is also unique and global, without any assumption of smallness of the initial data. AMS classification: 76U5, 76D5, 35Q35 Keywords: Primitive system, quasigeostrophic system, anisotropy, Strichartz inequalities 1 Introduction In this paper we will consider the primitive system (or primitive equations: (P E t U + v. U LU + 1 AU = 1 ( Φ, div v = U /t= = U. The unknowns are U and Φ. We denote by U a pair (v, θ where v is a vector field on R 3 (three dimensional velocity, θ a scalar function (the density fluctuation : in the case of the atmosphere it depends on the scalar (potential temperature and in the case of the ocean it depends on the temperature and the salinity, and Φ the pressure, all of them depending on (t, x. 1

2 1 INTRODUCTION The operator L is defined by and A by: A def = LU def = (ν v, ν θ 1 1 F 1 F 1 This system is obtained by combining the effects of the Coriolis force and those of the vertical stratification induced by the Boussinesq approximation (one can see [7] as well as [1], about rotating fluids. The coefficient > denotes the Rossby number, ν > is the viscosity and ν > the heat diffusivity (which we will also call a viscosity in the following. As the characterisic displacement of a particle in the ocean within a day is very small compared to the displacement caused by the rotation of the earth, the Rossby number is supposed to be small, about 1 1 to 1 3, and we focus on the limit of a strong rotation ( goes to zero. The coefficient F is called the Froude number. We refer to [6] for a study of the case F = 1, to [11] for the periodic case, and to [4] for the case ν ν, F 1 (convergence of Leray solution. Let us also refer to [] for the case ν = ν and F 1. We refer to [3], [8], [9], [1], [14], [16], and [17] for a discussion on this model, and its derivation. The fact that the parameter goes to zero gives high importance to the term AU, which is said to be penalized. But the term 1 doesn t play any role in the energy estimates thanks to the skew-symmetry of A. Definition 1.1 If s is a real number, the homogenous (resp. inhomogenous Sobolev space of order s, which we will denote by Ḣs (resp. H s, is defined as the space of tempered distributions u S (R 3 whose Fourier transform û is locally integrable and has the following property: u def = ξ s û(ξ dξ < (resp. u def Ḣ s H s = (1 + ξ s û(ξ dξ <. R 3 R 3 Remark 1.1 For more generality one can add an external force to (P E belonging to the space L (R +, L, but with a wish of simplification we will abstrain from it. The Leray theorem (see [15] is true for the primitive system: if the initial data U L (R 3, then there exists for all > a Leray solution of the system (P E, U, globally defined in time, belonging to L (R +, L (R 3 L (R +, Ḣ1 (R 3 and satisfying the following energy inequality (let ν = min(ν, ν > : t R +, U (t L (R 3 + ν. U (t L (R 3 dt U L (R 3. We refer to [4] where we studied the limit of Leray solutions when, the Rossby number, goes to zero and introduced the following notations and results in the case of weak solutions: the potential vorticity is defined by

3 1 INTRODUCTION 3 Ω def = 1 v v 1 F 3 θ. Then from this, we define the quasigeostrophic part of U, which is very close to the solution of the limit system (obtained by a formal method in the beginning of [4] and given in ( QG below, called system of the quasigeostrophic equations: def U,QG = F 1 Ω 1 F 1 Ω F 3 F 1 Ω, with F def = F 3. The other part is called the oscillating part and has a very different behaviour: U,osc def = U U,QG = v 1 + F 1 Ω v 1 F 1 Ω v 3 θ + F 3 F 1 Ω It is important to look at this decomposition in terms of orthogonal projections: if we denote by P the orthogonal projector onto the potential vorticity free vector fields (which is built the same way as the orthogonal projector P on divergence free vector fields, also called the Leray projector and Q = Id P, then in the Fourier variable, these two operators are written: (1 QU = 1 ξ F PU = 1 ξ F ξ ξ 1 ξ F ξ ξ 3 ξ 1 ξ ξ 1 F ξ 1 ξ 3 F ξ ξ 3 F ξ 1 ξ 3 F ξ 3. Û ξ 1 + F ξ 3 ξ 1 ξ F ξ ξ 3 ξ 1 ξ ξ + F ξ 3 F ξ 1 ξ 3 ξ F F ξ ξ 3 F ξ 1 ξ 3 ξ 1 + ξ Û, where ξ def F = ξ1 + ξ + F ξ3. So this decomposition U = U,QG + U,osc where U,osc = PU and U,QG = QU involves two homogeneous pseudo differential operators of degree zero. We also have the following obvious properties: ( PU = U Ω(U =, and QU = U PU = U is of the form ( Φ, 1 Φ,, F 3 Φ. The main result in [4] is the determination of the asymptotic behaviour of the Leray solution when goes to zero: Theorem 1 [4] If the initial data U belongs to L (R 3 and if we consider a sequence of Leray solutions (U then, there exists a subsequence that converges for all q ], 6[ in the

4 1 INTRODUCTION 4 space L loc (R +, L q loc (R3 to a function ŨQG which satisfies QŨQG = ŨQG and solution of the system: ( QG t Ũ QG ΓŨQG = 1 F 1 (ŨQG. Ω QG F 3 Ũ QG/t= = U,QG = Q(U L (R 3, where U. U is a simplified notation for v. U if U = (v, θ, and where Γ is the operator of order two defined by (3 Γ def = F 1 (ν 1 + ν + ν F 3. We can easily show that if U QG is a vector field such that QU QG = U QG, U QG. Ω QG = Ω(U QG. U QG which easily allows us to rewrite system ( QG into: (4 { t Ũ QG ΓŨQG + Q(ŨQG. ŨQG = Ũ QG/t= = U,QG. Remark 1. We can notice that ΓŨQG = QLŨQG, and that the projection of the advection term PP(ŨQG. Ω QG has a zero potential vorticity. In this paper we will focus on the case of a more regular initial data: U Ḣ1 (R 3 Ḣ 1 (R 3 and U,QG L (R 3, with the aim of studying solutions which are unique. Even if there is no scale invariance for this system, it is very natural to choose initial data in Ḣ 1. Indeed such a space is scale invariant for the three dimensional Navier- Stokes equations and as A is skewsymmetric and vanishes from any energy estimates we can, as for the Leray theorem, adapt the Fujita-Kato theorem to the primitive system (see for instance [1] or [5]: there exist a unique maximal time T >, and a unique solution U C([, T [, Ḣ 1 L loc ([, T [, Ḣ 3. Moreover, if T is finite, then we have T U (t dt = + ; finally there exists a constant c such that if U Ḣ 3 Ḣ (R 3 1 cν (R 3 then T = +. Contrary to the Leray solutions, the solutions are unique but we do not know whether they are global in general. The Fujita-Kato theorem also works on system ( QG, and again, it does not say whether the unique solution is global if we do not have a small initial data. In the following, we show like in [11] or [9] (the only difference is that we are in the whole space and not in the periodic case, but that does not change anything in the arguments that the quasigeostrophic system has a unique and global solution without any assumption on the smallness of the initial data: Theorem Assume U,QG H 1 (R 3, then the limit system ( QG has a unique solution, global in time in the space L (R +, H 1 L (R +, H.

5 THE LIMIT SYSTEM ( QG, PROOF OF THEOREM 5 Note that U Ḣ1 (R 3 Ḣ 1 (R 3 and U,QG L (R 3 imply U,QG H 1 (R 3. The global lifespan holds thanks to the special form of the system satisfied by the potential vorticity Ω QG, where like in -D Navier-Stokes, and unlike in 3-D Navier-Stokes, there is no stretching term such as Ω. U. The main point of this paper is to show, using dispersion phenomena induced by the operator L + 1 A, that if is small enough, then system (P E has a unique solution, global in time, and which converges to ŨQG in a certain sense, as goes to zero. This is the purpose of: Theorem 3 Assume that U Ḣ1 (R 3 Ḣ 1 (R 3 and U,QG L (R 3. Let us define for s R, E s def = L (R +, Ḣs L (R +, Ḣs+1 and let W be a solution of the following linear system: t W LW + 1 (5 PAW = G W /t= = U,osc = P(U (6 with G def = PP(ŨQG. ŨQG F (ν ν F Then we have the following results: F 3 F 1 3 ( W exists globally and is unique in the space E s for every s [ 1, 1]. Moreover W L (R +,L as. Ω QG. If we denote by γ def = U ŨQG W, then if is small enough, γ E s and converges to zero in this space E s for every s [ 1, 1]. Finally if is small enough U is defined for all time in E s and U ŨQG = γ + W goes to zero as goes to zero, in the space L (R +, L. This paper is structured in the following way: Section will be devoted to the quasigeostrophic system ( QG (Theorem, and in Section 3 we will prove Theorem 3. We will use many results or computations developped in [4], and some technical results are given in an appendix. The limit system ( QG, proof of Theorem The following method is exactly the same as for the periodic case ([11] or [9] so we will explain it without giving details. As we have seen we can study equivalently ( QG or (4. It is classical that there exist a unique maximal time T and a unique solution in C([, T [, H 1 L loc ([, T [, H. In our case, the Ḣ1 regularity of the initial data gives enough regularity to the potential vorticity Ω QG C([, T [, L L loc ([, T [, Ḣ1, so that we can compute a scalar product in L and obtain that for all t < T (ν = min(ν, ν > :

6 3 CONVERGENCE OF THE STRONG SOLUTIONS 6 It follows that Ω QG (t L + cν Ω QG (s L ds = Ω QG ( L C U. Ḣ 1 ŨQG(s Ḣ ds C Then from this, we easily find that : T Ω QG (s L ds <. ŨQG(t Ḣ 1 dt <, which by the usual blow-up criterion implies that ( QG has a unique, global solution when U,QG H 1, moreover: (7 t R +, ŨQG(t + cν Ḣ s ŨQG(t dt C(U Ḣ s+1, and Theorem is proved. 3 Convergence of the strong solutions In this section we will prove the main theorem of this paper : Theorem 3. We will first reformulate the limit system and define the systems on which we will work. Then we will provide energy estimates for these auxiliary systems, and we will finally be able to conclude. 3.1 The different systems : strategy of proof Let us rewrite the primitive system, using the Leray projector on the divergence-free vectorfields : t U + P(U. U LU + 1 (8 PAU = U /t= = U Let us begin by noticing that ( QG is equivalent to the following system: (9 t Ũ QG LŨQG + 1 PAŨQG = P(ŨQG. ŨQG + G Ũ QG/t= = U,QG Ḣ1 (R 3 L (R 3, (1 where G = PP(ŨQG. ŨQG F (ν ν F F 3 F 1 3 ( Ω QG. It is important to notice that G is the sum of two terms, both divergence-free and whose potential vorticity is zero (it is an application of remark 1. for the first term, and a simple computation for the second one, which will be important in the following.

7 3 CONVERGENCE OF THE STRONG SOLUTIONS 7 def As in [7] for rotating fluids, we realize that the study of the simple difference V = U ŨQG will not give any clue for the convergence. Indeed, on the one hand if we compute the difference of the two systems we obtain a new system that has a fixed external force which prevents us from getting any convergence to zero by Gronwall methods, and on the other hand, we need truncations in frequency space to use dispersion results. So in the spirit of [7] we will use the solution of the linear oscillating system in order to make the term G oscillate, in the sense that we consider the linearized system, with the term G as an external force. And, in order to use the Strichartz estimates proved in the appendix we are led to introducing the following systems : (11 (1 { t W LW + 1 PAW = G W /t= = U,osc t W r,r LW r,r + 1 r,r PAW = P r,r P 3+4 G W r,r /t= = P r,r P 3+4 U,osc, where P 3+4 is the projection onto the last two eigenvectors of the matrix L 1 PA (see in the appendix, and where P r,r is a frequency cut-off : (13 P r,r = χ( D R (1 χ( D 3 r, with χ is a fixed C function whose support is included in [ 1, 1] and equal to 1 in [ 1, 1 ], and (F 1 is the inverse Fourier transform: χ( D f = F (χ( ξ 1 ˆf(ξ and χ( D 3 f = F (χ( ξ 1 3 ˆf(ξ. As we will see in the following, with this external force, when we compute the equation satisfied by U ŨQG W r,r, the new external force will be G P r,r P 3+4 G which we will be able to make arbitrary small. Then we define the two following quantities : (14 def δ = U ŨQG W r,r def = W W r,r. δ In the rest of this paper we will state and prove energy estimates for (11 and (1, then we will use them to bound the energies of δ and δ, and show they are small. It is here that we see the reason to use system (1: truncation of the frequencies in C r,r allows us not only to use the estimates from [4] and the Strichartz estimates, but also to control the initial data and the external force in the energy estimates of δ and δ : once r and R are fixed to make these two terms small enough (smaller than an arbitrary η, we can show thanks to the given by the Strichartz estimates that these two energy estimates are also smaller than η. So γ, which is the difference between δ and δ, is finally smaller than 4η, for small enough (depending on r and R. We will next deduce that W goes to zero in L (R +, L, and using the Sobolev injection Ḣ (R 3 L (R 3 we will finally get that U ŨQG goes to zero in the space L (R +, L.

8 3 CONVERGENCE OF THE STRONG SOLUTIONS 8 3. Energy estimates for the linear systems Until the end of the paper we will use the following notation for constants: a constant which only depends on U, ν or ν ν shall be written C, and if it also depends on r and R it shall be written C r,r. Using the fact that ŨQG is globally defined and belongs to L (R +, H 1 L (R +, H (recall Theorem, we shall show that systems (11 and (1 have a unique global solution in E s = L (R +, Ḣs L (R +, Ḣs+1 for all s [ 1, 1]. From now on we take U Ḣ 1 (R 3 Ḣ1 (R 3 and require that its quasigeostrophic part U,QG belongs to L (R 3. Lemma 3.1 There exists a constant C such that, for all s [ 1, 1] and t R +, the solutions of (11 and (1 satisfy respectively (with ν = min(ν, ν > : W r,r (t Ḣ s + ν and W r,r (t Ḣ s dt C (1 + ν ν C r,r W (t + ν Ḣ s W (t dt C. Ḣ s Proof: We will prove the first estimate (the other one is proved using the very same computation and is in fact simpler. Taking the inner product (in Ḣs of (1 with and using Lemma 4.4, we get: W r,r 1 d dt W r,r Ḣ s + ν W r,r where we have defined (1 + ν ν C Ḣ s r,r G b Ḣs W r,r Ḣs (15 G b = PP(ŨQG. ŨQG and G l = F (ν ν F +(1 + ν ν C r,r G l Ḣs 1 W r,r Ḣs+1, F 3 F 1 3 ( Ω QG. Using the classical inequality ab 1 (a + b, multiplying by e R t Gb (τ Ḣs dτ and integrating, we find : W r,r (t Ḣ s + ν W r,r (t R t e Ḣ s +(1 + ν ν C r,r t G b (τ Ḣs dτ dt W r,r ( R t e Ḣ s Gb (τ Ḣs dτ ( G b Ḣs + 1 G l R t e ν Ḣ s 1 t G b (τ Ḣs dτ dt. Let us state the following lemma which will conclude the proof of these energy estimates: Lemma 3. There exists a constant C such that for all s [ 1, 1] we have: G b (t Ḣsdt + G l (t Ḣ s 1 dt C

9 3 CONVERGENCE OF THE STRONG SOLUTIONS 9 Proof of Lemma 3.: Because of the conditions about the orders when we compute a product in Sobolev spaces, we have to distinguish the case when s = 1 from the case when s ] 1, 1] and obtain: ŨQG G b L Ḣsdt C (R +,Ḣ1 ŨQG L (R +,Ḣ if s = 1 if s ] 1, 1]. ŨQG L (R +,Ḣs+1 Then, thanks to the energy estimate (7, and classical interpolation arguments : G b L 1 (R +,Ḣs C ν ( U,QG L + U,QG Ḣ1. Similarly, using (6 we also obtain the estimate for G l and this concludes the L (R +,Ḣs 1 proof of Lemma 3.. Let us go back to the proof of Lemma 3.1: thanks to Lemma 4.4 and the interpolation results in Sobolev spaces, we can estimate W r,r ( Ḣs and using Lemma 3., for all t R +, W r,r (t Ḣ s + ν W r,r (t Ḣ s dt C (1 + ν ν C r,r, which proves Lemma 3.1 (the computation for W being identical. 3.3 Energy estimates for δ and δ The aim of this subsection is the proof of the following energy estimates: Proposition 3.1 There exist constants C, C r,r, a function V L 1 (R and a function I(r, R (also depending also on ν ν, ν and U such that if is small enough (in a neighbourhood of zero which depends on r and R for all s in [ 1, 1] and for all t, and δ (t + ν Ḣ s δ (t dt Ḣ s ( I(r, R + C r,r e C +C r,r, δ (t Ḣ s + R t (ν C δ (t Ḣs δ (t e Ḣ s t V (τdτ dt ( I(r, R C r,r e C +C r,r, where we have the estimate V L 1 (R + C + C r,r and I(r, R goes to zero as r goes to zero and R goes to Notation and estimates From systems (11 and (1, and using that for any function g (see (13 for the definition of P r,r and definition 4. in the appendix for P, g P r,r P 3+4 g = ( 1 P r,r g + Pr,R P g,

10 3 CONVERGENCE OF THE STRONG SOLUTIONS 1 we can write that δ satisfies the following system: (16 with { t δ Lδ + 1 PAδ = f 1 + f δ /t= = δ, f 1 = ( 1 P r,r G b P r,r P G b, f = ( 1 P r,r G l P r,r P G l, δ = ( 1 P r,r U,osc + P r,r P U,osc. Remark 3.1 Remember that G l, G b and U,osc are divergence free, and have a zero potential vorticity (see the expression of G in (9 and of G l and G b in (15. From systems (8, (9, and (1, we get the system satisfied by δ : (17 with (18 t δ Lδ + 1 PAδ = P(δ. δ P(δ. (ŨQG + W r,r P((ŨQG + W r,r. δ +f 1 + f δ /t= = δ, { f 1 = f 1 f = f P(W r,r. ŨQG P(ŨQG. W r,r P(W r,r. W r,r The aim of the following subsections is the proof of Proposition 3.1: we will successively establish estimates on δ and δ, estimates on the external forces and then we will end the proof Energy for δ Let us take the inner product in Ḣs of (16 with δ (where s [ 1, 1], use the usual argument (ab 1 a + 1 b followed by a use of a Gronwall lemma as in the proof of def Lemma 3.1, we obtain that, noting u L p Ḣ = u s L p (R +,Ḣs (R 3, t : (19 δ (t + ν Ḣ s δ (t dt Ḣ s Energy for δ The inner product in Ḣs, yields: ( δ Ḣ s + f 1 L 1 Ḣ s + 1 ν f (t L Ḣs 1 e f 1 L 1Ḣs. 1 d dt δ + ν Ḣ s δ (P(δ Ḣ s. δ δ Ḣs + (P(δ. (ŨQG + W r,r δ Ḣs + (P((ŨQG + W r,r. δ δ Ḣs + (f 1 δ Ḣs + (f δ Ḣs

11 3 CONVERGENCE OF THE STRONG SOLUTIONS 11 Lemma 3.3 There exists a constant C such that the following estimates are true for all s [ 1, 1]: ( (P(δ. δ δ Ḣs C δ Ḣs δ Ḣ s (1 (P(δ. (ŨQG + W r,r δ Ḣs ν 6 δ Ḣ s+1 + C ν (ŨQG + W r,r Ḣ 1 δ Ḣ s ( (P((ŨQG + W r,r. δ δ Ḣs ν 6 δ + e Ḣ s+1 s (t δ Ḣ s C ν where e s (t = 3 ŨQG + W r,r Ḣ 1 ŨQG + W r,r if s = 1 Ḣ 3 C ν ŨQG + W r,r if s ] 1 Ḣ s+1, 1]. Proof: we use the fact that δ is divergence-free together with Sobolev product laws as well as the fact that if s > 1, Ḣs+1 is a Banach algebra. For example let us prove the last estimate ( : (P((ŨQG + W r,r. δ δ Ḣs (ŨQG + W r,r. δ Ḣs δ Ḣs. It is necessary to separate cases relatively to the Sobolev order (and using Sobolev product laws: ((P(ŨQG + W r,r C ŨQG + W r,r Ḣ1 δ Ḣ 1 δ Ḣ1 if s = 1. δ δ Ḣs C div ((ŨQG + W r,r δ Ḣs δ Ḣs if s ] 1, 1] Using Sobolev inclusions and interpolation, together with the fact that Ḣs+1 is a Banach algebra when s > 1 ap and the classical inequality ab p + bq q if 1 p + 1 q = 1 with p = and p = 4 leads to (e s is as announced in the lemma : (P(ŨQG + W r,r. δ δ Ḣs ν 6 δ Ḣ s+1 + e s (t δ Ḣ s The other terms are similar and that proves Lemma 3.3. Let us go back to the energy estimate: for all t, 1 d dt δ + ν Ḣ s δ C δ Ḣ s Ḣs δ + ν Ḣ s 6 δ Ḣ s+1 + C (ŨQG + W r,r δ ν Ḣ 1 + ν Ḣ s 6 δ + e Ḣ s+1 s (t δ Ḣ s Posing V (t = C ν (ŨQG + W r,r Ḣ 1 implies that for all t : + 1 f 1 Ḣs + 1 f 1 Ḣs δ Ḣ s + 1 ν f Ḣ s 1 + ν 6 δ Ḣ s+1. + e s (t + f 1 Ḣs, the usual Gronwall argument

12 3 CONVERGENCE OF THE STRONG SOLUTIONS 1 (3 δ (t Ḣ s + (ν C δ (t Ḣs δ (t e Ḣ s t V (τdτ dt R t ( δ ( Ḣ s + f 1 L 1 Ḣ s + 1 ν f L Ḣs 1 e V L 1 (R +, In the following we will obtain that if is in a small neighbourhood of zero (depending on the radiuses r and R we can bound all these quantities (including the L 1 norm of V Estimates on the external forces The estimates concerning the external forces are the following: Lemma 3.4 There exist a constant C, a function α(r, R bounded by C, and going to zero as r goes to zero, and R goes to infinity, and a constant C r,r such that if < 1 we have the following estimates f 1 L 1 (R +,Ḣs + f L (R +,Ḣs 1 + δ Ḣ s α(r, R + C r,r, and if is in a small neighbourhood of zero (depending on r and R, f L (R +,Ḣs 1 α(r, R C r,r. Proof: The function α is easily given by the Lebesgue Theorem together with Lemma 3., precisely: α(r, R = ( 1 P r,r G b L 1 Ḣ s + ( 1 P r,r G l L Ḣs 1 + ( 1 P r,r U,osc Ḣ s. Lemmas 4.3 and 3. give the first estimate. In order to get the last estimate, all we have to do is to estimate the last three terms in the expression of f (because all the other terms have just been estimated. By interpolation, we have: ŨQG. W r,r Ḣs 1 ŨQG. W r,r 1 r,r ŨQG. W Ḣs 1 L. Thanks to the product laws in Sobolev spaces (s 1 + s 3 = s and s [ 1, 1] we can write that: ŨQG. W r,r Ḣs 1 C ŨQG 1 Ḣ s 1 W r,r 1 Ḣs ŨQG 1 L W r,r 1 L. Using the energy estimates for ŨQG and W r,r as well as the fact that s 1 [, 1 ] and the Bernstein Lemma ( W r,r L CR W r,r L, we obtain (where the notation L p L q means L p (R +, L q (R 3 and similarly L p Ḣ s means L p (R +, Ḣs (R 3 : ŨQG. W r,r L (R +,Ḣs 1 C R ŨQG 1 L Ḣs 1 W r,r 1 L Ḣs+1 ŨQG 1 L L W r,r 1 L L. Let us now state the Strichartz estimates whose proof is given in the appendix:

13 3 CONVERGENCE OF THE STRONG SOLUTIONS 13 Lemma 3.5 There exists a constant C r,r such that if is small enough (the neighbourhood of zero depends on r, and R, we have the following estimate: W r,r L (R +,L (R C r,r. Using these Strichartz estimates together with Lemma 3.1, and (7 we obtain : (4 ŨQG. W r,r L (R +,Ḣs C r,r. Similarly : (5 W r,r. ŨQG L (R +,Ḣs C r,r and W r,r. W r,r L (R +,Ḣs C r,r. This concludes the proof of Lemma Conclusion: end of the proof of Proposition 3.1 We are now able to achieve the proof of Proposition 3.1. First let us bound the norm of function V introduced with (3: there exists a constant C such that: V L 1 (R + C (1 + ν ν C r,r 4 + G b r,r L 1 Ḣ s + C r,r which we can rewrite into (using Lemma 3.4: V L 1 (R + C + C r,r. Using the estimates given by (3, (19, and the ones on the external forces in Lemma 3.4 we can write that for all t : δ (t + ν Ḣ s δ (t ds Ḣ s ( I(r, R + ( C r,r + C r,r + 1 ν C r,r e C +C r,r, and if is close to zero (in a neighbourhood of zero depending on r and R: δ (t Ḣ s + (ν C δ (t Ḣs δ (t e Ḣ s t V (τdτ dt R t (I(r, R + ( C r,r + C r,r + 1 ν 1 16 C r,r e C +C r,r where we define (see Lemma 3.4 for the definition of α: I(r, R def = ( + 1 ν α(r, R, since Lemma 3.4 implies that I(r, R goes to zero when r goes to zero and R goes to +, and that 1, we finally obtain all the results stated in Proposition 3.1.

14 3 CONVERGENCE OF THE STRONG SOLUTIONS Convergence of solutions : proof of theorem 3 Let < η ν C and fix r small and R large such that we have (see Lemma 3.4: (6 I(r, Re C η 4. From now on and until the end of the paper, r and R are fixed like this. Moreover, in order to simplify notation in the following computations, we will write directly the estimate on δ instead of an approximation (such as for example the one given by the Friedrichs scheme as should be done. Let us fix < (ν, ν, U, r, R 1 such that, C r,r C and let us define the time T = sup{t such that δ C([, t], Ḣs and t t, δ (t Ḣ s η}. Let T 1 be a lifespan of a solution δ in E 1 = L ([, T 1 ], Ḣ 1 L ([, T 1 ], Ḣ 3 : with our initial data an adaptation of the Fujita and Kato Theorem with external force proves that such a T 1 > exists and on this interval the solution is in E s for all s [ 1, 1]. Let us fix < 1 (ν, ν, U, r, R such that 1, δ ( Ḣ s I(r, R + C r,r η 6 + Cr,R η. So consequently, T is well defined and strictly positive: < t < T ν C δ Ḣs ν. By (6 t < T, the energy estimate on δ becomes: δ (t Ḣ s + ν δ (t Ḣ s dt η C r,r e C. The definition of 1 also implies that the estimate for δ becomes: δ (t + ν Ḣ s δ (t dt η Ḣ s 4 + Cr,R e C. So there exists = (ν, ν, ν, U, r, R 1 such that and t < T, δ (t Ḣ s + ν δ (t Ḣ s dt η ν 4C and δ (t Ḣ s +ν δ (t Ḣ s dt η. This implies that, T = + (bootstrap and also that δ δ = U ŨQG W satisfies for all and t : (δ δ (t Ḣ s + ν (δ δ (s Ḣ s ds η + η = η and that achieves the proof of the convergence of γ. As W = δ + W r,r, and knowing that, thanks to the Strichartz estimate given in Lemma 3.5, if is close to zero: L (R +,L (R C r,r, we can define 3 = 3 (ν, ν, ν, U, r, R such that 3 and t, W r,r L (R +,L (R 3 η but with the case s = 1, we also have the estimate δ (t + ν Ḣ 1 δ (t Ḣ 1dt η, so as the dimension is 3, Ḣ L, we have δ L (R +,L η ν and finally: W r,r

15 4 APPENDIX 15 W L (R +,L (R 3 ( ν η Which gives (remember that U ŨQG = γ + W and use another time the injection Ḣ (R 3 L (R 3 that U ŨQG L (R +,L (R 3 goes to zero and achieves the proof of Theorem 3. 4 Appendix 4.1 Some technical precisions We won t give details for the proofs of the lemmas from this section. Definition 4.1 If < r < R, let C r,r def = {ξ R 3 such that ξ R, and ξ 3 r}. Let us denote by B(ξ, the following matrix (in Fourier variables: B(ξ, def = L 1 PA = ν ξ + ξ 1ξ ξ + ξ 3 ξ 1 ξ 3 ξ ξ F ξ ξ 1 + ξ 3 ξ ν ξ ξ 1ξ ξ ξ 3 ξ F ξ ξ ξ 3 ξ ξ 1ξ 3 ξ ν ξ ξ 1 + ξ F ξ 1 ν ξ F Then we have the following lemmas (we refer to [4] for the proofs:. Lemma 4.1 The matrix B is diagonalizable with four distinct eigenvalues we will denote the following way: µ = ν ξ µ which is real λ and λ. Lemma 4. When ξ C r,r, and is close to zero (the neighbourhood of zero depends on r and R, µ = (νξ1 + νξ + ν F ξ3 ξ ξ + I( F where the function τ is defined by: λ = τ(ξ ξ + i ξ F F ξ + is(ξ, + S (ξ, (7 τ(ξ def = ν ( 1 + F ξ 3 ξ F + ν ( 1 F ξ 3 ξ F ν > ξ C r,r and I, S, and S are functions of and ξ uniformly bounded in C r,r with respect to.

16 4 APPENDIX 16 Remark 4.1 The leading part of the asymptotic expansion of µ is exactly the expression in terms of the Fourier variable of operator Γ. A basis of eigenvectors W 1 (ξ,,..., W 4 (ξ, corresponding respectively to the previous eigenvalues is given by the following matrix of change of basis (noting A def = µ + ν ξ and B def = λ + ν ξ : ξ ξ 3 ξ 3 (ξ 1 A + ξ ξ 3 (ξ 1 B + ξ ξ 3 (ξ 1 B + ξ ξ 1 ξ 3 ξ 3 (ξ A ξ 1 ξ 3 (ξ B ξ 1 ξ 3 (ξ B ξ 1 F (ν ν ξ ξ3 A(ξ1 + ξ B(ξ 1 + ξ B(ξ 1 + ξ F ξ3 F ( ξ A + ξ3 F ( ξ B + ξ3 F ( ξ B + ξ3 As W 1 is not divergence free, a divergence free vector field has no coordinate along this vector, so Û,osc depends only on the last three eigenvectors : Û,osc = K W +K 3 W 3 + K 4 W 4. Let us give the following definition Definition 4. If g is a divergence free vector field (with four components but depending on x R 3, we can write: g = 4 i= P ig, where for i {, 3, 4}, P i g = F 1 (K i W i is the inverse Fourier of the projection onto the span of W i. Let us now state the following lemma showing that P has a small norm: Lemma 4.3 Let < r < R, then there exists a constant c r,r > such that for all functions g satisfying Pg = g and Pg = g, (i.e with zero divergence and zero potential vorticity, and supp(ĝ C r,r, we have: ξ C r,r, P g(ξ c r,r ν ν ĝ(ξ and consequently: s R, P g Ḣs c r,r ν ν g Ḣs. Let us state a useful lemma which is a consequence of Lemma 4.3: we define χ a C function whose support is included in [ 1, 1] and equal to 1 in [ 1, 1 ]. Lemma 4.4 There exists a constant C r,r such that for all s R and for all g divergence free and with a zero potential vorticity, denoting by P 3+4 the projector P 3 + P 4 (see 13 for the definition of operator P r,r : P r,r P 3+4 g Ḣs (1 + ν ν C r,r g Ḣs. 4. Proof of the Strichartz estimates We refer to [13] for Strichartz estimates in a more general case for the wave equation, to [7] for Strichartz estimates suited for the rotating fluid system, and to [4] for the case of the primitive system. Let us recall that r and R were fixed in (6. In this section we will prove the Strichartz estimates given in Lemma 3.5 about W r,r which satisfies system (1. Let us begin by stating the corresponding dispersion inequality (see [4] for the proof. Let us denote by K the following operator (see definition 4. for the notations: K(, t, z = ψ(ξe tτ(ξ ξ t ξ +i F F ξ +its(ξ,+ ts (ξ,+iz.ξ dξ, R 3 where

17 4 APPENDIX 17 the function ψ C (R3 is radial, supported in C r,r, and equals 1 near C r,r. ξ C r,r, τ(ξ ν >, according to (7. S, S and their derivatives with respect to ξ are bounded on C r,r by a constant A r,r. Then, according to [4] there exists a constant C r,r such that if is small enough: sup K(, t, z C r,r e ν 16 r t ( z R 3 t 1. If we note S λ the semigroup associated with (1 and: (8 w = P r,r P 3+4 U,osc g b = P r,r P 3+4 G b g l = P r,r P 3+4 G l then the Duhamel formula writes: W r,r (t = S λ (tw r,r ( + S λ (t t g b (t dt + S λ (t t g l (t dt. So it depends only on the last two Fourier eigenvalues, and like in [4] (section we will deal with the computation for the projection P 3 (the computation is the same for P 4 and use Lemma 4.4 to conclude. In the following we will state the estimates separately for each of the three terms in the previous expression. We will only prove the last one which differs a little from the classical proof Homogeneous case and bilinear inhomogeneous case Lemma 4.5 There exists a constant C r,r (r and R are fixed such that if is small enough, ν 3 t L 8 ( S λ (tw + S λ (t t g b (t dt C r,r 1 8 ( w L + g b L 1 L. (R +,L (R Linear inhomogeneous case Here we will adapt slightly the estimates of [4] for our purposes. Lemma 4.6 There exists a constant (r and R are fixed such that if is small enough, ν 3 t L 4 S λ (t t g l (t dt C r,r 1 4 g l L L (R +,L (R 3 Proof: This case is a little more complicated than those studied in [4] because we want a different bound on the right-hand side and it is necessary for our study. If we note B = {φ C (R + R 3 such that φ L (R +,L 1 (R 3 1}, then we have: S λ (t t g l (t dt L (R +,L (R 3 = sup φ B R 3 ( S λ (t t g l (t dt φ(t, xdxdt.

18 4 APPENDIX 18 Then, using the Fubini and Plancherel theorems (and denote by O the term (t sis(ξ, + (ts (ξ, + ss (ξ, as this term is easily dealt within the proof of the dispersion estimate: L S λ (t t g l (t dt (R +,L (R 3 C sup φ B t R 3 e (t t τ(ξ ξ +i(t t ξ F F ξ +O ĝ l (t φ(t, ξχ(ξdξdtdt Another use of the Fubini interversion theorem, the Cauchy-Schwarz inequality (in space bring us to: L S λ (t t g l (t dt C sup (R +,L (R 3 φ B ( e (t+s t τ(ξ ξ ξ +i(t s F R 3 t t g l (t L (R 3 1 F ξ +O χ(ξ φ(t, ξ φ(s, ξdtdsdξ dt. The Cauchy-Schwarz inequality (in time followed by the Fubini and Plancherel theorems give: sup φ B S λ (t t g l (t dt L (R +,L (R 3 C g l (t L (R +,L (R 3 ( t t [ R 3 ( e ix.ξ e (t+s t τ(ξ ξ ξ +i(t s F F ξ +O χ(ξ dξ R 3 1 ] (φ(t φ(s(xdx dtdsdt. Where f g(x = R f(x yg(ydy is the usual convolution. Then, the dispersive estimate, and the fact that L 1 L 1 L 1 imply: and sup φ B where S λ (t t g l (t dt L (R +,L (R 3 C g l (t L (R +,L (R 3 ( t t C r,r e ν 16 r (t+s t 1 t s 1 φ(t L 1 (R 3 φ(s L 1 (R 3 dtdsdt 1, S λ (t t g l (t dt L (R +,L (R 3 C r,r g l (t L (R +,L (R 3 sup φ B ( 1 I 1,

19 4 APPENDIX 19 I def = t t e ν 16 r (t+s t t s 1 φ(t L 1 (R 3 φ(s L 1 (R 3 dtdsdt All we need is to compute this integral. The Fubini theorem implies: I = e ν 16 r (t+s t s 1 φ(t L 1 (R 3 φ(s L 1 (R 3 ( Calculating the integral in t, which is exactly min(s,t e ν 8 r t dt gives I e ν 16 r (t+s t s 1 As s + t min(s, t = t s, we have: φ(t L 1 (R 3 φ(s L 1 (R 3 1 {t t}1 {t s}e ν 16 r ( t dt dtds ( e ν 16 r min(s,t ν 8 r dtds I 8 r ν e ν 16 r t s t s 1 8 r ν φ(t L 1 (R 3 φ(s L 1 (R 3 dtds φ(t L 1 (R 3 ( Let us define the two following functions: ν a : y e 16 r y L 1 (R y 1 b : y φ(y L 11 {y } L (R Then a b L and the Cauchy-Schwarz inequality gives: I enough, e ν 16 r t s φ(s L 1 (R 3 ds dt C r 3 ν 3. So, if is small (9 L S λ (t t g l (t dt 1 4 C r,r g l (R +,L (R 3 ν 3 L L Conclusion According to the previous results and the Duhamel formula, we can write that: (3 W r,r L (R +,L (R 3 C r,r 1 8 ν 3 8 ( w L + g b L 1 L + C r,r 1 4 ν 3 4 g l L L In order to achieve the proof of the Strichartz estimates we still have to estimate the norms of w, g g, and g l (see (8 for their definition: as supp(ŵ C r,r, there exists a constant C such that: w L C r w Ḣ1 and using Lemma 4.4 and (8, we have: w L C r (1 + ν ν C r,r U Ḣ1. The same argument, and the use of estimates close to those from Lemma 3.4 imply : g b L 1 L C rν (1 + ν ν C r,r U Ḣ 1. And similarly we also obtain that gl L L C ν (1 + ν ν C r,r U Ḣ1.

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