La lecture en ligne est gratuite
Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

Partagez cette publication

Abstract.In this paper we show that the quasigeostrophic system is well approximated by the primitive systems. More precisely, we prove that if the initial data are weakly well-prepared then the maximal time existence of the regular solution of the primitive system goes to infinity and the regular solution goes to the solution of the quasigeostrophic sys-tem, strongly on an arbitrary time interval. By weakly well-prepared initial data we mean that the initial data of the primitive systems is converging to an initial data with zero oscillating part, without any assumptions on the speed.
R´esum´e.-peiansebtelquysesemt`uaeq´gistsoehporeuqiDansceattrcielnoomtner proxim´eparlessyst`emesprimitifs.Pluspre´cis´ement,onmontreque,danslecasdes donn´eesinitialesfaiblementbienpr´epar´ees,letempsmaximaldexistencedelasolution r´eguli`eredusyste`meprimitiftendverslinnietlasolutionre´guli`eredusyst`emeprimitif tendverslasolutiondusyst`emequasig´eostrophique,etcefortementsurtoutintervallede tempsborn´e.Pardonne´esinitialesfaiblementbienpr´epar´ees,oncomprenddesdonn´ees initialesquiconvergentversunedonne´einitialeaveclapartieoscillantenulle,sansaucune hypothe`sesurlavitesse.
Introduction The well-known quasigeostrophic system (QG) has been extensively used in oceanogra-phy and meteorology for modeling and forecasting mid-latitude oceanic and atmospheric circulation. This system is obtained by taking the limit onεin a family of primitive systems. The primitive models are given by 1 1 tU+v∙ rU+AU= (−rΦ,0) ε ε (P Eε) divv= 0 U=U0 t=0 3 whereU(t, x) = (v(t, x), T(t, x)), vis a vector field onRdepending on the time,Tis a scalar function and   01 0 0 1 0 0 0 A=.   0 0 0 1 0 01 0 Physically,vis the velocity,Tis the potential temperature andεis proportional to the Rossby number. When the Rossby number is small, the fluid is highly rotating. For further details about the physical significance of these systems, see [14]. 1
Before taking the limit onεin (P Eε) we need to define the potential vorticity Ω =1v22v13T and the oscillating (or ageostrophic) part Uosc= (vosc, Tosc) given by 1 11 v=v+2Δ Ω osc 2 21 =vΩ vc1Δ os 3 3 v=v osc 1 Tosc=T+3Δ Ω. Some formal calculus show that taking the limit onεimplies that the oscillating part of the limit is vanishing, therefore the limit of the primitive systems forε0 may be written as the following quasigeostrophic system (see [3], [6]): tΩ +v∙ rΩ = 0 11 v= (2Δ Ω, ∂1Δ Ω,0) (QG)1 T=3Δ Ω U0,osc= 0. It is easy to see that the first equation in (QG) implies the conservation of theL-norm of 2 Ω and that, in general, the (QG) system behaves like the Euler system inRmethods. The 2 used to prove the well-posedness of the Euler system inRare easily adjustable in order to prove that the (QGFor the () system has a global strong solution, too. P Eε) system the classical theory for quasi-linear, symmetric hyperbolic systems shows the local existence of strong solutions. The problem which appears is whether the solution of (P Eε) converges to the solution of (QGBeale and A. Bourgeois considered in [3] the same problem with). T. periodic boundary conditions in the horizontal directions and rigid boundary conditions ε in the vertical direction. They proved that ifUgoes toU0whereU0,osc= 0 with the 0 ε O(ε)-speed (which means thattUis bounded independently ofε) then the maximal t=0 ε time existence of the regular solution goes to infinity whenε0 and thatU, the solution of (P Eε), goes to the solutionUQGof (QG) with theO(ε)-speed on any bounded time interval. Here we show that, in order to obtain that the maximal time existence of the ε0 ε regular solution goes to infinity, it suffices to assume thatU−→0 and if, in addition, 0,osc ε0 ε −→Ut U0 0hen the solutionUεgoes toUQGHence, weon any bounded time interval. allow oscillations in time of the initial data. Also, our regularity assumptions on the initial data are more general than those of T. Beale and A. Bourgeois, that is we use the space s5 5 H,s >, instead ofH. 2 J.-Y. Chemin in [6] and E. Grenier in [8] are studying the same problem of convergence for models which are modified in the sense that in the equation satisfied byUappears a viscosity termνΔU,ν >0.The ideas of [6] and the methods used to solve the Euler equations are the foundation of this paper. 2
S. Schochet proved in [15] a general theorem of convergence for a class of systems of the same type but with arbitrary convergent initial data non-necessarily well-prepared. He derived a limit system which, in the well-prepared case for the primitive systems that we consider, is the quasigeostrophic system. However, his theorem is valid only on the torus and this hypothesis seems to be important. In the same case of periodic boundary conditions, P. Embid and A. Majda in [7] and A. Babin, A. Mahalov, B. Nicolaenko, Y. Zhou in [1], [2] considered, between other problems, the particular case of primitive systems with an arbitrary convergent initial data non-necessarily well-prepared. They studied in detail the limit system, who was already deduced by S. Schochet. In [1] and [2] a study of small divisors leads to similar results as proved in this paper but in the periodic case and not well-prepared initial data. For the mathematical modeling which leads to these systems we refer to [9], [10], [11] and [12].
1.Notations and assertions s s1 2s3 ˆ We denote by Δ the operator given by ΔU=F(|ξ|U(ξwork in)). We Rand we s3 shall useH, the space of tempered distributions valued inRwhich satisfy Z 1 def 2s2 ˆ2 kUks= ( (1 +|ξ|)|U(ξ)|)<, 3 R where| ∙ |is the usual length. It is easy to see that the two normskUUosck0+kΩks1+ kUoscksandkUksare equivalent. From now on,swill denote a real number strictly greater 5 than . 2 s Definition 1.1.We say thatUis a regular solution on[0,T[ifUC([0,T[;H).
ε s Theorem 1.1.Let(U)ε>0be a bounded family of divergence free vector fields inHsuch 0 that ε2 limU= 0inL . 0,osc ε0 Then there existsTε>0such that ε ε initial dataU i) there existsUa regular solution of(P Eε)on[0,Tε)with0, ii)limTε= +. ε0
ε s Theorem 1.2.Let(U)ε>0be a bounded family of divergence free vector fields inHsuch 0 s that there existsU0,QGHwith(U0,QG)osc= 0and ε2 limU=U0,QGinL . 0 ε0 ε ε Then the solutionsUof(P E)with initial dat e solution of(QG), εaU0converge toUQG, th σ strongly inL(]0,T[;H),for allT<andσ < s. 3tt We shall constantly use the fact that ift >thenHL,His an algebra and 2 kuvktC(kuktkvkL+kvktkukL).In the following we denote byCa constant which ∞ ∞ 3
depends only onsis not difficult to see that the system (. It P Eε) is equivalent to the system ( 1 tU+v∙ rU+AUosc=GP(U, U) g (P Eε)ε U=U0 t=0 where  ! X 1i j GP(U, U) = (−rp,0) =rijΔ (v v),0 i,j or, equivalently, tΩ +v∙ rΩ =q(Uosc, U) 1 (DP Eε)tUosc+v∙ rUosc+AUosc=Q(U, U) ε U=U0 t=0 where
3 3 2 3 1 , U) =∂ v(∂ v∂ v)∂ v∂ v +∂ v∂ v q(Uosc3osc1 2 12 1 osc3 2osc3 +3vosc∙ rT+3(vvosc)∙ rTosc and   11 2Δq(Uosc, U) + [v∙ r, ∂2Δ ]Ω1p 11 1Δq(Uosc, U)[v∙ r, ∂1Δ ]Ω2p   Q(U, U) =,   3p 11 3Δq(Uosc, U) + [v∙ r, ∂3Δ ]Ω (see [6] for the detailed computations). In the estimates onqwe shall use thatq(Uosc, U) is a sum of products of derivatives ofUandUoscplus derivatives ofUoscmultiplied by derivatives ofUUosc.
2.Proofs Proof of theorem 1.1 ε LetM= supkUks.In the following we shall denote byC(M) a constant which ε>0o depends only onsandMand byoM(ε) a constant which depends ons, Mandεwith the property that limoM(ε) = 0 for each fixedM <. LetUbe a regular solution of system ε0 (P Eεthe matrix). Since Ais antisymmetric, the singular term vanishes while making s energy estimates. Hence, applying the operator (Idthe equation verified byΔ) to U 2 s and multiplying by (IdΔ)Uyields 2 s s 2 2 2 2 k tkUk ≤ |<(IdΔ) (v∙ rU)|(IdΔ)U >| ≤CkrUkkUs. s L (for the proof of the last inequality see, for instance, [13]). Gronwall’s lemma yields  Z t (1)kU(t)ks≤ kU(0)ksexpCkrU(τ)kLdτ . 0 4
Z t Letf(t) =CkrU(τ)kLdτ.We find that 0 (2)kU(t)ksMexpf(t). Hence, in order to controlU, we need the control offwill be done by deriving a. This differential inequality forfstart by writing. We (3)krU(t)kL≤ krUosc(t)kL+kr(UUosc)(t)kL. ∞ ∞
1 From the definition ofUoscwe see thatr(UUosc) is a sum of terms of the typeijΔ Ω. But it is well-known that   1kΩks1 kijΔ ΩkLCkΩkLloge+ +CkΩk2, ∞ ∞ L kΩkL (see, for instance, [4], [5]). We also have kΩks1CkUksCMexpf. The definition of the oscillating part shows thatr(UUosc) has as components functions 1 which are vanishing or are of the typeijΔ Ω. Therefore, the two inequalities above yield   CMexpf (4)kr(UUosc)kLCkΩkLloge+ +CkΩk2. ∞ ∞ L kΩkL CM If1, then we find from the relation above kΩkL kr(UUosc)kLCkΩkLlog(e+ expf) +CkΩk2. ∞ ∞ L CM If1, we use the fact that, for allα0, the function kΩkL   α xxloge+ x is increasing to deduce from (4) that kr(UUosc)kLCMlog(e+ expf) +CkΩk2. L In both cases, the following inequality is true kr(UUosc)kLCmax{M,kΩkL,kΩk2}log(e+ expf). ∞ ∞ L 3σ1Now, letσ] + 1, s[. Returning to inequality (3), using thatHLand the above 2 estimate, we obtain (5)krU(t)kL≤ kUosc(t)kσ+Cmax{M,kΩ(t)kL,kΩ(t)k2}log(e+ expf(t)). ∞ ∞ L We need to control theLIndeed, theBut this quantity is almost conserved. norm of Ω. equation of Ω tΩ +v∙ rΩ =q(Uosc, U), 5
together with the particular form ofqimply
Z t  kΩ(t)kL≤ kΩ0kL+kq Uosc(τ), U(τ)kL∞ ∞ 0 Z t ≤ kΩ0kL+CkrUosc(τ)kL(krUosc(τ)kL+kr(UUosc)(τ)kL)∞ ∞ ∞ ∞ 0 Z t ≤ kΩ0kL+CkUosc(τ)kσkU(τ)ksdτ. 0 Using again the basic estimate (2) and the monotonicity off, we obtain kΩ(t)kL≤ kΩ0kL+CM texp(f(t)) supkUosc(τ)kσ. ∞ ∞ 0τt A similar estimate holds forkΩk2. SincekΩ0kLCMand log(e+ expf)C(1 +f)L Cexpf, inserting the above inequality in (5) gives (6)krU(t)kL≤ kUosc(t)kσ+CM(1 +f(t)) +CM texp(2f(t)) supkUosc(τ)kσ. 0τt We shall now estimateUosc. By differentiation of (P Eε) we get 1 1 ttU+vrtU+A∂tU= (−rtΦ,0)tvrU. ε ε Taking the scalar product withtUimplies 1 2 2 k∂ Uk2≤ |< ∂k∂ Uk2. t t L tvrU|tU >| ≤ krUkLL t 2 Thus 22 ktU(t)kL≤ ktUkLexpf(t). t=0 g 2 TheL-norm applied to (P Eε) at timet= 0 gives 1 1 kUk+kvk2+kGP(U0, U)k2≤ kUk+C(M). ktUkL0,osc L0∙ rU0L0L0,osc L 2 2 2 t=0 ε ε 2s1g Taking theL-norm in (P Eε) and using thatHLwe get 1 kAUosc(t)kL≤ kv(t)∙ rU(t)k2+kGP(U(t), U(t))k2+ktU(t)k 2 2 L L L ε 2 CkU(t)ks+ktU(t)kL 2   1 C(M) exp(2f(t+)) 1 kU0,osckL 2 ε 1 C(M) exp(2f(t))(ε+kU0,osckL). 2 ε Since the matrixAis invertible andkU0,osckL0 we have 2 kUosc(t)kLoM(ε) exp(2f(t)). 2 6
Un pour Un
Permettre à tous d'accéder à la lecture
Pour chaque accès à la bibliothèque, YouScribe donne un accès à une personne dans le besoin