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The resolution of the Navier Stokes equations in anisotropic spaces

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28 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
The resolution of the Navier-Stokes equations in anisotropic spaces Dragos¸ IFTIMIE Universite de Rennes 1 IRMAR Campus de Beaulieu 35042 Rennes Cedex France Abstract In this paper we prove global existence and uniqueness for solutions of the 3-dimensional Navier-Stokes equations with small initial data in spaces which are H?i in the i-th direction, ?1 + ?2 + ?3 = 12 , ? 1 2 < ?i < 1 2 and in a space which is L 2 in the first two directions and B 1 2 2,1 in the third direction, where H and B denote the usual homogeneous Sobolev and Besov spaces. Resume Dans cet article on montre l'existence et l'unicite globale des solutions des equations de Navier-Stokes tridimensionnelles pour des donnees initiales petites dans des espaces qui sont H?i dans la ieme direction, ?1 +?2 +?3 = 12 , ? 1 2 < ?i < 1 2 ou dans un espace qui est L2 dans les deux premieres directions et B 1 2 2,1 dans la troisieme direction, ou H et B sont les espaces de Sobolev et de Besov homogenes habituels. Introduction In this paper we study the problem of global existence and uniqueness for so- lutions of the 3-dimensional Navier-Stokes equations. These equations are the following: (N-S) ? ?? ?? ∂tU + U · ?U ? ?∆U = ??P div U(t, ·) = 0 for all t ≥ 0 U |t=0

  • besov spaces

  • let

  • sobolev

  • navier stokes equations

  • p3 ≤

  • supported smooth


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The resolution of the Navier-Stokes equations in anisotropic spaces
Dragos IFTIMIE ¸
Universite´deRennes1 IRMAR Campus de Beaulieu 35042 Rennes Cedex France
Abstract In this paper we prove global existence and uniqueness for solutions of the 3-dimensionalNavier-Stokese´quationswithsmallinitialdatainspaceswhichare Hδiin the i-th direction,δ1+δ2+δ3=12,12< δi<21and in a space which isL2 1 in the first two directions andB22,1in the third direction, whereHandBdenote the usual homogeneous Sobolev and Besov spaces.
R´ ´ esume Danscetarticleonmontrelexistenceetlunicit´eglobaledessolutionsdes ´equationsdeNavier-Stokestridimensionnellespourdesdonn´eesinitialespetites dans des espaces qui sontHδidans la i`emedirection,δ1+δ2+δ3=21,12< δi<12 1 ou dans un espace qui estL2mirexpeusdlensdaceitnoese`erdsritB22,1dans la troisi`emedirection,o`uHetBntsoesselevoseBedtne`gomohdeesacspveleboSo habituels.
Introduction
In this paper we study the problem of global existence and uniqueness for so-lutions of the 3-dimensional Navier-Stokes equations. These equations are the following: tU+U∙ rUνΔU=−rP (N-S)divU(t) = 0 for allt0. U|t=0=U0. Here,U(t x) is a time-dependent three-dimensional vector-field. The goal of this work is to solve these equations in the spaces Hδ123 δ1+δ2+δ321=12< δi<12and in the space1 HB0,0,
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where the first space isHδiin the i-th direction and the second space isL2in 1 the first two directions andB22,1in the third direction, whereHs, respectively Bspq, are We, denote the usual homogeneous Sobolev, respectively Besov, spaces. using homogeneous spaces because they are more easy to handle in the case of the Navier-Stokes equations and, in addition, they are larger than the classical ones, so we obtain more general results. By solving(N-S)in the spaceXwe mean proving the global existence and uniqueness of solutions for small initial data inXand the local existence and uniqueness of solutions for arbitrary initial data inX. The first paragraph is devoted to the study of the spacesHs1,s2,s3, essentially the proof of a product theorem in these spaces. A somewhat similar theorem was provedbyM.Sable´-Tougeronin[10]fortheH¨ormanderspaces. The second paragraph contains the resolution of(N-S)in Hδ123 δ1+δ2+δ3=1212< δi<12. The methods used here are inspired from a paper of J.-Y. Chemin and N. Lerner (see [4]). The case when one of theδiequals21is important but it cannot be studied through our results becauseH21(R difficulty is) is not an algebra. This 1 partially avoided by replacingH21(R) withB22,1(R) which has the property to cancel this critical case. And this is how we come to solve(N-S)in the space HB0,0,21 The same method of replacingduring the third paragraph.HswithBs2,1 may be used in the resolution of general hyperbolic symmetric systems. These systems can be solved in the spaceHs(Rd) > sd2 but the case+ 1s=d2+ 1 cannot be proved unless we replaceHd2+1withB22d,11+(a short proof is given in the Appendix). Finally, the last paragraph makes a comparison between this article and the results which are known. We shall see there that the spaceHB0,0,21is not imbed-3 ded in any of the spaces introduced by H. Kozono and M. Yamazaki in [8],Np,q1,+p, provided that 1qp <32q,p > are not able to prove an imbedding or3. We a nonimbedding ifp32q space. TheHδ123is also interesting if we remark, for instance, that we allow negative values forδi. The results of this article can be easily extended to an arbitrary dimension, d here we considerR3 fact, if we work inonly for sake of simplicity. InR, we can solve(N-S)in the spaces Hδ12,∙∙∙d δ1+δ2+∙ ∙ ∙+δd=d1δ <1 212<i2and in the space ,2 HB0,∙∙∙,01 where the first space isHδiin the i-th direction and the second space isL2in the 1 firstn1 directions andB2 solve the 2Din the last one an 2,1 instance, we c. For
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