Uniform regularity in the low Mach number and inviscid limits for the full Navier-Stokes system in domains with boundaries

Motivated by the studies on the low Mach number limit problem, this manuscript establishes uniform regularity estimates with respect to the Mach number for the non-isentropic compressible Navier-Stokes system in smooth domains with Navier-slip boundary conditions, in the general case of ill-prepared initial data. The thermal conduction is taken into account and the large variation of temperature is allowed. Moreover, the obtained regularity estimates are also uniform in the Reynolds number $\text{Re}\in[1,+\infty),$ P\'eclet number $\text{Pe}\in [1,+\infty),$ provided $$\big|\frac{1}{\text{Re}}-\frac{\iota_0}{\text{Pe}}\big|\lesssim \frac{1}{\text{Pe}^{\frac{1}{2}}}\frac{1}{\text{Re}},$$ where $\iota_0$ is a fixed constant independent of Mach number, Reynolds number and P\'eclet number. The convergence to the limit system when the Mach number tends to zero is then justified for an exterior domain outside a smooth compact set in $\mathbb{R}^3$ in the spirit of \cite{MR2106119}.Comment: Comments are welcome

Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg--de Vries equations

In this note, we extend the detailed study of the linearized dynamics obtained for cnoidal waves of the Korteweg--de Vries equation in \cite{JFA-R} to small-amplitude periodic traveling waves of the generalized Korteweg-de Vries equations that are not subject to Benjamin--Feir instability. With the adapted notion of stability, this provides for such waves, global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. When doing so, we actually prove that such results also hold for waves of arbitrary amplitude satisfying a form of spectral stability designated here as dispersive spectral stability.Comment: 15 page

Stabilité uniforme pour certains systèmes issus de la mécanique des fluides et de la physique des plasmas

This thesis is devoted to the uniform stability and singular limit problems for some fluid systems arising from plasma physics and fluid mechanics. In the first part, we investigate the uniform (with respect to the Reynolds number) stability problem for the Navier-Stokes-Poisson (NSP) system, which is a model describing the dynamics of plasmas. More precisely, for 3d NSP, we construct a unique global solution around the constant equilibrium with a smallness assumption on the perturbation which is independent of the Reynolds number except for the curl part of the velocity. Under a similar assumption, we also obtain the lifespan estimate for 2d NSP which is more delicate due to the weaker dispersion. The ’space-time resonance’ method and the classical parabolic estimates are the main ingredients of the proof. Moreover, the ’splitting’ idea proposed in these works can also be used to get the lower bound for the lifespan of solutions to some fluid systems. In the second part of the thesis, we study the low Mach number limit problem for the isentropic compressible Navier-Stokes equations (CNS) in a domain with fixed and free boundaries. We establish the uniform (with respect to Mach number) high regularity estimates and justify that as the Mach number vanishes, the strong solution of (CNS) converges (in some suitable sense) to that of the incompressible Navier-Stokes equations. In the domain with fixed boundaries, we obtain uniform estimates in a general setting of ill-prepared initial data where the simultaneous appearance of boundary layer effects and the fast oscillation are the main obstacle of the proof. For the (CNS) with a free surface, due to the extra difficulty arising from the uniform regularity of the surface, we prove uniform estimates by allowing the initial data to be slightly well-prepared.Cette thèse est consacrée aux problèmes de stabilité uniforme et de limites singulières des systèmes fluides issus de la physique des plasmas et de la mécanique des fluides. Dans la première partie, nous étudions le problème de stabilité uniforme (par rapport au nombre de Reynolds) pour le système de Navier-Stokes-Poisson (NSP), qui est un modèle décrivant la dynamique des plasmas. Plus précisément, pour 3D NSP, nous construisons une solution globale unique autour de l'équilibre constant avec une hypothèse de petitesse sur la perturbation qui est indépendante du nombre de Reynolds sauf pour la partie rotationnelle de la vitesse. Sous hypothèse similaire, nous obtenons également l'estimation de la durée de vie pour 2d NSP qui est plus délicate en raison de la dispersion plus faible. La méthode de la "résonance espace-temps" et les estimations paraboliques classiques sont les principaux ingrédients de la preuve. De plus, l'idée de "splitting" proposée dans ces travaux peut également être utilisée pour obtenir la borne inférieure de la durée de vie des solutions de certains systèmes fluides. Dans la seconde partie de la thèse, nous étudions le problème de la limite du bas nombre de Mach pour les équations de Navier-Stokes compressibles isentropiques (CNS) dans des domaines à frontières fixes et libres. Nous établissons des estimations uniformes (par rapport au nombre de Mach) à haute régularité et justifions que la solution forte de (CNS) converge (dans un sens approprié) vers celle des équations incompressibles de Navier-Stokes. Dans le domaine aux frontières fixes, on obtient des estimations uniformes dans un cadre général de données initiales mal préparées où l'apparition simultanée d'effets de couche limite et l'oscillation rapide sont le principal obstacle de la preuve. Pour le (CNS) avec une surface libre, en raison de la difficulté supplémentaire résultant de la régularité uniforme de la surface, nous prouvons des estimations uniformes en permettant aux données initiales d'être légèrement bien préparées

Uniform regularity for the compressible Navier-Stokes system with low Mach number in domains with boundaries

We establish uniform with respect to the Mach number regularity estimates for the isentropic compressible Navier-Stokes system in smooth domains with Navier-slip condition on the boundary in the general case of ill-prepared initial data. To match the boundary layer effects due to the fast oscillations and the ill-prepared initial data assumption, we prove uniform estimates in an anisotropic functional framework with only one normal derivative close to the boundary. This allows to prove the local existence of a strong solution on a time interval independent of the Mach number and to justify the incompressible limit through a simple compactness argument.Nous établissons des estimations de régularité uniformes par rapport au nombre de Mach pour le système de Navier-Stokes compressible isentropique dans les domaines réguliers avec condition de Navier au bord dans le cas général de données initiales mal préparées. Pour être cohérent avec les effets de couche limite dus aux oscillations rapides et à l'hypothèse de données initiales mal préparées, nous prouvons des estimations uniformes dans un cadre fonctionnel anisotrope avec une seule dérivée normale proche du bord. Ceci permet de prouver l'existence locale d'une solution forte sur un intervalle de temps indépendant du nombre de Mach et de justifier la limite incompressible par un argument de compacité simple

Incompressible limit for the free surface Navier-Stokes system

93 pages, comments are welcome!We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size $\sqrt{\varepsilon}$, $\varepsilon$ being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations

Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg--de Vries equations

15 pagesInternational audienceIn this note, we extend the detailed study of the linearized dynamics obtained for cnoidal waves of the Korteweg--de Vries equation in \cite{JFA-R} to small-amplitude periodic traveling waves of the generalized Korteweg-de Vries equations that are not subject to Benjamin--Feir instability. With the adapted notion of stability, this provides for such waves, global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. When doing so, we actually prove that such results also hold for waves of arbitrary amplitude satisfying a form of spectral stability designated here as dispersive spectral stability

Spectral instability of small-amplitude periodic waves of the electronic Euler-Poisson system

36 pages, any comments are welcome!The present work shows that essentially all small-amplitude periodic traveling waves of the electronic Euler-Poisson system are spectrally unstable. This instability is neither modulational nor co-periodic, and thus requires an unusual spectral analysis. The growth rate with respect to the amplitude of the background waves is also provided when the instability occurs. We expect that the corresponding newly devised arguments could have a broad scope of application