49 research outputs found
On the inviscid Boussinesq system with rough initial data
We deal with the local well-posedness theory for the two-dimensional inviscid
Boussinesq system with rough initial data of Yudovich type. The problem is in
some sense critical due to some terms involving Riesz transforms in the
vorticity-density formulation. We give a positive answer for a special
sub-class of Yudovich data including smooth and singular vortex patches. For
the latter case we assume in addition that the initial density is constant
around the singular part of the patch boundary.Comment: 26 page
On the V-states for the generalized quasi-geostrophic equations
We prove the existence of the V-states for the generalized inviscid SQG
equations with These structures are special rotating simply
connected patches with fold symmetry bifurcating from the trivial solution
at some explicit values of the angular velocity. This produces, inter alia, an
infinite family of non stationary global solutions with uniqueness.Comment: 54 page
Doubly connected V-states for the generalized surface quasi-geostrophic equations
In this paper, we prove the existence of doubly connected V-states for the
generalized SQG equations with They can be described by
countable branches bifurcating from the annulus at some explicit "eigenvalues"
related to Bessel functions of the first kind. Contrary to Euler equations
\cite{H-F-M-V}, we find V-states rotating with positive and negative angular
velocities. At the end of the paper we discuss some numerical experiments
concerning the limiting V-states.Comment: 65 page
An analytical and numerical study of steady patches in the disc
In this paper, we prove the existence of -fold rotating patches for the
Euler equations in the disc, for both simply-connected and doubly-connected
cases. Compared to the planar case, the rigid boundary introduces rich dynamics
for the lowest symmetries and . We also discuss some numerical
experiments highlighting the interaction between the boundary of the patch and
the rigid one.Comment: 56 page
Axisymmetric Incompressible Viscous Plasmas: Global Well-Posedness and Asymptotics
This paper is devoted to the global analysis of the three-dimensional
axisymmetric Navier--Stokes--Maxwell equations. More precisely, we are able to
prove that, for large values of the speed of light , for
some threshold depending only on the initial data, the system in
question admits a unique global solution. The ensuing bounds on the solutions
are uniform with respect to the speed of light, which allows us to study the
singular regime and rigorously derive the limiting
viscous magnetohydrodynamic (MHD) system in the axisymmetric setting.
The strategy of our proofs draws insight from recent results on the
two-dimensional incompressible Euler--Maxwell system to exploit the
dissipative--dispersive structure of Maxwell's system in the axisymmetric
setting. Furthermore, a detailed analysis of the asymptotic regime
allows us to derive a robust nonlinear energy estimate which holds uniformly in
. As a byproduct of such refined uniform estimates, we are able to describe
the global strong convergence of solutions toward the MHD system.
This collection of results seemingly establishes the first available global
well-posedness of three-dimensional viscous plasmas, where the electric and
magnetic fields are governed by the complete Maxwell equations, for large
initial data as .Comment: 52 page
Dynamics of vortex cap solutions on the rotating unit sphere
In this work, we analytically study the existence of periodic vortex cap
solutions for the homogeneous and incompressible Euler equations on the
rotating unit 2-sphere, which was numerically conjectured by Dritschel-Polvani
and Kim-Sakajo-Sohn. Such solutions are piecewise constant vorticity
distributions, subject to the Gauss constraint and rotating uniformly around
the vertical axis. The proof is based on the bifurcation from zonal solutions
given by spherical caps. For the one--interface case, the bifurcation
eigenvalues correspond to Burbea's frequencies obtained in the planar case but
shifted by the rotation speed of the sphere. The two--interfaces case (also
called band type or strip type solutions) is more delicate. Though, for any
fixed large enough symmetry, and under some non-degeneracy conditions to avoid
spectral collisions, we achieve the existence of at most two branches of
bifurcation.Comment: 40 pages, 4 figure
Dynamique des tourbillons pour quelques modèles de transport non-linéaires
In this dissertation, we are concerned with the study of some non-linear evolution models arising in fluid mechanics. We distinguish three independent parts. The first part of the thesis deals with the existence of the rotating vortex patches (called also V-states) for an inviscid quasi-geostrophic model. Our study is divided into two chapters dealing with different topological structures of the V-states. In the first chapter we study the simply connected case and we prove the existence of such structures in a neighborhood of the Rankine vortices by using the bifurcation theory. In the second chapter we discuss the doubly connected case where the patches admit only one hole. More precisely, close to a given annulus we describe this family by countable branches bifurcating from this annulus at some explicit angular velocities related to Bessel functions of the first kind. Our theoretical study was completed by numerical simulations on the limiting V-states and a number of interesting numerical observation were formulated opening new research perspectives. The second part of the thesis concerns the local well-posedness theory for the inviscid Boussinesq system with rough initial data. The problem is in some sense critical due to some terms involving Riesz transforms in the vorticity-density formulation. We give a positive answer for a special sub-class of Yudovich data including smooth and singular vortex patches. In the last part we address the problem of the incompressible limit for the 2D isentropic fluids associated to ill-prepared initial data and for which the vortices are not necessarily bounded and belong to some weighted BMO spaces. We mainly use two ingredients: On one hand, the Strichartz estimates to control the acoustic part and prove that it does not contribute for low Mach number. On the other hand, we use the transport compressible structure of the vorticity and we establish a linear propagation estimate in the spirit of a recent work of Bernicot and Keraani conducted in the incompressible case. The first part of the thesis deals with the existence of the rotating vortex patches (called also V-states) for an inviscid quasi-geostrophic model. Our study is divided into two chapters dealing with different topological structures of the V-states. In the first chapter we study the simply connected case and we prove the existence of such structures in a neighborhood of the Rankine vortices by using the bifurcation theory. In the second chapter we discuss the doubly connected case where the patches admit only one hole. More precisely, close to a given annulus we describe this family by countable branches bifurcating from this annulus at some explicit angular velocities related to Bessel functions of the first kind. Our theoretical study was completed by numerical simulations on the limiting V-states and a number of interesting numerical observation were formulated opening new research perspectives. The second part of the thesis concerns the local well-posedness theory for the inviscid Boussinesq system with rough initial data. The problem is in some sense critical due to some terms involving Riesz transforms in the vorticity-density formulation. We give a positive answer for a special sub-class of Yudovich data including smooth and singular vortex patches. In the last part we address the problem of the incompressible limit for the 2D isentropic fluids associated to ill-prepared initial data and for which the vortices are not necessarily bounded and belong to some weighted BMO spaces. We mainly use two ingredients: On one hand, the Strichartz estimates to control the acoustic part and prove that it does not contribute for low Mach number. On the other hand, we use the transport compressible structure of the vorticity and we establish a linear propagation estimate in the spirit of a recent work of Bernicot and Keraani conducted in the incompressible case.Cette thèse est consacrée à l'étude théorique de quelques modèles d'évolution non-linéaires issus de la mécanique des fluides. Nous distinguons trois parties indépendantes. La première partie de la thèse traite essentiellement de l'existence des poches de tourbillon en rotation uniforme (appelées aussi V-states) pour un modèle quasi-géostrophique non visqueux. Notre étude est répartie sur deux chapitres où les poches présentent des structures topologiques différentes. Dans le premier chapitre nous étudions le cas simplement connexe et nous validons l'existence de ces structures dans un voisinage du tourbillon de Rankine en utilisant des techniques de bifurcation. Dans le deuxième chapitre nous abordons le cas doublement connexe où la poche admet un seul trou. Plus précisément, proche d'un anneau donné, nous décrivons cette famille par des branches dénombrables bifurquant de cet anneau à certaines valeurs explicites des vitesses angulaires liées aux fonctions de Bessel. Notre étude théorique a été complétée par des simulations numériques portant sur les V-states limites et un bon nombre de constatations ont été formulées ouvrant la porte à de nouvelles perspectives de recherche. La seconde partie concerne l'étude du problème de Cauchy pour le système de Boussinesq non visqueux 2D avec des données initiales de type Yudovich. Le problème est dans un certain sens critique à cause de quelques termes comportant la transformée de Riesz dans la formulation tourbillon-densité. Nous donnons une réponse positive pour une sous-classe comprenant les poches de tourbillon régulières et singulières. Dans la dernière partie nous analysons le problème de la limite incompressible pour les équations d'Euler isentropiques 2D associées à des données initiales très mal préparées et pour lesquelles les tourbillons ne sont pas forcément bornés mais appartiennent plutôt à des espaces de type ''BMO'' à poids. On utilise principalement deux ingrédients: d'un côté les estimations de Strichartz pour contrôler la partie acoustique. D'un autre côté, on se sert de la structure de transport compressible du tourbillon et on démontre une estimation de propagation linéaire dans l'esprit d'un travail récent de Bernicot et Keraani mené dans le cas incompressible