Search

This thesis is basically devoted to the study of some mathematical properties of three equations: the Navier-Stokes equations, the Boussinesq equations and the Hall-magnetohydrodynamic equations in the three-dimensional context. It is worth noting that, for the time being, the question of the global solvability of the 3D Navier-Stokes equations remains as an open problem: In particular we do not know if a regular initial data would whether uniquely generate a global in time regular solution or if a formation of a singularity at finite time can occur. Thus, it is clear that such a problem is still open for all the equations studied in this work. The goal of this thesis is to study first some conditions related to the blow-up phenomenon of the 3D Navier-Stokes equations that can occur at finite time Chapter 2. Then, Chapters 3, 4 and 5 are devoted to the question of the well-posedness of the Boussinesq system with different cases of dissipation. More precisely, in Chapter 3 we establish the global well-posedness of the completely viscous Boussinesq equations when the initial data is axisymmetric belonging to the critical Lebesgue spaces. On the other hand, in the case when there is no dissipation in the temperature’s equation, it is hard to work in critical spaces. However, in Chapter 4 we solve the problem globally in time in the situation where the initial data is axisymmetric and critical with respect to the velocity's scaling. The main result in Chapter 5 is the uniqueness of the solution (in some class of anisotropic spaces) to the Boussinesq system with partial dissipation in all the equations. We use then this latest to improve some known results. Finally, in Chapter 6 we establish several results of the local (or global for small initial data) well-posedness and large time behavior of the global solutions (in several functional spaces) to a model of Hall-MHD.; Cette thèse est essentiellement consacrée à l'étude de certaines propriétés mathématiques de trois équations : les équations de Navier-Stokes, les équations de Boussinesq et les équations de Hall magnétohydrodynamique, dans le contexte tridimensionnel. Il convient de noter que, pour le moment, la question de la solvabilité globale des équations de Navier-Stokes 3D reste un problème ouvert : nous ne savons pas si des données initiales régulières génèrent, de manière unique, une solution régulière et globale en temps, ou bien si une formation d'une singularité peut se produire à temps fini. Il est donc clair qu'un tel problème est aussi ouvert pour toutes les équations à l'étude dans ce travail. Le but de cette thèse est d'étudier d'abord certaines conditions liées au phénomène de l'explosion à temps fini des solutions des équations de Navier-Stokes 3D (Chapitre 2). Les Chapitres 3, 4 et 5 sont consacrés à la question du caractère bien posé du système de Boussinesq avec différents cas de dissipations. Nous consacrerons le Chapitre 3 aux équations de Boussinesq complètement visqueuses avec des données initiales axisymétriques appartenant aux espaces critiques de Lebesgue. Pour le meme système mais sans dissipation dans l'équation de la température, travailler dans des espaces critiques est plus délicat. Toutefois dans le Chapitre 4 nous parvenons à traiter le cas des données initiales axisymmetriques qui sont critiques par rapport à l'échelle de la vitesse. Le résultat principal du Chapitre 5 est l'unicité des solutions (dans une classe d'espaces anisotropes) du système de Boussinesq avec des dissipations partielles (horizontales) dans toutes les équations. Nous utilisons ensuite ce résultat pour améliorer certains théorèmes déjà établis. Enfin, dans le Chapitre 6 nous établissons quelques résultats du critère bien posé (local en temps ou global pour les données initiales petites) et du comportement à long temps des solutions globales (dans certains espaces fonctionnels) pour un modèle de Hall-MHD.

This thesis is basically devoted to the study of some mathematical properties of three equations: the Navier-Stokes equations, the Boussinesq equations and the Hall-magnetohydrodynamic equations in the three-dimensional context. It is worth noting that, for the time being, the question of the global solvability of the 3D Navier-Stokes equations remains as an open problem: In particular we do not know if a regular initial data would whether uniquely generate a global in time regular solution or if a formation of a singularity at finite time can occur. Thus, it is clear that such a problem is still open for all the equations studied in this work. The goal of this thesis is to study first some conditions related to the blow-up phenomenon of the 3D Navier-Stokes equations that can occur at finite time Chapter 2. Then, Chapters 3, 4 and 5 are devoted to the question of the well-posedness of the Boussinesq system with different cases of dissipation. More precisely, in Chapter 3 we establish the global well-posedness of the completely viscous Boussinesq equations when the initial data is axisymmetric belonging to the critical Lebesgue spaces. On the other hand, in the case when there is no dissipation in the temperature’s equation, it is hard to work in critical spaces. However, in Chapter 4 we solve the problem globally in time in the situation where the initial data is axisymmetric and critical with respect to the velocity's scaling. The main result in Chapter 5 is the uniqueness of the solution (in some class of anisotropic spaces) to the Boussinesq system with partial dissipation in all the equations. We use then this latest to improve some known results. Finally, in Chapter 6 we establish several results of the local (or global for small initial data) well-posedness and large time behavior of the global solutions (in several functional spaces) to a model of Hall-MHD.; Cette thèse est essentiellement consacrée à l'étude de certaines propriétés mathématiques de trois équations : les équations de Navier-Stokes, les équations de Boussinesq et les équations de Hall magnétohydrodynamique, dans le contexte tridimensionnel. Il convient de noter que, pour le moment, la question de la solvabilité globale des équations de Navier-Stokes 3D reste un problème ouvert : nous ne savons pas si des données initiales régulières génèrent, de manière unique, une solution régulière et globale en temps, ou bien si une formation d'une singularité peut se produire à temps fini. Il est donc clair qu'un tel problème est aussi ouvert pour toutes les équations à l'étude dans ce travail. Le but de cette thèse est d'étudier d'abord certaines conditions liées au phénomène de l'explosion à temps fini des solutions des équations de Navier-Stokes 3D (Chapitre 2). Les Chapitres 3, 4 et 5 sont consacrés à la question du caractère bien posé du système de Boussinesq avec différents cas de dissipations. Nous consacrerons le Chapitre 3 aux équations de Boussinesq complètement visqueuses avec des données initiales axisymétriques appartenant aux espaces critiques de Lebesgue. Pour le meme système mais sans dissipation dans l'équation de la température, travailler dans des espaces critiques est plus délicat. Toutefois dans le Chapitre 4 nous parvenons à traiter le cas des données initiales axisymmetriques qui sont critiques par rapport à l'échelle de la vitesse. Le résultat principal du Chapitre 5 est l'unicité des solutions (dans une classe d'espaces anisotropes) du système de Boussinesq avec des dissipations partielles (horizontales) dans toutes les équations. Nous utilisons ensuite ce résultat pour améliorer certains théorèmes déjà établis. Enfin, dans le Chapitre 6 nous établissons quelques résultats du critère bien posé (local en temps ou global pour les données initiales petites) et du comportement à long temps des solutions globales (dans certains espaces fonctionnels) pour un modèle de Hall-MHD.

Novel features of turbulent flows have been analyzed recently, for example: (1) the possibility of an ideal invariant, such as the energy, to be transferred both to the small scales and to the large scales, in each case with a constant flux; (2) the existence of non-Gaussian wings in Probability Distribution Functions of kinetic, magnetic, and temperature fluctuations, together with their gradients, thus displaying large-scale as well as small-scale intermittency; and (3) the linear dependence on the control parameter of the effective dissipation in turbulence when non-linear eddies and waves interact. We shall briefly review these results with examples stemming from Solar Wind data, the atmosphere and the ocean with either magnetic fields, stratification, and/or rotation. In a second part, we shall examine numerically the inverse cascades of magnetic and of generalized helicity for Hall-MHD in the presence of forcing. These helical invariants in the ideal non-dissipative case involve various cross-correlations between the velocity and vorticity, the magnetic field, and the magnetic potential. For an ion inertial length larger than the forcing scale, the effect of the waves is significant. It leads to an exponential attenuation of the inverse cascade to large scales, since, through the velocity and vorticity, small scales play an increasing dynamical role for a strong Hall current.

We study the propagation of nonlinear waves in a Hall-magnetohydrodynamic model. An asymptotic method is used to derive the Gardner-Burgers equation for fast magnetosonic waves; here, the flux function is nonconvex with both quadratic and cubic nonlinearities, and the evolution equation involves both second- and third-order derivatives representing diffusion and dispersion terms, respectively. Effects of Hall parameter are discussed on the evolution of waves and their interaction by solving a pair of Riemann problems both analytically and numerically. It is shown that the Hall parameter is responsible for shock splittinga phenomenon that is completely absent in ideal magnetohydrodynamic; indeed, the Hall parameter plays a significant role in deciding about the structure of the solution that involves undercompressive shocks and their interaction with refracted waves and the Lax shocks. It is found that increasing Hall parameter means increasing dispersion that triggers the physical mechanism causing speed and strength of an undercompressive shock to increase and the wave-fan width to decrease; numerical solutions substantiate these features predicted by the analytical solution.

Vlasov simulation is a method to solve time evolution of a plasma by directly time advancing the distribution function in the position-velocity phase space. Unlike conventional PIC (particle-in-cell)simulations using finite number of particles, the Vlasov simulation is free from thermal (numerical) noise, and thus is advantageous in analyzing fine details of nonlinear plasma phenomena. With this background in mind, we have developed a new Vlasov simulation code (1-d in space, 3-d in the velocity space), in order to study basic properties of nonlinear evolution of magnetohydrodynamic (MHD) waves in the solar wind. In contrast to traditional Vlasov simulations in which electron waves are of major concern, our simulation code focuses on solving plasma behavior around the ion scales, assuming the massless electron fluid. Since we mainly deal with low frequency MHD waves propagating quasi-parallel to the background magnetic field, cyclotron coupling can be assumed to be weak for parameters typical to the solar wind. Thus the Vlasov equation is solved only along the longitudinall direction whereas the MHD equations are solved for the transverse directions. Propagation of Alfven and ion acoustic waves in the simulation is shown to satisfy theoretically obtained dispersion relations. Some results on parametric decay instability of Alfven waves are also presented.

En términos generales existen dos perspectivas para modelar la dinámica de los plasmas. Por un lado están los modelos cinéticos y por el otro los modelos de medios continuos (o modelosde fluidos). La teoría cinética describe a los plasmas desde la naturaleza microscópicadel sistema. Las teorias de fluídos por otro lado describen de manera natural los fenómenos aescalas macroscópicas. La mayor complicación del estudio de la turbulencia magnetohidrodinámica (y también lahidrodinámica) es que es un problema de multiescalas. Los rangos de escala que están en losextremos (macro y micro) son claros dominios de una y otra teoría, sin embargo las escalas quese encuentran entre las escalas MHD y la escala de Kolmogorov (o de disipación) son un rangocontroversial al respecto. En esta tesis se introducen efectos cinéticos en la magnetohidrodinámica de medios continuos,a traves de modelos de dos fluidos que consideran la separación entre iones y electrones, En particular, se desarrolla un modelo aproximado de dos fluidos para plasmas con campomagnético fuerte que presenta importantes ventajas computacionales. El interes en el efectodel término Hall con campo magnético fuerte está motivado por las observaciones geofísicas,astrofísicas y la implementación tecnológica de confinamiento por medio de guías magnéticas. El primer paso consistió en testear el modelo aproximado frente al modelo Hall-MHDgeneral. Luego desarrollamos un estudio detallado que incluyó múltiples puntos de vista. Mostramos que el efecto Hall afecta los valores de las las magnitudes globales y sus tiemposcaracterísticos. La distribución de energía por escalas se ve también modificada, incrementándoseel rango de escalas disipativas. Se modifican las estructuras del flujo cambiando su formay tamaño en las escalas comprendidas entre el ion skin depth (o escala de Hall) y la escala de Kolmogorov. Se reduce la intermitencia espacial y la autosimilaridad del flujo se incrementatendiendo a lamonofractalidad. Se analiza el efecto sobre la formación y estructura de las hojasde corriente. Se analizan las propiedades estadísticas del flujo, en particular la fractalidad y elefecto sobre la dimensionalidad de las estructuras responsables de la disipación. Finalmente,extendemos los primeros efectos cinéticos mas allá del efecto Hall introduciendo la inerciaelectrónica y presentamos resultados preliminares que muestran que la masa electrónica afectala dinámica mas allá de los efectos del término Hall. There are broadly two approaches for modeling the dynamics of plasma. On one side arethe kinetic theory models and on the other continuum models (or fluid models). The kinetictheory describes the plasmas from a microscopic point of view. On the other hand, the fluidmodels describe phenomena at macroscopic scales. The major complication in the study of magnetohydrodynamic turbulence (and hydrodynamic)is that it is a multi-scale problem. The scale ranges that are at the extremes (macro andmicro) are clearly domains of one theory or the other, however, scales between MHD scale and Kolmogorov (or dissipation) scales are controversial. In this thesis we introduce the first kineticeffects into the continuous media magnetohydrodynamic description, considering a two-fluidmodel, which thakes into accounr the separation between ions and electrons. In particular, we develop an approximate model of two fluids for plasmas with a strongmagnetic fields, where variations along the direction of the magnetic field are smoother thantransverse variations. This approximate model shows imortant computational advantagesagainst the fully general tridimensional Hall MHD model. The interest in the effect of the Hall term in a strong magnetic field is in turn motivated by geophysical and astrophysicalobservations, and technological implementation of confinement through magnetic guides. The first step was to test the approximante model with the full general Hall-MHD model. We then developed a detailed study which included multiple points of view. We showed thatthe Hall effect affects the values global quantities and their characteristic times. It also modifiesthe distribution of energy among scales, increasing the range of dissipative scales. It affectsflow structures changing its shape and size between the ion skin depth (Hall scale) and Kolmogorov scale. The intermittency is reduced and self-similarity of the flow increases approachingmonofractality. We analyzed the effects on the formation and structure of currentsheets (important for the energy dissipation). We studied the statistical properties of the flow,particularly the fractality and the effect on the dimensionality of the structures responsible forthe dissipation. Finally, we extend the first kinetic effects beyond the Hall effect, considering electron inertiaand introduce preliminary results showing that the electronic mass affects the dynamicsbeyond the effects of the Hall term. Fil: Martín, Luis N.. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

In this work we numerically test a model of Hall magnetohydrodynamics in the presence of a strong mean magnetic field: the reduced Hall magnetohydrodynamic model RHMHD derived by Gomez et al., Phys. Plasmas 15, 102303 2008 with the addition of weak compressible effects. The main advantage of this model lies in the reduction of computational cost. Nevertheless, up until now the degree of agreement with the original Hall MHD system and the range of validity in a regime of turbulence were not established. In this work direct numerical simulations of three-dimensional Hall MHD turbulence in the presence of a strong mean magnetic field are compared with simulations of the weak compressible RHMHD model. The results show that the degree of agreement is very high when the different assumptions of RHMHD, such as spectral anisotropy, are satisfied. Nevertheless, when the initial conditions are isotropic but the mean magnetic field is maintained strong, the results differ at the beginning but asymptotically reach a good agreement at relatively short times. We also found evidence that the compressibility still plays a role in the dynamics of these systems, and the weak compressible RHMHD model is able to capture these effects. In conclusion the weak compressible RHMHD model is a valid approximation of the Hall MHD turbulence in the relevant physical context. © 2010 American Institute of Physics. Fil: Martin, Luis Nicolas. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentina Fil: Dmitruk, Pablo Ariel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentina Fil: Gomez, Daniel Osvaldo. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentina

International audience

We obtain some regularity criteria for the wave map, a liquid crystals model, and the Hall-MHD with ion-slip effect.