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11	Contributions aux équations d'évolutions non locales en espace-temps / Contributions to non local evolution equations in space-time Dannawi, Ihab 11 September 2015 (has links) Dans cette thèse, nous nous intéressons à l'étude de quatre équations d'évolution non-locales. Les solutions de ces quatre équations peuvent exploser en temps fini. Dans la théorie des équations d'évolution non-linéaires, une solution est qualifiée de globale si elle est définie pour tout temps positif. Au contraire, si une solution existe seulement sur un intervalle de temps [0; T) borné, elle est dite locale. Dans ce dernier cas et quand le temps maximal d'existence est relié à une alternative d'explosion, on dit aussi que la solution explose en temps fini. Dans un premier travail, nous considérons l'équation de Schrödinger non-linéaire avec une puissance fractionnaire du laplacien, et nous obtenons l'explosion de la solution en temps fini Tmax > 0 pour toute condition initiale positive et non-triviale dans le cas d'exposant sous-critique. Ensuite, nous étudions une équation des ondes amorties avec un potentiel d'espace-temps et un terme non-linéaire et non-local en temps. Nous obtenons un résultat d'existence locale d'une solution dans l'espace d'énergie sous des conditions restrictives sur les données initiales, la dimension de l'espace et la croissance du terme non-linéaire. De plus, nous obtenons l'explosion de la solution en temps fini pour toute condition initiale de moyenne strictement positive. De plus, nous étudions un problème de Cauchy pour l'équation d'évolution avec un p- Laplacien avec une non linéarité non-locale en temps. Dans ce cadre, nous nous intéressons à l'étude de l'existence locale d'une solution de cette équation ainsi qu'un résultat de non-existence de solution globale. Finalement, nous étudions l'intervalle maximal d'existence des solutions de l'équation des milieux poreux avec un terme non-linéaire non-local en temps. / In this thesis, we study four non-local evolution equations. The solutions of these four equations can blow up in finite time. In the theory of nonlinear evolution equations, a solution is qualified as global if it isdefined for any time. Otherwise, if a solution exists only on a bounded interval [0; T), it is called local solution. In this case and when the maximum time of existence is related to a blow up alternative, we say that the solution blows up in finite time. First, we consider the nonlinear Schröodinger equation with a fractional power of the Laplacien operator, and we get a blow up result in finite time Tmax > 0 for any non-trivial non-negative initial condition in the case of sub-critical exponent. Next, we study a damped wave equation with a space-time potential and a non-local in time non-linear term. We obtain a result of local existence of a solution in the energy space under some restrictions on the initial data, the dimension of the space and the growth of nonlinear term. Additionally, we get a blow up result of the solution in finite time for any initial condition positive on average. In addition, we study a Cauchy problem for the evolution p-Laplacien equation with nonlinear memory. We study the local existence of a solution of this equation as well as a result of non-existence of global solution. Finally, we study the maximum interval of existence of solutions of the porous medium equation with a nonlinear non-local in time term. Equations de Schrödinger Explosion en temps fini Laplacien Fractionnaire Equation d'onde amortie non linéaire Exposant sous critique Problème de Cauchy Equation de p-Laplacien Existence locale Non existence globale Equation hyperpolique Solutions douces et faibles Estimation de Strichartz Schrödinger equations Blow-up Fractional Laplacian Nonlinear damped wave equation Subcritical potential Cauchy problem P-Laplacian equation Local existence Global nonexistence Hyperbolic equation Mild and weak solutions Strichartz estimate
12	High order numerical methods for a unified theory of fluid and solid mechanics Chiocchetti, Simone 10 June 2022 (has links) This dissertation is a contribution to the development of a unified model of continuum mechanics, describing both fluids and elastic solids as a general continua, with a simple material parameter choice being the distinction between inviscid or viscous fluid, or elastic solids or visco-elasto-plastic media. Additional physical effects such as surface tension, rate-dependent material failure and fatigue can be, and have been, included in the same formalism. The model extends a hyperelastic formulation of solid mechanics in Eulerian coordinates to fluid flows by means of stiff algebraic relaxation source terms. The governing equations are then solved by means of high order ADER Discontinuous Galerkin and Finite Volume schemes on fixed Cartesian meshes and on moving unstructured polygonal meshes with adaptive connectivity, the latter constructed and moved by means of a in- house Fortran library for the generation of high quality Delaunay and Voronoi meshes. Further, the thesis introduces a new family of exponential-type and semi- analytical time-integration methods for the stiff source terms governing friction and pressure relaxation in Baer-Nunziato compressible multiphase flows, as well as for relaxation in the unified model of continuum mechanics, associated with viscosity and plasticity, and heat conduction effects. Theoretical consideration about the model are also given, from the solution of weak hyperbolicity issues affecting some special cases of the governing equations, to the computation of accurate eigenvalue estimates, to the discussion of the geometrical structure of the equations and involution constraints of curl type, then enforced both via a GLM curl cleaning method, and by means of special involution-preserving discrete differential operators, implemented in a semi-implicit framework. Concerning applications to real-world problems, this thesis includes simulation ranging from low-Mach viscous two-phase flow, to shockwaves in compressible viscous flow on unstructured moving grids, to diffuse interface crack formation in solids. High order ADER Discontinuous Galerkin DG Finite Volume FV Finite Element FE FEM DGFEM CFD Voronoi Delaunay Unstructured meshe Fortran Fluid mechanic Solid Mechanic Continuum Mechanic Baer Nunziato Surface tension Multiphase flow Compressible flow Navier Stoke Semi-implicit GLM cleaning Partial differential equation Numerical method Stiff source Finite rate stiff relaxation Hyperbolic equation Metodi d'alto ordine ADER Discontinous Galerkin Volumi Finiti Elementi Finiti Meccanica dei fluidi Meccanica dei solidi Meccanica dei continui Fluidi multifase Fluidi comprimibili Metodi semi-impliciti Equazioni alle derivate parziali Metodi numerici Equazioni iperboliche Griglie non strutturate

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Contributions aux équations d'évolutions non locales en espace-temps / Contributions to non local evolution equations in space-time

High order numerical methods for a unified theory of fluid and solid mechanics