# Etude de Certaines Equations aux Dérivées Partielles

Abstract : The first part of this work concerns non-coercive elliptic equations. We first prove existence and uniqueness of a weak solution in the usual energy space $H^1(\Omega)$ for a class of linear convection-diffusion equations in which the convection term entails the loss of coercivity. We prove Hölder regularity results for the solutions of these equations, and this allows us to solve the same equations with a measure right-hand side. We also extend the existence and uniqueness results to the variational nonlinear noncoercive case. We study then, for a linear noncoercive elliptic equation, the convergence of a finite volume scheme. The second part concerns the uniqueness of solutions to nonlinear elliptic problems with a measure right-hand side. In the third part, we study the hyperbolicity condition for first order systems with constant coefficients. We prove a necessary and sufficient condition for such a system to have solutions for any initial condition of Riemann type (a natural initial condition in the study of numerical schemes for such systems). Thanks to a particular system, we study the difference between our condition and the several hyperbolicity conditions of the literature, and we then prove that the solution of a hyperbolic system is not always stable with respect to the flux. The fourth part gathers some other works. The first work concerns the density in $W^{1,p}(\Omega)$ of regular functions satisfying a Neumann condition. The second is the study of a Mixed Finite Element---Finite Volume scheme for a two-fluids flow through a porous media. The third and last is the study of measures on $]0,T[\times \Omega$ that do not charge sets of null parabolic capacity and the application of this study to a nonlinear parabolic equations with measure right-hand side.
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Mathematics [math]. Université de Provence - Aix-Marseille I, 2001. French
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https://tel.archives-ouvertes.fr/tel-00001180
Contributor : Jérôme Droniou <>
Submitted on : Friday, March 1, 2002 - 1:33:27 PM
Last modification on : Friday, March 1, 2002 - 1:33:27 PM

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• HAL Id : tel-00001180, version 1

### Citation

Jérôme Droniou. Etude de Certaines Equations aux Dérivées Partielles. Mathematics [math]. Université de Provence - Aix-Marseille I, 2001. French. <tel-00001180>

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