Guy Métivier

The Incompressible Limit of the Non-Isentropic Euler Equations

Abstract: We study the Euler equations for slightly compressible fluids, that is, after rescaling, the limits of the Euler equations of fluid dynamics as the Mach number tends to zero. In this paper, we consider the general non-isentropic equations and general data. We first prove the existence of classical solutions for a time independent of the small parameter. Then, on the whole space ℝd, we prove that the solution converges to the solution of the incompressible Euler equations.

Stability of Multidimensional Shocks

This series of lectures is devoted to the study of shock waves for systems of multidimensional conservation laws. In sharp contrast with one-dimensional problems, in higher space dimensions there is no general existence theorem for solutions which allow discontinuities. Our goal is to study the existence and the stability of the simplest pattern of a single wave front ∑, separating two states u + and u -, which depend smoothly on the space-time variables x. For example, our analysis applies to perturbations of planar shocks. They are special solutions given by constant states separated by a planar front. Given a multidimensional perturbation of the initial data or a small wave impinging on the front, we study the following stability problem. Is there a local solution with the same wave pattern? Similarly, a natural problem is to investigate the multidimensional stability of one-dimensional shock fronts. However, the analysis applies to much more general situations and the main subject is the study of curved fronts.

Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems

Introduction Linear stability: the model case Pieces of paradifferential calculus

The Block Structure Condition for Symmetric Hyperbolic Systems

L^2

Focusing at a point and absorption of nonlinear oscillations

and conormal estimates near the boundary Linear stability Nonlinear stability Appendix A. Kreiss symmetrizers Appendix B. Para-differential calculus Appendix Bibliography.

Global Solutions to Maxwell Equations in a Ferromagnetic Medium

In the analysis of hyperbolic boundary value problems, the construction of Kreiss’ symmetrizers relies on a suitable block structure decomposition of the symbol of the system. In this paper, we show that this block structure condition is satisfied by all symmetrizable hyperbolic systems of constant multiplicity. In [2], H.O.Kreiss proved a maximal L2 energy estimate for the solutions of mixed boundary-initial value problems for strictly hyperbolic systems and boundary conditions which satisfy the uniform Lopatinski condition (see also [8] for systems with complex coefficients). The proof is based on the construction of a symmetrizer. Thanks to the pseudodifferential calculus, the proof is reduced to the construction of an algebraic symmetrizer for the symbol of the equation (see e.g. [1]). The result extends to the case where the coefficients have finite smoothness ([3]) and, using the paradifferential calculus of J.M.Bony-Y.Meyer, to Lipschitzean coefficients ([7],[6]). Kreiss’ analysis is extended to a class of characteristic boundary value problems in [5]. However, many interesting physical examples of hyperbolic systems are not strictly hyperbolic. For instance, Euler’s equations of gas dynamics, Maxwell’s equations or the equations of elasticity are not strictly hyperbolic. In the construction of Kreiss’ symmetrizer, the strict hyperbolicity assumption is used at only one place, to prove that the symbol of the system has a suitable block decomposition near glancing modes (see Lemmas 2.5, 2.6 and 2.7 in [2]). In [5] and [3], it is shown that this block structure condition is satisfied by several nonstrictly hyperbolic systems such as the linearized shock front equations of gas dynamics ([3]), Maxwell’s equations or the linearized shallow water equations ([5]). However, due to the lack of a simple criterion, one had to check the condition for each system separately. The aim of this paper is to prove that the block structure assumption is satisfied for a large class of systems of physical interest which contains the examples above : the class of symmetrizable hyperbolic systems of constant multiplicity. As a corollary, continuing the analysis as in [2], [1], [5] or [3], this implies the local well posedness of boundary value problems for linear symmetric (or

Averaging theorems for conservative systems and the weakly compressible Euler equations

Several recent papers give rigorous justifications of weakly nonlinear geometric optics. All of them consider oscillating wave trains on domains where focusing phenomena do not exist, either because the space dimension is equal to one, or thanks to a coherence assumption on the phases. This paper is devoted to a study of some nonlinear effects of focusing. In a previous paper, the authors have given a variety of examples which show how focusing in nonlinear equations can spoil even local existence in the sense that the domain of existence shrinks to zero as the wavelength decreases to zero. On the other hand, there are many problems for which global existence is known and in those cases it is natural to ask what happens to oscillations as they pass through a focus. The main goal of this paper is to present such a study for some strongly dissipative semilinear wave equations and spherical wavefronts which focus at the origin. We show that the strongly nonlinear phenomenon which is produced is that oscillations are killed by the simultaneous action of focusing and dissipation. Our study relies on the analysis of Young measures and two-scale Young measures associated to sequences of solutions. The main step is to prove that these measures satisfy appropriate transport equations. Then, their variances are shown to satisfy differential inequalities which imply a propagation result for their support.

Fonction spectrale et valeurs propres d'une classe d'operateurs non elliptiques

Abstract. We study the Cauchy problem for the Landau-Lifschitz model in ferromagnetism without exchange energy. Once existence of global finite energy solutions is obtained, we study additional uniqueness and regularity properties of these solutions.

Paralinearization of the Dirichlet to Neumann Operator, and Regularity of Three-Dimensional Water Waves

Abstract A generic averaging theorem is proven for systems of ODEs with two-time scales that cannot be globally transformed into the usual action-angle variable normal form for such systems. This theorem is shown to apply to certain Fourier-space truncations of the non-isentropic slightly compressible Euler equations of fluid mechanics. For the full Euler equations, we derive formally the generic limit equations and analyze some of their properties. In the one-dimensional case, we prove a generic converic convergence result for the full Euler equations, analogous to the result for ODEs. By making use of special properties of the one-dimensional equations, we prove convergence to the solution of a more complicated set of averaged equations when the genericity assumptions fail.

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

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