Jean-François Coulombel

The strong relaxation limit of the multidimensional isothermal Euler equations

We construct global smooth solutions to the multidimensional isothermal Euler equations with a strong relaxation. When the relaxation time tends to zero, we show that the density converges towards the solution to the heat equation.

Weak Stability of Nonuniformly Stable Multidimensional Shocks

The aim of this paper is to investigate the linear stability of multidimensional shock waves that violate the uniform stability condition derived by Majda [Mem. Amer. Math. Soc., 41 (1983)]. Two examples of such shock waves are studied: (1) planar Lax shocks in isentropic gas dynamics and (2) phase transitions in an isothermal van der Waals fluid. In both cases we prove an energy estimate on the resulting linearized system. Special attention is paid to the losses of derivatives arising from the failure of the uniform stability condition.

On the transition to instability for compressible vortex sheets

We study the linear stability of a vortex sheet in a limit case that corresponds to a transition between a weakly stable regime and a violently unstable regime. We prove an energy estimate that reflects the high degeneracy of the uniform Kreiss–Lopatinskii condition.

Note on a paper by Robinet, Gressier, Casalis & Moschetta

In Robinet et al. (2000), an instability for shock waves in gas dynamics is exhibited. The fluid was assumed to obey the polytropic perfect gas pressure law. We show in this note that such an instability does not occur, as proved in the extensive work of Majda (1983). Two arguments are developed: we give first a mathematical result that is violated by the conclusions of Robinet et al. (2000). Then we detail the results of Robinet et al. (2000) and show why they do not yield any conclusions. Finally, we develop a general calculation and show that an instability cannot occur.

Stability of multidimensional undercompressive shock waves

This paper is devoted to the study of linear and nonlinear stability of undercompressive shock waves for first order systems of hyperbolic conservation laws in several space dimensions. We first recall the framework proposed by Freistuhler to extend Majda’s work on classical shock waves to undercompressive shock waves. Then we show how the so-called uniform stability condition yields a linear stability result in terms of a maximal L2 estimate. We follow Majda’s strategy on shock waves with several improvements and modifications inspired from Metivier’s work. The linearized problems are solved by duality and the nonlinear equations by mean of a Newton type iteration scheme. Finally, we show how this work applies to phase transitions in an isothermal van der Waals fluid.

ENTROPY-BASED MOMENT CLOSURE FOR KINETIC EQUATIONS: RIEMANN PROBLEM AND INVARIANT REGIONS

We study a nonlinear hyperbolic system of balance laws that arises from an entropy-based moment closure of a kinetic equation. We show that the corresponding homogeneous Riemann problem can be solved without smallness assumption, and we exhibit invariant regions.

Stability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems

We study the stability of finite difference schemes for hyperbolic initial boundary value problems in one space dimension. Assuming stability for the dicretization of the hyperbolic operator as well as a geometric regularity condition, we show that the uniform Kreiss-Lopatinskii condition yields strong stability for the discretized initial boundary value problem. The present work extends results of Gustafsson, Kreiss, Sundstrom and a former work of ours to the widest possible class of finite difference schemes by dropping some technical assumptions. We give some new examples of numerical schemes for which our results apply.

Semigroup stability of finite difference schemes for multidimensional hyperbolic initial-boundary value problems

We develop a simple energy method to prove the stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems. In particular we extend to several space dimensions a crucial result by Goldberg and Tadmor. This allows us to give two conditions on the discretized operator that ensure that stability estimates for zero initial data imply a semigroup stability estimate for general initial data. We then apply this criterion to several numerical schemes in two space dimensions.

From gas dynamics to pressureless gas dynamics

This paper is devoted to the convergence of solutions of the compressible Euler equations towards solutions of the pressureless gas dynamics system, when the pressure tends to 0. The goal is to prove accurate uniform bounds for particular solutions of the Euler equations.

Weak Stability of Multidimensional Shocks

In [8], A. Majda studied the linear stability of multidimensional shock waves for general systems of conservation laws. His analysis relied on two main assumptions: the shock wave was assumed to satisfy the so-called uniform stability condition (we shall recall it in the sequel of this paper) and the system was assumed to satisfy a block structure condition. The linear study performed in 8 enabled Majda to prove the (local) existence of multidimensional shock waves in this context, see [7, 9, 14] for an overview. Up to our knowledge, the study of multidimensional shock waves has known little progress since these oustanding breakthroughs. Let us mention however two significant results. To carry out the study of the variable coefficients linear systems in [8], Majda used an H 8 version of pseudodifferential calculus (essentially based on Moser’s inequalities). Another approach for this type of problems is the paradifferential calculus of Bony and Meyer [2, 12]. Using these techniques, Mokrane precised the regularity assumptions under which Majda’s theorems still hold, see [13] and [11]. More recently, Metivier has shown in [10] that the block structure condition defined by Majda in [8] was met by all symmetric hyperbolic systems with constant multiplicity. Whether the block structure condition is met by the magnetohydrodynamics equations (or by other nonconstant multiplicity systems) is still an open (but interesting) question.