Denis Serre

Large amplitude variations for the density of a compressible viscous fluid.

Abstract The flow of a compressible viscous fluid is governed by the Navier-Stokes equations. This system is of mixed parabolic-hyperbolic type. The hyperbolic part is associated with a linear degeneracy so that initial large-amplitude high-frequency waves can propagate along the particle paths. However, the parabolic part kills the oscillations of the velocity field. We give a formal relaxed (i.e. homogenized) system in any dimension for the Eulerian formulation. In the 1 − d case, we prove the relevance of this system in the equivalent Lagrangian formulation.

Alternate Evans Functions and Viscous Shock Waves

The Evans function is known as a helpful tool for locating the spectrum of some variational differential operators. This is of special interest regarding the stability analysis of traveling waves, ...

Relaxations semi-linéaire et cinétique des systèmes de lois de conservation

Abstract R. DiPerna (1983) proved the convergence of the approximate solutions given by the vanishing viscosity method, towards an entropy solution of the underlying hyperbolic system. He used two main assumptions: the existence of convex positively invariant domains (in the sense of K.N. Chuey et al. (1977)) and genuine nonlinearity. We prove below that, under the same assumptions together with the sub-characteristic condition, the approximate solutions given by the semi-linear relaxation converge too. Actually, our result stands for a more general approximation, first introduced by R. Natalini (1998).

Spectral Stability of Periodic Solutions of Viscous Conservation Laws: Large Wavelength Analysis

Abstract We complete and unify the works by Oh and Zumbrun (2003a) and by the author (1994), about the spectral stability of traveling waves that are spatially periodic, in systems of n conservation laws. Our context is one-dimensional. These systems are of order larger than one, in general. For instance, they could be viscous approximations of first-order systems that are not everywhere hyperbolic. However, modelling considerations often lead to higher order terms, like capillarity in fluid dynamics; our framework remains valid in this more general setting. We make generic assumptions, saying in particular that the set of periodic traveling waves is a manifold of maximal dimension, under the restrictions given by the conserved quantities. The spectral stability of a periodic traveling wave is studied through Floquets theory. Following Gardner (1993), we introduce an Evans function D(λ, θ), being λ the Laplace frequency and θ the phase shift. The large wavelength analysis is the description of the zero set of D around the origin. Our main result is that this zero set is described, at the leading order, by a characteristic equation This formula involves a flux F, which enters into a first-order system of conservation laws, describing the slow modulation of the periodic traveling waves. Its size N is in practice larger than n. The important consequence is that hyperbolicity of the latter system is a necessary condition for spectral stability of periodic traveling waves. Finally, we show that a similar treatment works for coupled map lattices obtained by discretizing systems of conservation laws.

Convergence of a relaxation scheme for hyperbolic systems of conservation laws

Summary. This paper concerns the study of a relaxation scheme for

Richness and the Classification of Quasilinear Hyperbolic Systems

N\times N

Unstable Godunov Discrete Profiles for Steady Shock Waves

hyperbolic systems of conservation laws. In particular, with the compensated compactness techniques, we prove a rigorous result of convergence of the approximate solutions toward an entropy solution of the equilibrium system, as the relaxation time and the mesh size tend to zero.

Generic types and transitions in hyperbolic initial–boundary-value problems

Rich quasilinear hyperbolic systems are those which possess the largest possible set of entropies. Such systems have a property of global existence of weak solutions, whatever large is the bounded initial data. Although the full gas dynamics is not rich, many physically meaningful systems are. One gives below new examples and properties of the fully linearly degenerate case.

Shock Reflection in Gas Dynamics

Given a system of n conservation laws ut+f(u)x=0, the steady shock waves, when processed by the Godunov scheme, admit rather simple discrete profiles. One shows that the linear stability of these profiles depends only on the location of the eigenvalues of some endomorphism of an (n-1)-dimensional space. Applying our theory to the gas dynamics with the perfect gas law p=(\ga-1)\rho e

Oscillations non linéaires des systèmes hyperboliques: méthodes et résultats qualitatifs

, we construct unstable profiles for values of