Waves in Systems of Conservation Laws: Large vs Small Scales

Several results about the stability of waves in systems of conservation/balance laws, disseminated in the litterature, obey to a common rule. The linear/spectral stability of the microscopic pattern (the internal structure of the wave) implies the well-posedness of a macroscopic Cauchy problem for an other system of conservation laws. The latter is often obtained by retaining only the conservation laws of the former system and dropping the higher order terms. But recent examples display a more complicated “average system”. 1. Dispersive/Dissipative Mechanisms for Conservation Laws In general, we are interested in systems of conservation laws (1) ∂tu + divxq = 0, u =   u1 .. un   , q = (q)α=1,...,d, q(x, t) ∈ R. In an ideal modelling, the fluxes q are determined by vector fields: q = F(u). However, in more realistic situations, there is a discrepancy between the actual fluxes q and the one at equilibrium F(u). Among the various forms of physical mechanisms, we know the following: Dissipation: Here, q = Q(u,∇xu) where some entropy is dissipated. The mechanism is not reversible, contrary to that described by (1). For instance, one has q = F (u)−∇xu (artificial viscosity). An important example is given by the Navier–Stokes equations for a compressible fluid. It is the occasion to notice that even in presence of dissipation, the system (1) is not really parabolic, but displays both hyperbolic and parabolic features. This is the This research is partially supported by the European IHP project “HYKE”, contract # HPRN– CT–2002-00282.