Systems of conservation laws with dissipation

2 Systems of conservation laws have the form ∂ t u+div x q = 0. They describe processes in many situations of the real life, including continuum mechanics. A system is in closed form when the flux q is given in terms of u. An algebraic equation of state q := f (u) yields a quasilinear first-order system. But a more accurate description involves a dissipative mechanism, through various additional terms in q − f (u): viscosity, relaxation, hyperbolic-elliptic coupling. Many well-known equations, as Navier-Stokes and Boltzman, are relevant from this framework. The goal of this course is to give an overview of the basic properties of such dissipative systems: local and global well-posedness, asymptotic behaviour, singular limit,... Key tools are reduced first-order systems and the Kawashima–Shizuta condition. Even in the the singular limit (for instance, vanishing viscosity), a small amount of dissipation is still present, in the admissibility condition for shock waves. Acknowledgements. These notes are the expanded version of a course delivered at S.A. for its warm hospitality and for giving the occasion to deliver a course on this topic. Special thanks go to Stefano Bianchini for his invitation. At the time when this course was given, I had not had enough time to organize the notes below. They were raw meat, but contained some (even fresh) information. Expectedly, successive versions will be more complete and better presented. Roughly speaking, dissipation is the opposite to conservation. More precisely, though not that much, dissipation is something added to a conservative time-dependent differential equation, which has the effect to damp the solutions and to let them tend to some equilibrium manifold. Let us take a basic example. We start with a linear conservative ODE. The conserved quantity is a positive definite quadratic form that we can take to define a norm. Thus the system writes ˙ u + Au = 0 where A ∈ M n (R) is skew-adjoint: A T = −A. Dissipation will come from an additional term Bu; the new system is ˙ u + (A + B)u = 0. Of course we may assume that B is symmetric. Otherwise put its skew-symmetric part in A. Let us assume that the norm of u is dissipated, meaning that t → |u(t)| 2 is non-increasing in the evolution. This amounts to saying that B is semi-definite positive. We have two basic questions. The first one is the time-asymptotics: …