Undercompressive Shocks in Systems of Conservation Laws

Abstract

In this paper, we describe recent progress in our understanding of Riemann problems that involve undercompressive shock waves for 2 × 2 systems of nonstrictly hyperbolic conservation laws. A 2 × 2 system of conservation laws $$ {U_t} + F{(U)_x} = 0 $$ ((1.1)) , U = U(x,t) ∈ R 2, F: R 2 → R 2, is nonstrictly hyperbolic if the eigenvalues λ1(U) ≤ λ2(U) of dF(U) are real, but not distinct for every U. As defined in [4], system (1.1) has an umbilic point at U = U* if dF(U*) is a multiple of the identity. Hyperbolic equations with an isolated umbilic point can be classified locally according to properties of the quadratic map d 2 F(U*). Since linear changes of coordinates do not affect the shocks or rarefaction waves for quadratic nonlinearities F, the general family of quadratic nonlinearities F with a unique umbilic point can be reduced to a two parameter family, which we write as $$ Q(u,v) = d(a{u^3}/3 + b{u^2}v + u{v^2}),a \ne 1 + {b^2} $$ ((1.2)) , where d denotes gradient with respect to U = (u,v).KeywordsBifurcation DiagramRiemann ProblemQuadratic NonlinearitySaddle ConnectionMelnikov FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.