Existence of traveling waves associated with Lax shocks which violate Oleinik's entropy criterion. Issue 5 (4th April 2017) - Hiep, Nguyen Huu; Thanh, Mai Duc; Huy, Nguyen Dinh
Abstract : This paper answers to the question whether a shock wave in conservation laws satisfying the Lax shock inequalities but not Oleinik's entropy criterion is admissible under the vanishing viscosity-capillarity effects. Such a shock appears in van der Waals fluids when a secant line meets the graph of the flux function at four distinct points, and the shock jumps between the two farthest points. The existence of the corresponding traveling waves would justify the admissibility of the shock. For this purpose, we will first show that the corresponding traveling waves satisfy a system of differential equations with two saddle points and two asymptotically stable points. Second, we estimate the domains of attraction of the asymptotically stable equilibrium points, relying on Lyapunov's stability theory. Third, we investigate the circumstances when an unstable trajectory leaving the saddle point corresponding to the left-hand state of the shock will ever enter the domain of attraction of each of the two asymptotically stable equilibrium points. Finally, we establish the existence of traveling waves associated with a Lax shock but violating the Oleinik's entropy criterion.