Viscous–capillary traveling waves associated with classical and nonclassical shocks in van der Waals fluids

Abstract

We present in this work existence results for various types of diffusive–dispersive traveling waves associated with shock waves with the same shock speed in van der Waals fluids. For any shock satisfies a single entropy inequality, there exist corresponding traveling waves with suitable choices of viscosity and capillarity coefficients. The shock may be a classical shock, a nonclassical shock satisfying Lax’s shock inequalities, or a nonclassical shock violating Lax’s shock inequalities. Therefore, shock waves satisfying a single entropy inequality are also admissible under the criterion of vanishing viscosity/capillarity effects.