Entropy solutions for diffusion-convection equations with partial diffusivity

Abstract:

We consider the Cauchy problem for the following scalar conservation law with partial viscosity \[ {u_t} = {\Delta _x}u + {\partial _y}(f(u)),\quad (x,y) \in {{\mathbf {R}}^N},t > 0.\] The existence of solutions is proved by the vanishing viscosity method. By introducing a suitable entropy condition we prove uniqueness of solutions. This entropy condition is inspired by the entropy criterion introduced by Kruzhkov for hyperbolic conservation laws but it takes into account the effect of diffusion.