Vanishing Viscosity Limits of Scalar Equations with Degenerate Diffusivity

TL;DR

This paper proves the existence and convergence of solutions for a scalar degenerate parabolic equation with source terms, showing they tend to the Kruzhkov entropy solution as viscosity vanishes, using H-measure compactness.

Contribution

It establishes the existence of solutions for degenerate scalar equations with sources and demonstrates their convergence to entropy solutions in multiple dimensions.

Findings

Solutions exist for initial data with bounded variation.

Solutions converge to Kruzhkov entropy solutions as viscosity approaches zero.

H-measure compactness is used to prove convergence in several space dimensions.

Abstract

We consider a scalar, possibly degenerate parabolic equation with a source term, in several space dimensions. For initial data with bounded variation we prove the existence of solutions to the initial-value problem. Then we show that these solutions converge, in the vanishing-viscosity limit, to the Kruzhkov entropy solution of the corresponding hyperbolic equation. The proof exploits the H-measure compactness in several space dimensions.