On the Theory of Discontinuous Solutions to Some Strongly Degenerate Parabolic Equations

TL;DR

This paper investigates discontinuous solutions to strongly degenerate parabolic equations similar to Burgers' equation, establishing existence and uniqueness results for generalized solutions with bounded variation, accommodating unbounded velocity gradients.

Contribution

It introduces a new framework for defining and analyzing generalized solutions to degenerate parabolic equations with unbounded gradients, including existence and uniqueness theorems.

Findings

Existence of generalized solutions in bounded variation classes.

Uniqueness proven for piecewise smooth solutions with regular discontinuity lines.

A priori estimates focus on diffusion flux, allowing arbitrary local growth of velocity gradient.

Abstract

It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux. Such equations degenerate to hyperbolic ones as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted. In the paper the definition of the generalized solution is given and the existence theorem is established in the classes of functions close to ones of bounded variation. The main feature of used a priori estimates is the fact that one needs to estimate only the diffusion flux, which allows to have in fact arbitrary local growth of the velocity gradient. The uniqueness theorem is proven for essentially narrower class of piecewise smooth functions with regular behavior of discontinuity lines.