Generalized Young Measure Solutions for a Class of Quasilinear Parabolic Equations with Linear Growth

TL;DR

This paper extends Young measure solution theory to certain quasilinear parabolic equations with linear growth, proving existence, uniqueness, and equivalence to strong solutions for specific gradient flows.

Contribution

It introduces the concept of generalized Young measure solutions and establishes their existence, uniqueness, and equivalence to strong solutions in relevant cases.

Findings

Existence of generalized Young measure solutions

Uniqueness of these solutions

Equivalence to strong solutions for gradient flows

Abstract

Using the generalized Young measure theory, we extend the theory of Young measure solutions to a class of quasilinear parabolic equations with linear growth, and introduce the concept of generalized Young measure solutions. We prove the existence and uniqueness of the generalized Young measure solutions. In addition, for the gradient flow of convex parabolic variational integral, we show that the generalized Young measure solutions are equivalent to the strong solutions.