Well-posedness of nonlinear parabolic equations with unbounded drift via nonlinear evolution theory

TL;DR

This paper introduces a nonlinear evolution framework for parabolic equations with unbounded drift in Lorentz spaces, establishing well-posedness and long-term behavior of solutions.

Contribution

It constructs uniformly m-accretive operators using Lorentz-Sobolev embeddings, enabling the application of the Crandall-Liggett theorem for these equations.

Findings

Proves existence and uniqueness of solutions.

Shows solutions are consistent across mild and weak formulations.

Analyzes long-time asymptotic behavior of solutions.

Abstract

We develop a nonlinear evolution framework for nonlinear parabolic equations with unbounded drift terms formulated in Lorentz spaces. The main contribution lies in the construction of uniformly m-accretive operators based on Lorentz-Sobolev embeddings, which allows us to apply the Crandall-Liggett generation theorem for nonlinear evolution equations. Within this framework, we establish existence, uniqueness, and stability of mild solutions. We further show that these mild solutions coincide with weak solutions, ensuring consistency with the variational formulation. Finally, we investigate the long-time asymptotic behavior of solutions.