An application of global gradient estimates in Lorentz-Morrey spaces: The existence of stationary solution to degenerate diffusive Hamilton-Jacobi equations

TL;DR

This paper establishes the existence of renormalized solutions for degenerate diffusive Hamilton-Jacobi equations with measure data, utilizing global gradient estimates in Lorentz-Morrey spaces, advancing the understanding of solutions in complex mathematical physics models.

Contribution

It introduces new existence results for stationary solutions of degenerate Hamilton-Jacobi equations using Lorentz-Morrey space estimates, extending previous gradient estimate techniques.

Findings

Proved existence of solutions with finite measure data.

Applied Lorentz-Morrey space estimates to degenerate equations.

Extended previous gradient estimate methods to new equation classes.

Abstract

In historical mathematics and physics, the Kardar-Parisi-Zhang equation or a quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi equation in growing interface and universality classes, is also known by the different name as the quasilinear Riccati type equation. The existence of solutions to this type of equation under some assumptions and requirements, still remains an interesting open problem at the moment. In our previous studies \cite{MP2018, MPT2019}, we obtained the global bounds and gradient estimates for quasilinear elliptic equations with measure data. There have been many applications are discussed related to these works, and main goal of this paper is to obtain the existence of a renormalized solution to the quasilinear stationary solution to the degenerate diffusive Hamilton-Jacobi equation with the finite measure data in Lorentz-Morrey spaces.