Gradient flows with jumps associated with nonlinear Hamilton-Jacobi equations with jumps

TL;DR

This paper studies gradient flows with jumps generated by specific vector fields, analyzing their evolution and solutions to related Hamilton-Jacobi equations using gradient representations and measures.

Contribution

It introduces a novel analysis of gradient flows with jumps associated with involutive vector fields and their connection to Hamilton-Jacobi equations using Radon measures.

Findings

Describes the evolution of bounded gradient flows with jumps.

Provides a unique solution for the integral equation involving the flow.

Connects gradient flows with jumps to Hamilton-Jacobi equations through gradient representation.

Abstract

We analyze gradient flows with jumps generated by a finite set of complete vector fields in involution using some Radon measures $u\in \mathcal{U}_a$ as admissible perturbations. Both the evolution of a bounded gradient flow $\{x^u(t,\l)\in B(x^*,3\g)\subseteq \mbn: \,t\in[0,T],\,\l\in B(x^*,2\g)\}$ and the unique solution $\l=\psi^u(t,x)\in B(x^*,2\g)\subseteq \mbn$ of integral equation $x^u(t,\l)=x\in B(x^*,\g), \,t\in[0,T]$, are described using the corresponding gradient representation associated with flow and Hamilton-jacobi equations.