Existence of mild solutions for the Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces

TL;DR

This paper proves the existence of global mild solutions for a Hamilton-Jacobi equation with critical fractional viscosity in Besov spaces, and describes their asymptotic behavior resembling the Poisson kernel.

Contribution

It establishes the existence of global solutions under small initial data in Besov spaces and analyzes their asymptotic behavior, which was not previously known for this equation.

Findings

Existence of global mild solutions for small initial data.

Solutions behave asymptotically like multiples of the Poisson kernel.

Applicable to Hamilton-Jacobi equations with critical fractional dissipation.

Abstract

We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ \partial_t u + (-\Delta)^{ 1/2} u = |\nabla u|^p, \quad x \in \mathbb R^N, t > 0, \qquad u(x,0) = u_0(x) , \quad x \in \mathbb R^N, $$ where $p > 1$ and $u_0 \in B^1_{r,1}(\mathbb R^N) \cap B^1_{\infty,1} (\mathbb R^N)$ with $r \in [1,\infty]$. We show that for sufficiently small $u_0 \in \dot B^1_{\infty,1}(\mathbb R^N)$, there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel.