Global existence and smoothness for solutions of viscous Burgers equation. (2) The unbounded case: a characteristic flow study

TL;DR

This paper proves the global existence and smoothness of solutions to the viscous Burgers equation with unbounded initial data, using characteristic flow estimates and probabilistic methods.

Contribution

It extends known results by establishing global smooth solutions for unbounded initial conditions using a characteristic flow approach and probabilistic estimates.

Findings

Global smooth solutions exist for initial data growing slower than linearly at infinity.

The proof employs Feynman-Kac representation and Schauder estimates.

The results apply to certain large initial conditions with stable characteristic flows.

Abstract

We show that the homogeneous viscous Burgers equation $(\partial_t-\eta\Delta) u(t,x)+(u\cdot\nabla)u(t,x)=0,\ (t,x)\in{\mathbb{R}}_+\times{\mathbb{R}}^d$ $(d\ge 1, \eta>0)$ has a globally defined smooth solution if the initial condition $u_0$ is a smooth function growing like $o(|x|)$ at infinity. The proof relies mostly on estimates of the random characteristic flow defined by a Feynman-Kac representation of the solution. Viscosity independent a priori bounds for the solution are derived from these. The regularity of the solution is then proved for fixed $\eta>0$ using Schauder estimates. The result extends with few modifications to initial conditions growing abnormally large in regions with small relative volume, separated by well-behaved bulk regions, provided these are stable under the characteristic flow with high probability. We provide a large family of examples for which this loose criterion may be verified by hand.