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

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.

arXiv comment37 pages

References in corpus (2)

• Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications
• Global existence for strong solutions of viscous Burgers equation. (1) The bounded case

Cited by in corpus (1)

• Stationary solutions to the stochastic Burgers equation on the line