Global existence and smoothness for solutions of viscous Burgers equation. (2) The unbounded case: a characteristic flow study
Résumé
We show that the homogeneous viscous Burgers equation (∂t−ηΔ)u(t,x)+(u⋅∇)u(t,x)=0, (t,x)∈R+×Rd (d≥1,η>0) has a globally defined smooth solution if the initial condition u0 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 η>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.