2015/10/06 by Jérémie Unterberger, Unterberger, Jeremie
Economics, Econometrics and Finance · Mathematics · #35A01 #35B45 #35B50 #35K15 #35L65 #35Q30 #35Q35 #76N10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1510.01539
openalex publication_date 2015/10/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.