Pré-Publication, Document De Travail Année : 2015

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.

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Dates et versions

hal-01304557 , version 1 (19-04-2016)

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  • HAL Id : hal-01304557 , version 1

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Jérémie Unterberger. Global existence and smoothness for solutions of viscous Burgers equation. (2) The unbounded case: a characteristic flow study. 2015. ⟨hal-01304557⟩
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