Abstract

We provide a lower bound for the blow up time of the H2 norm of the entropy solutions of the inviscid Burgers equation in terms of the H2 norm of the initial datum. This shows an interesting symmetry of the Burgers equation: the invariance of the space H2 under the action of such nonlinear equation. The argument is based on a priori estimates of energy and stability type for the (viscous) Burgers equation.

Highlights

  • Consider the Cauchy problem for the inviscid Burgers equation:∂tu + u∂xu = 0, 0 < t < T, x ∈ R, (1)u(0, x) = u0(x), x ∈ R, Citation: Coclite, G.M.; di Ruvo, L

  • The goal of this paper is to investigate the blow-up of the H2 norm of the solution of the Burgers equation

  • We prove an easy to verify relation between the H2 norm of the initial datum and the blow-up time for the H2-norm of the solution

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Summary

Introduction

Consider the Cauchy problem for the inviscid Burgers equation:. u(0, x) = u0(x), x ∈ R, Citation: Coclite, G.M.; di Ruvo, L. A more refined tool that can be used for the analysis of the geometric structure and the large-time behavior of the solution of (1) is the Hopf-Cole [13,14,15] transformation that turns the (inviscid) Burgers equation into the linear heat equation. We provide a lower bound for the maximal time T of existence of an H2 solution of (1) in terms of the H2 norm of the initial datum This result shows a symmetry of the Burgers equations: the invariance of the space H2 under the action of that nonlinear equation in the time interval [0, T]. Our argument is based on a priori estimates on the smooth solution uε of the (viscous) Burgers equation of energy and stability type (see [16,17,18]):.

Proof of Theorem 1
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