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Estimates for solutions of a low-viscosity kick-forced generalized Burgers equation

Published online by Cambridge University Press:  18 March 2013

Alexandre Boritchev*
Affiliation:
Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France ([email protected])

Abstract

We consider a non-homogeneous generalized Burgers equation

Here, ν is small and positive, f is strongly convex and satisfies a growth assumption, while ηω is a space-smooth random ‘kicked’ forcing term. For any solution u of this equation, we consider the quasi-stationary regime, corresponding to t ⩾ 2. After taking the ensemble average, we obtain upper estimates and time-averaged lower estimates for a class of Sobolev norms of u. These estimates are of the form with the same values of β for bounds from above and from below. They depend on η and f, but do not depend on the time t or the initial condition.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2013

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