Optimal  parameter-dependent bounds for Kuramoto-Sivashinsky-type equations

Ralf W. Wittenberg

doi:10.3934/dcds.2014.34.5325

Article Contents

2014, Volume 34, Issue 12: 5325-5357. Doi: 10.3934/dcds.2014.34.5325

This issue Previous Article Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions Next Article Realization of tangent perturbations in discrete and continuous time conservative systems

Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations

Ralf W. Wittenberg^1,

1.
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6

Received: August 2010

Revised: May 2013

Published: December 2014

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Abstract

We derive a priori estimates on the absorbing ball in $L^2$ for the stabilized and destabilized Kuramoto-Sivashinsky (KS) equations, and for a sixth-order analog, the Nikolaevskiy equation, and in each case obtain bounds whose parameter dependence is demonstrably optimal. This is done by extending a Lyapunov function construction developed by Bronski and Gambill (Nonlinearity 19 , 2023--2039 (2006)) to take into account the dependence on both large and small parameters in the system. In the case of the destabilized KS equation, the rigorous bound lim $\sup_{t \to \infty}|| u || \leq K \alpha L^{3/2}$ is sharp in both the large parameter $\alpha$ and the system size $L$. We also apply our methods to improve previous estimates on a nonlocal variant of the KS equation.

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Mathematics Subject Classification: Primary: 35B45, 35K30; Secondary: 37L30.

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