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

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


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