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Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivashinsky equation in the general case. (English) Zbl 0858.35017

Summary: We derive estimates on the lowest dimension in various Sobolev spaces of inertial manifolds for the Kuramoto-Sivashinsky equation: \[ {{\partial u}\over{\partial t}}+\nu D^4u+D^2u+uDu=0 \] for solutions which are periodic with period \(L\). Contrary to earlier results by C. Foiaş, B. Nicolaenko, G. R. Sell and R. Temam [J. Math. Pures Appl. 67, No. 3, 197-226 (1988; Zbl 0694.35028)] and other works, there is no requirement on the antisymmetry of the initial data. Our results are: 1. the lowest dimension of inertial manifolds in the Sobolev space \(H^m\) is bounded by a universal constant times \(\widetilde{L}^{0.82m+2.05}\); 2. the lowest dimension of inertial manifolds in \(L^2\) is bounded by a universal constant times \(\widetilde{L}^{1.64}(\ln \widetilde{L})^{0.2}\), where \(\widetilde{L}= L/2\pi\sqrt{\nu}\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35B45 A priori estimates in context of PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations

Citations:

Zbl 0694.35028
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