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Applied Math Seminar
L_2 Bounds and uncertainty for the Kuramoto-Sivashinsky equation
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The Kuramoto Sivashinski (KS) equation $u_t + u u_x = -u_{xxxx} - u_{xx}$ is a canonical model for spatio-temporal chaos in PDEs. It is known from results of Temam and collaborators that global in time bounds on the L_2 norm of solutions of the KS equation give estimates on the dimension of the attractor. There are various results (by Nicolaenko, Scheurer and Temam, Collet, Eckmann, Epstein and Stubbe, Goodman, and others) that give various L_2 bounds for the KS equation. All of these results work in basically the same way, by constructing a potential in a fourth order linear eigenvalue problem so that the ground state energy is as large as possible. We give a simple argument based on uncertainty estimates which indicates the best estimate that should be possible, and construct a potential which realizes this. This idea of extremizing the ground state energy as a function of the potential is related to Lieb-Thirring inequalities, and to some older work of Keller on the strength of the strongest column. (This is joint work with Tom Gambill (Univ. Illinois)) |