Nonlinear damping estimates and stability of large-amplitude periodic wave trains

Mercredi 6 mai 2015 14:15-15:15 - Kevin Zumbrun - Indiana University

Résumé : In the stability theory for shock waves of partially dissipative equations such as compressible Navier-Stokes, MHD, etc., two delicate points are the establishment of high-frequency resolvent bounds, and the handling of regularity issues in the absence of (strictly) parabolic smoothing. Both of these can be handled together in a straightforward way by the use of linear and nonlinear damping estimates obtained by « Kawashima-type’’ energy estimates, which for small-amplitude (slowly-varying) shocks are straightforward by the original ideas of Kawashima, but for large-amplitude shocks require the additional ingredient of a »Goodman-type’’ exponential weight penalizing transverse convection. We first review these ideas in the shock wave context, then give a recent analog established with L.M. Rodrigues in the context of periodic wave trains of the Saint Venant equations for inclined thin film flow. As in the shock case, this gives for the first time a nonlinear stability result for large-amplitude spectrally stable waves, or, what turns out to be equivalent, waves with large Froude number. This includes waves of Froude number in the range physically relevant for applications in hydraulic engineering, which were inaccessible by previously existing theory.

Lieu : bât. 425 - 117-119

Nonlinear damping estimates and stability of large-amplitude periodic wave trains  Version PDF