Systems of Nonlinear Hyperbolic PDE's and Applications to Continuum Mechanics
speaker: Thomas Sideris (California University Santa Barbara)abstract: These lectures present recent results on the global existence of solutions to the initial value problem for 3d nonlinear elastodynamics close to the reference configuration, in both the compressible and incompressible cases. The equations of motion are presented using a field theoretic approach with the least action principle, allowing for a natural introduction to energy methods based on the Galilean invariance. Dispersive estimates are derived via the combination of generalized Sobolev inequalities and a new series of weighted L2 estimates. A detailed analysis of the nonlinear structure of the equations uncovers a null condition which is necessary to cancel the resonances of each wave family. The results are explained within the context of nonlinear hyperbolic pde's with an emphasis on the evolution of techniques used therein.
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