Introduction to geometric methods in the study of quasilinear hyperbolic systems
speaker: Serge Alinhac (Université Paris-Sud)abstract: - Introduction: cones, free solutions of the wave equation, $Z$-fields. - Perturbation techniques for quasilinear wave equations } : Klainerman's method, Klainerman's inequality. The Klainerman-Sideris method. - Some elements of Riemanian geometry} : connexion, divergence, hessian, deformation tensor, etc. - Energy inequalities for the wave equation} : standard, conformal, KSS inequalities ; ghost weight method Geometry associated to an optical function } Geometric approach to energy inequalities and applications } : standard, improved standard and conformal inequalities ; applications to quasilinear equations satisfying null conditions. Energy inequalities for Maxwell and Bianchi equations Estimates for the components of $\pi $ : transport and ellipticity The commutation problem } : formula for the scalar case ; the tensorial case. The quasiradial point of view. Application to the study of the behavior of free solutions to the wave equation on a curved background} Application to the study of the "blowup at infinity" model case $$\partial t2u=(1+u)\Delta u.$$
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