Phase Space Analysis of Partial Differential Equations

Effectively hyperbolic Cauchy problem

speaker: Tatsuo Nishitani (Department of Mathematics, Graduate School of Science, Osaka University)

abstract: Multiple characteristics Necessary conditions of Ivrii-Petkov Effective hyperbolicity Generalized flows Generalized characteristic curves Generalized bicharacteristics Microlocal hyperbolic a priori estimate, a heuristic argument Local hyperbolic a priori estimate Uniqueness results Existence theorem Deriving a priori estimate, a heuristic argument Metric with large parameters associated to a surface Symbols and weights A lemma for calculus Specializing symbols Regularization of time function $T$ Symbol $\Lambda=M\log T$ Composition formula Composition $\langle{\xi}\rangle_{\gamma}^{-a\rho}\#P\#\langle{\xi} \rangle_{\gamma}^{a\rho}$ with a parabolic $\rho$ Composition $T^{-M}\#P\#T^M=P_{T^M}$ Symbol $P_{T^M}$ Definition of $p(z;\zeta)$, $Q(z)$ Hyperbolicity of $p(z;\zeta)$ Estimating $\Lambda$ $Q(z)$ separates $p(z;\zeta)$ Estimating $P_j$ Estimating $P_{T^M}$ Estimating $Q(z)$ Estimating $S(z)=({\bar Q}\#P_{T^M}-\overline{ P_{T^M}}\#Q)2i Estimating $S(z)^w$ from below Hyperbolic polynomials Semi-continuity of hyperbolic cones Applications of semi-continuity

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