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Interaction of Semilinear Conormal Waves (joint work with Yiran Wang)

Banff International Research Station for Mathematical Innovation and Discovery

We study the local propagation of singularities of solutions of $P(y,D) u= f(y,u),$ in $R^3,$ where $P(y,D)$ is a second order strictly hyperbolic operator and $f\in C^\infty.$ We choose a time function $t$ for $P(y,D)$ and assume that $f(y,u)$ is supported on $t>-1$ and that for $t<-2,$ $u$ is assumed to be the superposition of three conormal waves that intersect transversally at a point $q$ with $t(q)=0.$ We show that, provided the incoming waves are elliptic conormal distributions of appropriate type and $(\p_u^3 f)(q, u(q))\not=0,$ the nonlinear interaction will produce singularities on the light cone for $P$ over $q.$ Melrose and Ritter, and Bony, had independently shown that the solution $u$ is a Lagrangian distribution of an appropriate class associated with the light cone over $q$ and we show that under this non-degeneracy condition, $u$ is an elliptic Lagrangian distribution and we compute its principal part.

Mathematics; Global analysis, analysis on manifolds; Dynamical systems and ergodic theory

video/mp4

Author affiliation: Purdue University

2019-10-16

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/71923

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/