IUMJ

Title: Approximation of pseudo-differential flows

Authors: Benjamin Texier

Issue: Volume 65 (2016), Issue 1, 243-272

Abstract:

Given a classical symbol $M$ of order zero, and associated semiclassical operators $\operatorname{op}_{\epsilon}(M)$, we prove that the flow of $\operatorname\{op}_{\epsilon}(M)$ is well approximated, in time $O(|\ln\epsilon|)$, by a pseudo-differential operator, the symbol of which is the flow $\exp(t M)$ of the symbol $M$. A similar result holds for non-autonomous equations, associated with time-dependent families of symbols $M(t)$. This result was already used, by the author and co-authors, to give a stability criterion for high-frequency WKB approximations, and to prove a strong Lax-Mizohata theorem. We give here two further applications: sharp semigroup bounds, implying nonlinear instability under the assumption of spectral instability at the symbolic level, and a new proof of sharp G\aa rding inequalities.