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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

3 votes
1 answer
329 views

Wavefront set and Duhamel's principle

Consider the Cauchy problem: $$ \frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0, $$ where $A$ has real principal symbol $a …
1 vote
0 answers
69 views

Smoothing in linear hyperbolic equations

This is a bit fuzzy, but I've somewhere read or heard something like: "For linear hyperbolic equations smoothing in time leads to smoothing in space" Is this in any sense true? References, elabor …