Wavefront set and Duhamel's principle

Question

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 $a(x,\xi)$. This problem is discussed in a number of sources (Hormander v.iii, Taylor's $\Psi$DO, etc. ).

Let $S(t,s)$ be the propagator from $s$ to time $t$, then when $f=0$ one has $$ \operatorname{WF}(S(t,0)u_0) = \chi_t \operatorname{WF}(u_0) $$ where $\chi_t$ is the flow generated by the Hamiltonian $H_a$.

Now take $f \in C^0([0,T], H^s(\mathbb{R}^n))$, $u_0 = 0$ (for simplicity), then the equation is well-posed and the solution is given by: $$ u(t,x) = \int_0^t S(t,s) f(s) d s$$

My questions:

What is the wavefront set of $u$ as a distribution in $(0,T)\times\mathbb{R}^n$?

I'm missing references on the question.

My feeling is that maybe seeing integration as the push-forward of the projection $\pi(t,s,x) = (t,x)$ might work. But then I guess I should see $S(t,s)$ as an FIO in $(t,s,x)$ which sounds strange and probably there is a more elementary derivation

Answer 1

Your equation reads $Pu=f$ where $P=D_t+A(x,D_x)$ is a real principal type operator which is scalar and of first order. (I assume that $D_x=-i\partial_x$ is meant.) Moreover, $u=0$ for $t<0$. I claim that the wavefront set of $u$ is contained in the union of the wavefront set of $f$ and the bicharacteristics issuing in positive time direction from those points in the wavefront set of $f$ which are in the characteristic set of $P$. For a proof apply the forward parametrix $E^+$ in Theorem 26.1.14 in volume 4 of Hörmander's treatise. Precisely, let $\chi\in C_c^\infty$ be equal to $1$ in an open set $U$ over which we want to localise the analysis of the wavefront set of $u$. Then $\chi u\equiv E^+ P\chi u$ modulo $C^\infty$. The wavefront relation of (the Schwartz kernel of) $E^+$ is the union of the diagonal $\Delta^*$ and the forward bicharacteristic relation $C^+$. (I am using Hörmander's notations.) From known calculus rules for wavefront sets we derive the claim about $WF(u)$ over $U$ with $P\chi u$ instead of $f$. However, when the cutoff $\chi$ is chosen properly, depending on $U$, we see that $P\chi u$ and $f$ are microlocally equal along backward bicharacteristics, and the claim follows.