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

  • $\begingroup$ sorry, I have Hörmander's book in front of me, but I cannot find the theorem as you state it. Do you think you could give an exact reference? $\endgroup$ – Sascha Aug 8 '18 at 15:09
  • 1
    $\begingroup$ @Sascha Propagation of the singularities in $u_0$ is in theorem 23.1.4 of Hormander's Volume III. $\endgroup$ – F.M.R. Aug 8 '18 at 15:41

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.