Stationary Phase and Propagation Speed (Reference)

I'm trying to understand how one can make precise statements about propagation speed for various (linear and nonlinear) PDEs (in particular, ones with infinite propagation speed) and what, if anything, these precise statements can be used to prove (which isn't trivial by other methods). An example set of questions I'd be interested in learning more about is the following:

Suppose we have a nice initial data $\varphi \in \mathcal{S}(\mathbb{R}^n)$ which is frequency supported in $|\xi| \leq N$. Let $u$ be the resulting (strong) solution to a linear or nonlinear Schrodinger equation with initial data $\varphi$. In what sense does $u$ propagate with speed at most $N$ in physical space (as predicted by the formula ''speed of propagation equals minus the gradient of the dispersion relation'' together with the initial frequency localization of $u$)? If, for example, $u$ at time zero (i.e., $\varphi$) has most of its $L^2$ norm in the set $|x| \leq M$, does most of its $L^2$ norm remain in the set $|x| \leq M + TN$ when $|t| \leq T$? What about $H^k$ norms (for integral $k \geq 1$, say)? How is this behavior different in the linear and nonlinear cases? What if we only assume that $\varphi$ has ''most'' of its frequency support in $|\xi| \leq N$ (as quantified by some weighted $L^2$ norm in frequency space)?

Tao mentions in Nonlinear Dispersive Equations that one can use the method of stationary phase to give precise meaning to statements about propagation speed, but I don't believe he gives any references. So, I'd be interested in any references which cover questions like the ones indicated above.

1 Answer

Here is one response for the case of the linear Schrödinger equation. In this case, you can write a formula for solutions, namely $$u(t,x) = (4\pi i t)^{-d/2} \int_{\mathbb{R}^d} e^{i|x-y|^2/4t} \phi(y)\,dy$$ is the solution to $i\partial_t u + \Delta u =0$ with initial data $\phi$.

From this you can read off the "Fraunhofer" approximation $$u(t,x) \sim u_F(t,x):= (4\pi i t)^{-d/2} e^{i|x|^2/4t} \hat \phi(\tfrac{x}{2t}),$$ where $\hat\phi$ is the Fourier transform. You could alternatively derive this asymptotic by solving the PDE with the Fourier transform and then taking the stationary phase approximation in the resulting integral expression.

Comparing with the true formula (and doing a change of variables and using Plancherel), one has $$\| u(t) - u_F(t) \|_{L^2} = \|(1-e^{i|y|^2/4t})\phi\|_{L^2} \lesssim t^{-1}\|y^2\phi\|_{L^2} \lesssim t^{-1},$$ since we assume $\phi$ is Schwartz.

Now, if $\hat\phi$ is supported where $|\xi|\leq N$, then $u_F(t)$ is supported where $|x|\leq 2Nt$. So, using the approximation above, one finds that $$\|u(t)\|_{L^2(|x|>2Nt)} \lesssim t^{-1},$$ which gives an approximate finite speed of propagation statement (at least for large $t$).