Good morning everybody. I'd like to do a qualitative question about the Strichartz estimate. Let me recall it.
Let $u$ be the solution of the Schrodinger equation:
$$ i \partial_t u + \Delta u = f(x) $$ with initial data $u(0) = \phi \in H^1(\mathbb{R^d})$.
If the following relation of indexes holds $$ \frac{2}{q} + \frac{d}{r} = \frac{d}{2} $$ then the following inequality is true: $$ |u|_{L^q_t L^r_x} \leqslant |\phi|_{L^2_x} + |f|_{L^{q'_1}_t L_x^{r'_1}} $$ where $q_1$ and $r_1$ is a couple of Strichartz indexes and $q'_1, r'_1$ satisfie the relation: $$ \frac{1}{q}+\frac{1}{q'} = 1. $$
My question is not at all technicall, infact is really clear to me how the proof works and why these estimates are so usefull.
I really would like understand what is the physical meaning of these kind of inequality (if it exists) and how can I give an interpretation to these results with the probabilistic point of view of the solution as probability density for the position of the particle.
I think that a starting point is to consider the inequality given by the couple $q = + \infty$ and $r = 2$ (this couple works for every dimension!). I think it is very important since it gives an uniform bound in time of the integral of the density.
$$ 1 = |u|_{L^{\infty}L^2} \leqslant |\phi|_{L^2_x} + |f|_{L^{q'_1}_t L_x^{r'_1}} $$
This kind of inequality seems like full of information about some constraints between initial data and the potential (or something linked to it).
I've deeply studied the NLS and a lot of property of the structure of this equation but even in the Tao's books and cannot find any interpration to the huge amount of developed technologies.
Ciao AM