Strichartz estimates and Schrödinger equation with derivative

Question

Let

$$ iu_t + \Delta u = \Phi \cdot \nabla u$$

with initial data $u_0 \in H^s(\mathbb{R}^d)$ and $\Phi\colon \mathbb{R}^d \rightarrow \mathbb{R}^d$ sufficiently regular. Suppose I want to use Strichartz:

$$\|e^{it\Delta} u\|_{L^p W^{s,q}} \lesssim \|u_0\|_{H^s} \\ \|\int_0^t e^{i(t-s)\Delta} (\Phi \cdot\nabla u)(s) ds \|_{L^p W^{s,q}}\lesssim \|\Phi \cdot\nabla u\|_{L^{\tilde{p}'} W^{s,\tilde{q}'}}$$

where $(p,q),(\tilde{p},\tilde{q})$ admissible and $s\geq 0$ sufficiently large. Normally I want to close the argument in $L^pW^{s,q} \cap L^{\infty}H^1$ and use Banach's fixed point theorem. However I encounter a loss of derivative when trying to estimate

$$ \|\Phi \cdot\nabla u\|_{L^{\tilde{p}'} W^{s,\tilde{q}'}} $$

because of $\nabla u$ and so I always have to put $u$ in a space with $s+1$ derivatives. Is there a way to circumvent this and be able to use Strichartz estimates?

Answer 1

It is not clear to me how you plan to close your argument unless you assume that the coefficient $\Phi$ is sufficiently small. I think it is not easy to get rid of first order terms by a perturbative approach without smallness.

In some cases, large first order terms can be handled by including them in the operator. E.g., Strichartz estimates are available for some equations of the form $iu_{t}+Au=0$, where $A$ is a selfadjoint operator like $A=-\Delta+i \nabla \cdot(a u)+ia \cdot \nabla u$, provided $a(x)$ is smooth enogh and decays at infinity (but it can be large). However, your equation does not seem to fit into this form.

Let me also mention that there exists another class of estimates, the so called smoothing estimates, which ensure a gain of one full derivative w.r.to the forcing term, and may be possibly useful in your case. Note that smoothing estimates are weaker than Strichartz, indeed they only allow to control a weighted $L^{2}$ norm of the solution.