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?