# Existence and regularity for fractional elliptic problem with gradient term: $(-\Delta)^s u + v\cdot \nabla u = 0$ with $v \in \dot H^s$

Let us consider the problem
$$(-\Delta)^s u + v\cdot \nabla u = 0 \quad \text{ in } \mathbb R^n,$$ where $$s \in (0,1)$$, $$(-\Delta)^s$$ is the fractional Laplace operator and
$$v:\mathbb R^n \to \mathbb R^n$$ satisfies $$\|v\|_{\dot H^s(\mathbb{R}^n)}^2:=\iint\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy < +\infty$$

Question. Where can I find a result on existence, uniqueness and regularity for its solutions? In particular, I'm interested in the case $$n=2$$.

If I understand correctly the symbol of the differential operator in question is: $$p(x,\xi)=-|\xi|^{2s}+(\xi,v(x))$$ where $$(\cdot,\cdot)$$ stands for the inner product on $$\mathbb{R}^{n}$$. If $$s\in(\frac{1}{2},1)$$ then $$2s>1$$. By the regularity assumption on $$v$$, you can then use the Sobolev embedding theorem to obtain $$C^{r}_{*}$$-estimates on $$v$$, and then $$|D_{\xi}^{\alpha}p(x,\xi)|\leq C_{1}\langle\xi\rangle^{2s-|\alpha|} \\\|D_{\xi}^{\alpha}p(\cdot,\xi)\|_{C^{r}_{*}}\leq C_{2}\langle\xi\rangle^{2s-|\alpha|}$$ for some constants $$C_{1},C_{2}>0$$. The only nontrivial case seems to be when $$|\alpha|=1$$, as the $$(\xi,v(x))$$ term falls under further differentiations by $$D_{\xi}$$. Then, if again I understand correctly, the existence and regularity results in Chapter 14.4 of Taylor's book seem to apply.
• Thank you. What do you mean by $C^r_\ast$-estimate? Could you add some details on what is the existence and regularity result in Chapter 14.4 of Taylor's book? Commented Mar 5, 2021 at 11:36