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 $n=2$.

 A: Perhaps techniques from the theory of pseudodifferential operators might be of help here? I have a relatively brief acquittance with the subject, but share in hope that it will be of some help to you. You can find more details in e.g. Michael E. Taylor Partial Differential Equations part 3, chapters 13.8 and 14.4.
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.
