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On the differentiability of the solution to an equation with drift and fractional diffusion. (English) Zbl 1308.35042
Let $$s\in (0,1/2)$$, $$\alpha\in (0,2s)$$, and consider a solution $$u\in L^\infty([-1,0]\times B_1)$$ of the nonlocal drift-diffusion equation $\partial_t u + {\mathbf b}\cdot \nabla u + (-\Delta)^s u = f\;, \quad (t,x)\in (-1,0)\times \mathbb{R}^n\;,$ where the components of the vector field $${\mathbf b}$$ and the source term $$f$$ both belong to $$L^\infty([-1,0];C^{1-2s+\alpha}(B_1))$$. Here, $$B_r$$ is the ball of $$\mathbb{R}^n$$, $$n\geq 1$$, with radius $$r>0$$ centered at zero and $$(-\Delta)^s$$ denotes the fractional Laplacian. It is proved that the assumed regularity of $$({\mathbf b},f)$$ implies that $$u(t)$$ belongs to $$C^{1,\alpha}(B_{1/2})$$ for each $$t\in (-1,0]$$ and that $\|u\|_{L^\infty([-1/2,0];C^{1,\alpha}(B_{1/2}))} \leq C \left[ \|u\|_{L^\infty([-1,0]\times B_1)} + \|f\|_{L^\infty([-1,0];C^{1-2s+\alpha}(B_1))} \right]$ for some constant $$C$$ depending only on $$n$$, $$s$$, and $$\|{\mathbf b}\|_{L^\infty([-1,0];C^{1-2s+\alpha}(B_1))}$$. Under the weaker assumption that $${\mathbf b}$$ and $$f$$ enjoy the above regularity properties with $$\alpha=0$$, it was shown previously by the author that $$u(t)$$ is Hölder continuous for $$t\in (-1,0]$$ and the interesting outcome of the paper under review is that a mild increase in the regularity of $$({\mathbf b},f)$$ results in a rather strong improvement of the regularity of $$u$$. The proof relies on the fact that $$u$$ can be interpreted as the trace $$u(t,x,0)$$ of a function $$(t,x,y)\mapsto u(t,x,y)$$ as $$y\to 0$$ where $$u$$ solves the extended equation $\partial_t u(t,x,0) + {\mathbf b}(t,x)\cdot \nabla u(t,x,0) - c \lim_{y\to 0} y^{1-2s} \partial_y u(t,x,y) = f(t,x)$ for $$(t,x)\in (-1,0)\times \mathbb{R}^n$$, and $\mathrm{div}\left( y^{1-2s} \nabla u(t,x,y) \right) = 0\;, \quad (t,x,y)\in (-1,0)\times \mathbb{R}^n \times (0,\infty)\;,$ for some constant $$c>0$$ depending on $$n$$ and $$s$$. Such a transformation maps a nonlocal equation in $$\mathbb{R}^n$$ to a local one in $$\mathbb{R}^n\times (0,\infty)$$. Another step is the derivation of local estimates to the above extended equation when $${\mathbf b}\equiv 0$$ and their extension to suitably small vector fields $$\mathbf b$$.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35R11 Fractional partial differential equations 35K10 Second-order parabolic equations 35D40 Viscosity solutions to PDEs 35B45 A priori estimates in context of PDEs
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