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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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