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On the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion. (English) Zbl 1379.35049

Summary: We investigate the differentiability property of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical- or critical-type cases. Under the suitable conditions on the velocity field and the forcing term in terms of the spatial Hölder regularity, and for the initial data without regularity assumption, we show the a priori differentiability estimates for any positive time. If additionally the velocity field is divergence-free, we also prove that the vanishing viscosity weak solution is differentiable with some Hölder continuous derivatives for any positive time.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
35R11 Fractional partial differential equations
35B45 A priori estimates in context of PDEs
35R09 Integro-partial differential equations
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