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\(C^{\sigma,\alpha}\) estimates for concave, non-local parabolic equations with critical drift. (English) Zbl 1353.35083

Summary: Given a concave integro-differential operator \(I\), we study regularity for solutions of fully nonlinear, nonlocal, parabolic equations of the form \(u_t-Iu=0\). The kernels are assumed to be smooth but non necessarily symmetric, which accounts for a critical non-local drift. We prove a \(C^{\sigma +\alpha}\) estimate in the spatial variable and \(C^{1,\alpha}\) estimates in time assuming time regularity for the boundary data. The estimates are uniform in the order of the operator \(I\), hence allowing us to extend the classical Evans-Krylov result for concave parabolic equations.

MSC:

35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35D40 Viscosity solutions to PDEs
35K55 Nonlinear parabolic equations
35R09 Integro-partial differential equations
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