## Journal of Integral Equations and Applications

### $C^\sigma ,\alpha$ estimates for concave, non-local parabolic equations with critical drift

#### Abstract

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.

#### Article information

Source
J. Integral Equations Applications, Volume 28, Number 3 (2016), 373-394.

Dates
First available in Project Euclid: 17 October 2016

https://projecteuclid.org/euclid.jiea/1476706346

Digital Object Identifier
doi:10.1216/JIE-2016-28-3-373

Mathematical Reviews number (MathSciNet)
MR3562356

Zentralblatt MATH identifier
1353.35083

#### Citation

Lara, Héctor Chang; Dávila, Gonzalo. $C^\sigma ,\alpha$ estimates for concave, non-local parabolic equations with critical drift. J. Integral Equations Applications 28 (2016), no. 3, 373--394. doi:10.1216/JIE-2016-28-3-373. https://projecteuclid.org/euclid.jiea/1476706346

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