[Paper Review] Regularity for fully nonlinear nonlocal parabolic equations with rough kernels

Joaquim Serra|arXiv (Cornell University)|Jan 18, 2014

Nonlinear Partial Differential Equations6 references4 citations

TL;DR

This paper establishes space and time regularity for viscosity solutions of fully nonlinear nonlocal parabolic equations with rough kernels, using a novel Liouville-type theorem and blow-up/compactness arguments. It proves that bounded solutions are Hölder continuous in space with exponent $\beta < \min\{\sigma, 1+\alpha\}$ and in time with exponent $\beta/\sigma$, extending prior results to parabolic settings and $\sigma \leq 1$.

ABSTRACT

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $u_t = \I u$, where $\I$ is translation invariant and elliptic with respect to the class $\mathcal L_0(σ)$ of Caffarelli and Silvestre, $σ\in(0,2)$ being the order of $\I$. We prove that if $u$ is a viscosity solution in $B_1 imes (-1,0]$ which is merely bounded in $\R^n imes (-1,0]$, then $u$ is $C^β$ in space and $C^{β/σ}$ in time in $\overline{B_{1/2}} imes [-1/2,0]$, for all $β&lt; \min\{σ, 1+α\}$, where $α&gt;0$. Our proof combines a Liouville type theorem ---relaying on the nonlocal parabolic $C^α$ estimate of Chang and Dávila--- and a blow up and compactness argument.

Motivation & Objective

• To establish interior regularity estimates for fully nonlinear nonlocal parabolic equations with rough kernels in the class $\mathcal{L}_0(\sigma)$, where $\sigma \in (0,2)$.
• To extend the $C^{1+\alpha}$ regularity theory from elliptic to parabolic equations under the same rough kernel assumptions.
• To address the challenge of limited global control in $L^\infty$ norms for incremental quotients in nonlocal settings, which blocks classical iteration methods.
• To provide a new proof strategy based on Liouville-type theorems and blow-up/compactness arguments, applicable beyond the scope of prior methods.
• To prove regularity results for $\sigma \leq 1$, where classical $C^{1+\alpha}$ estimates fail due to strong influence of distant oscillations.

Proposed method

• Establish a Liouville-type theorem for global solutions of the nonlocal parabolic equation $u_t = \tilde{I}u$, showing that bounded global solutions with vanishing first-order moments are affine functions.
• Use a blow-up and compactness argument: rescale solutions near a point of interest and extract a limit solution from a sequence of rescaled functions.
• Apply the nonlocal parabolic $C^\alpha$ estimate from Chang and Dávila to control the growth of rescaled solutions and ensure convergence to a global solution.
• Use the optimality conditions of least squares (vanishing mean and linear terms) in the rescaling process to enforce symmetry and control the limit behavior.
• Control the growth of rescaled solutions in expanding parabolic cylinders $Q_R^{\sigma_m}$ via a recursive estimate on the oscillation of incremental quotients.
• Pass to the limit in the rescaled equations to show that the limit solution satisfies a translation-invariant, elliptic nonlocal equation, leading to a contradiction if the solution is not smooth.

Experimental results

Research questions

• RQ1 Can $C^{\beta}$ space and $C^{\beta/\sigma}$ time regularity be established for viscosity solutions of fully nonlinear nonlocal parabolic equations with rough kernels in $\mathcal{L}_0(\sigma)$?
• RQ2 Does the classical iteration method for improving Hölder regularity fail for nonlocal equations with rough kernels due to loss of global $L^\infty$ control on incremental quotients?
• RQ3 Can a Liouville-type theorem be used to bypass the failure of iterative methods in nonlocal settings and derive interior regularity estimates?
• RQ4 Is it possible to extend $C^{1+\alpha}$ regularity results to the case $\sigma \leq 1$ under rough kernel assumptions, where previous methods fail?
• RQ5 What is the sharp regularity threshold for solutions that are merely bounded in $\mathbb{R}^n \times (-1,0]$ in the context of nonlocal parabolic equations with rough kernels?

Key findings

• Solutions $u$ that are bounded in $\mathbb{R}^n \times (-1,0]$ and satisfy $u_t = \mathrm{I}u$ in $B_1 \times (-1,0]$ are $C^\beta$ in space and $C^{\beta/\sigma}$ in time in $\overline{B_{1/2}} \times [-1/2,0]$, for all $\beta < \min\{\sigma, 1+\alpha\}$.
• The regularity threshold $\beta < \min\{\sigma, 1+\alpha\}$ is sharp: for $\sigma \leq 1$, the best possible space regularity is $C^{\sigma - \epsilon}$, and for $\sigma > 1$, it is $C^{1+\alpha - \epsilon}$.
• The proof introduces a new method based on a Liouville theorem and blow-up/compactness, which avoids the failure of classical iterative schemes due to lack of global $L^\infty$ control.
• The method is flexible and can be extended to nonlocal equations with time dependence and to boundary regularity problems.
• The result holds uniformly in $\sigma \in (0,2)$, including $\sigma \leq 1$, where previous $C^{1+\alpha}$ estimates required smoother kernels.
• The compactness argument yields a contradiction when assuming the rescaled sequence does not converge to an affine function, proving the solution must be regular.