Regularity for parabolic integro-differential equations with very irregular kernels

Abstract

We prove Hölder regularity for a general class of parabolic integro-differential equations, which (strictly) includes many previous results. We present a proof which avoids the use of a convex envelop as well as give a new covering argument which is better suited to the fractional order setting. Our main result involves a class of kernels which may contain a singular measure, may vanish at some points, and are not required to be symmetric. This new generality of integro-differential operators opens the door to further applications of the theory, including some regularization estimates for the Boltzmann equation.arXiv:1412.3790v1 [math.AP] 11 Dec 2014Heuristically from the point of view of jump-diffusion stochastic processes, b records the drift, A records the local covariance (or √ A is the diffusion matrix), and µ records the jumps. The first Hölder regularity result for an equation of the form (1.1) was obtained in [6]. In that paper, the authors consider the elliptic equation (u constant in time), with symmetric kernels satisfying the pointwise bound (1.2) and without drift. Their proof uses probabilistic techniques involving a related Markov (pure jump) stochastic process. Other results using probabilistic techniques were [5] and [39], where different assumptions on the kernels are considered. The first purely analytical proof was given in [35]. This first generation of results are all only elliptic problems. They are not robust in the sense that as the order approaches two, the constants in the estimates blow up (hence not recovering the known second order results). Furthermore, they all require a pointwise bound below for the kernels as in (1.2).The first robust Hölder estimate for the elliptic problem was obtained in [10], which means that the estimate proved in [10] has constants that do not blow up as the order α of the equation goes to two. In that sense, [10] is the first true generalization of the theorem of Krylov and Safonov. It was the first of the series of papers [10], [12] and [11] recreating the regularity theory for fully nonlinear elliptic equations in the nonlocal setting. As above, these results are only for the elliptic problem, and they require symmetric kernels which satisfy the pointwise assumption (1.2).The first estimate for parabolic integro-differential equations, in non-divergence form, appeared, to the best of our knowledge, in [36] (the divergence case had some earlier results such as [3], [16]). In this case the kernels are symmetric and satisfy (1.2) with α = 1. The focus of [36] is on the interaction between the integro-differential part and the drift term. The proof can easily be extended to arbitrary values of α, but the estimate is not robust (it blows up as α → 2), and the details of this proof are explained in the lecture notes by one of the authors [33]. It is even possible to extend this proof to kernels which satisfy the upper bound in average like in our assumption (A2) below (see [38]). However, the estimates are not robust, and the lower bound in (1.2) is required....

Eulerian dynamics with a commutator forcing III. Fractional diffusion of order 0<α<1

“…Regularity of these equations has been the subject of active research in recent years. In particular, the result of Silvestre [16], see also Schwab and Silverstre [13], states that there exists a γ > 0 such that for all t > 0,…”

Further Time Regularity for Nonlocal, Fully Nonlinear Parabolic Equations

“…At face value, Theorem 1.1 applies to smooth solutions, but here we outline how to apply it to obtain information about viscosity solutions in several situations. We do not define viscosity solutions here, but rather refer to [7,20].…”

“…The estimate in Theorem 2.3 is one of the main tools we will use in Section 3 in order to prove our result. The same strategy for a more general class of operators, such as the one considered in the recent paper [20] or in other current research such as in [12,16,17], could be applied as long as a similar estimate, controlled in terms of sup tP. 1;0 ku.t/k L 1 , is available.…”

“…Given that u P Class./, it is enough that K be bounded for the integral to converge. In order for L K;b to be uniformly elliptic it suffices that K be bounded away from 0 (more general conditions on the kernel under which the elliptic theory can be developed was recently studied in [20]). Here is the family of linear operators we will be dealing with in this paper.…”

Fractional Fokker--Planck Equation with General Confinement Force

“…Nash type inequalities let appear the following family of ordinary differential inequalities that can be solved explicitly and lead to the growth in time given by the following application of Gronwall's inequality. We can now combine Lemma 4.2 with previous Nash type inequalities (37) and (38) to prove Proposition 4.1.…”