Well-posedness of fully nonlinear and nonlocal critical parabolic equations

Abstract

Abstract. In this paper we prove the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index. The proof relies on the apriori Hölder estimate for advection fractional-diffusion equation established by Silvestre [11]. Introduction and main resultIn this paper we are interested in solving the following fully nonlinear and nonlocal parabolic equation:where F (t, x, u, w, q) where F denotes the Fourier's transform, S(R d ) is the Schwartz class of smooth real-valued rapidly decreasing functions.Recently, in the sense of viscosity solutions, fully nonlinear and nonlocal elliptic and parabolic equations have been extensively studied (cf. [4,10,3,9], etc.). In [4], Caffarelli and Silvestre studied the following type of nonlocal equation:where α ∈ (0, 2), i, j ranges in arbitrary sets, c i j ∈ R and b i j ∈ R d , the kernel a i j (y) satisfiesThis type of equation appears in the stochastic control problems. In [4], the extremal Pucci operators are used to characterize the ellipticity, and the ABP estimate, Harnack inequality and interior C 1,β -regularity were obtained. In [11], Silvestre studied the following nonlocal parabolic equation with critical index α = 1:and established C 1,β -regularity of viscosity solutions. In particular, the following first order Hamilton-Jacobi equation is covered by the above equation when H is Lipschitz continuous:In [9], Lara and Davila extended Silvestre's result to the more general case, and in particular, focused on the uniformity of regularity as α → 2.However, it is not known how to solve the fully nonlinear and nonlocal equation (1.1) in Sobolev spaces. Let us fix the main idea of the present paper for solving (1.1). Assume that F does not depend on u. Taking the gradient with respect to x for equation (1.1), we have We make the following observation:

Heat kernel estimates for critical fractional diffusion operators

“…By the maximal principal of nonlocal equation (cf. [25] or [26,Theorem 2.3]), it follows that for all 0 s < t T 0 , we obtain (3.35) for small time by (2.17). For the large time, it follows by a standard time shift argument (see [3,22]).…”

Lp-Solvability of Nonlocal Parabolic Equations with Spatial Dependent and Non-smooth Kernels

“…We now prove the following main result of this paper. 2), and for ϕ ∈ W k+α− α p ,p , there exists a unique u ∈ X k+α,p satisfying equation (20). Moreover, for all t ∈ [0, 1],…”

“…The strategy is to prove the apriori estimate (30) and then use the continuity method (cf. [12,20]). (Step 1) Let us first rewrite equation (20) as…”

“…Suppose that a(t, x, y) = a(t, x) is independent of y and satisfies (H a ν ), and b satisfies (H b ). Let p > 1 and not equal to α α−1 if α ∈ (1, 2), and let f ∈ Y 0,p and u ∈ X α,p satisfy (20). Then for all t ∈ [0, 1],…”