L-estimates for time fractional parabolic equations with coefficients measurable in time

Abstract

We establish the Lp-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend a recent result in [6] to a large extent.2010 Mathematics Subject Classification. 35R11, 26A33, 35R05.

L-estimates for time fractional parabolic equations in divergence form with measurable coefficients

“…Besides that they are singular integral free, another advantage of these approaches is their flexibility: they can be applied to both divergence and non-divergence or even nonlocal equations with coefficients which are very irregular in some of the independent variables. See, for instance, [2,13,14,10,9,11,6] and the references therein. The class of coefficients considered in this paper was first treated by the second named author and Krylov in [13,14] for elliptic and parabolic equations in non-divergence form, and later also in [7] (without symmetric condition on a ij ), [3] (with symmetric a ij ), and [10,9,8,4] for elliptic and parabolic equations and systems in divergence form.…”

“…In this paper, we adapted the level set argument in [5] to equations of the form (1.1) with a nonlocal time derivative term, following the scheme in [6]. The main difficulty arises in the key step where one needs to estimate local L ∞ estimates of the gradient of solutions to locally homogeneous equations.…”

“…It is worth noting that to apply the Sobolev type embedding results, we need to estimate the H α,1 2 norm of the spatial derivatives of solutions. Compared to [6], here the main obstacle is that we cannot estimate the whole gradient D x u in the H α,1 2 space due to the lack of regularity of a ij in the x 1 direction. To this end, we also exploit an idea in [10] by considering D x ′ u and a certain linear combination of the first spatial derivatives, instead of D x u.…”

Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

“…To prove that J is a contraction, one has to argue as above, exploiting also the fact that for bounded z ∈ β V 2 p , p > d + 2 β , then Dz is bounded in L ∞ (Q τ ). Using [18,Lemma A.6] and using that (z 1 − z 2 )(0) = 0 we have…”

Optimal Regularity for a Dirichlet-Conormal Problem in Reifenberg Flat Domain

“…Compared to the purely Dirichlet or purely conormal problems, the approximation function in our problem is less regular, which is only W 1,4−ε , not Lipschitz. This situation is similar to [11], where dedicated decay rates of the level sets are required.…”