A Parabolic Problem with a Fractional Time Derivative

By Mark Allen, Luis Caffarelli, Alexis Vasseur. 2015

226 citations recorded by the source corpus.

We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi–Nash–Moser Hölder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/f0be695bb703f0267db62320cdc2d6a0d6a4c314

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, methodology): “- Parabolic equations with fractional derivative in time by Allen, Caffarelli and Vasseur [1].”
Extension properties and boundary estimates for a fractional heat operator (2015, methodology): “as a non-local operator in space as well. This type of equations has attracted considerable interest during the last years, mostly due to their applications in the modeling of anomalous diﬀusion, see [ACV], [KSVZ], [KSZ], and the references therein. Decisive progress in the study of the ﬁne properti”
L-estimates for time fractional parabolic equations with coefficients measurable in time (2018, background): “One can find De GiorgiNash-Moser type Hölder estimates for time fractional parabolic equations in [14], and for parabolic equations with fractional operators in both t and x in [1].”
Getting Acquainted with the Fractional Laplacian (2017, background): “in Section 2 of [5], it may be also convenient to constantly extend u in (−∞, 0).”
Time fractional diffusion equations: solution concepts, regularity, and long-time behavior (2019, methodology): “of equations in divergence form. In the following we will describe the main results of the De Giorgi-Nash-Moser theory for (16), which has been developed by the author, see [47, 48, 49, 53], see also [1, 2] for the fully non-local case. Throughout this section we will assume that α ∈ (0,1), u0 ∈ L2(”
Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview (2018, background): “, [5, 21, 71, 88, 74, 90, 55, 46, 4, 65, 101, 25]; see also the monograph [87] for closely related evolutionary integral equations.”
Maximum principles, extension problem and inversion for nonlocal one-sided equations (2015, background): “We just mention here the work [2] and the special issue [13], see also the references therein.”
Representation of solutions and large-time behavior for fully nonlocal diffusion equations (2015, background): “In a very recent paper Allen, Caffarelli and Vasseur [1] have studied the regularity of weak solutions to such problems.”