Local Regularity for Parabolic Nonlocal Operators

By Matthieu Felsinger, M. Kassmann. 2012

158 citations recorded by the source corpus.

Weak solutions to parabolic integro-differential operators of order α ∈ (α0, 2) are studied. Local a priori estimates of Hölder norms and a weak Harnack inequality are proved. These results are robust with respect to α↗2. In this sense, the presentation is an extension of Moser's result from [20].

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/5d150db054b3c1c0ab68f2efd9481e35a5d5b2b1

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Optimal results for the fractional heat equation involving the Hardy potential (2014, methodology): “For the proof of this result, we closely follow the work of Felsinger and Kassmann in [22], where the authors develope a weak parabolic Harnack inequality for a general type of nonlocal operators.”
Quantitative approximation properties for the fractional heat equation (2017, unknown): “For more refined results we refer to for instance [14, 21, 26, 3, 18].”
Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains (2015, background): “Still, the weak Harnack inequality is established for the general case f 2 L(1), and thus the proof of the Hölder regularity with a nonzero right-hand side f (as stated above) is just a minor modification of the one in [41].”
Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, background): “Actually, there has been lately a substantial literature dealing with nonlocal parabolic equations (see for instance [14, 20, 26] and references therein).”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, unknown): “See also [8, 55, 83, 101, 121, 219, 233], among many other references.”
Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation (2016, methodology): “More recently, the method has been extended to integral operators, such as fractional diffusion, in [10, 9] — see also the work of Kassmann [45] and of Kassmann and Felsinger [29].”
Representation of solutions and large-time behavior for fully nonlocal diffusion equations (2015, background): “For the regularity theory of nonlocal equations in case α = 1 or β = 2, we refer to [11, 7, 22, 4, 5, 30, 2, 49, 48] and the references therein.”
The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian (2015, background): “3-(i) in [21], there exist positive constants C1 and C2, depending on q, such that”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, background): “The results were extended to a much more general class of kernels in [34], [43] and [33].”
On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, background): “Local Hölder continuity is not a direct consequence of the Harnack inequality in the nonlocal case, see [3, 23].”
Nonlocal self-improving properties (2015, unknown): “We finally remark that, in recent times, there has been much attention to the regularity of solutions to nonlocal problems, especially in the basic case of kernels with measurable coefficients; see for instance [1, 3, 5, 7, 8, 14].”
Schauder estimates for solutions of linear parabolic integro-differential equations (2014, methodology): “c equations such as Ho¨lder estimates, Cordes-Nirenberg type estimates and EvansKrylov theorem were stablished. Regularity for parabolic integro-differential equations has been also studied, e.g., in [8, 9, 10, 13, 14, 23] and many others. In this paper, we prove optimal pointwise Schauder estimates”
Regularity estimates for elliptic nonlocal operators (2015, unknown): “See [14] and [23] for corresponding results in the parabolic case.”