On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion

By Luis Silvestre. 2009

100 citations recorded by the source corpus.

Abstract We prove that the Hamilton–Jacobi equation for an arbitrary Hamiltonian H (locally Lipschitz but not necessarily convex) and fractional diffusion of order one (critical) has classical C 1 , α solutions. The proof is achieved using a new Holder estimate for solutions of advection–diffusion equations of order one with bounded vector fields that are not necessarily divergence free.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/0dc0a220e165fb1d00d1433669ab3a9c54774521

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In Applied Mathematics

In Modeling and Simulation

In Public Health, Environmental and Occupational Health

Citations in the corpus

Regularity results for nonlocal parabolic equations (2013, methodology): “In [48], Silvestre proves that indeed the nonlocal term has enough influence to regularize and solutions will be classical.”
Global well-posedness for the two-dimensional Muskat problem with slope less than 1 (2017, unknown): “Then since ( K ( t , x , h ) + K ( t , x , − h ))/ 2 is a symmetric uniformly elliptic kernel, it follows that we have local C α bounds for α α 0 for some α 0 depending on ellipticity constants; see [Silvestre 2011].”
Lp-maximal regularity of nonlocal parabolic equations and applications (2013, unknown): “On the other hand, Hölder regularity theory for the viscosity solutions of fully non-linear and nonlocal elliptic equations was also developed by Caffarelli and Silvestre [8], and Barles, Chasseigne and Imbert [4] (see also [3] and the series of works of Silverstre [30,31, 33,32], etc. .”
On the Loss of Continuity for Super-Critical Drift-Diffusion Equations (2012, methodology): “Using the ideas from [33], the result was generalized to non divergence-free drifts in [32].”
Local Regularity for Parabolic Nonlocal Operators (2012, background): “See also [23] for related results in a critical nonlinear setting when = 1.”
Holder estimates for advection fractional-diffusion equations (2010, methodology): “The fractional Laplacian could be replaced by arbitrary integro-differential operators of order 2s, even with discontinuous coefficients, as in [11], but we chose to keep only the fractional Laplacian in order to make the article accessible to a larger audience.”
On the differentiability of the solution to an equation with drift and fractional diffusion (2010, background): “In the case of s = 1/2, this provides an improvement of the regularity obtained in [18].”
Boundary regularity for fully nonlinear integro-differential equations (2014, unknown): “This is why the same ideas have been used in other different contexts [16, 47].”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, unknown): “There are also many important results in this direction including [17], [60], [16], [15], [57], [22], [58], [13], [12], [19], [23], [25], [47], [26], [42] and [56].”
Regularity for solutions of non local parabolic equations (2011, background): “The ideas of the proof of this result can be found in Section 4 and 5 in [3] for the stationary problem and in the appendix of [13] for the time depending problem.”
Regularity for parabolic integro-differential equations with very irregular kernels (2014, background): “The focus of [36] is on the interaction between the integro-differential part and the drift term.”