On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion. (English) Zbl 1216.35165
The main objective of this interesting paper is to prove that the fractional Hamiltonian-Jacobi equation for an arbitrary Hamiltonian and involving certain fractional Laplacian has classical enough smooth solutions. The fractional Laplacian used by the author is defined through the corresponding Fourier transform as a hyper-singular, which is the classical inverse a Riesz potential operator.
Reviewer: Juan J. Trujillo (La Laguna)
MSC:
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A22 | Transform methods (e.g., integral transforms) applied to PDEs |
References:
| [1] | Barles, G.; Imbert, C., Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. Inst. H. Poincare Anal. Non Lineaire, 25, 3, 567-585 (2008) · Zbl 1155.45004 |
| [2] | Cabre, X.; Caffarelli, L. A., Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ., vol. 43 (1995) · Zbl 0834.35002 |
| [3] | Caffarelli, L.; Silvestre, L., Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62, 5, 597-638 (2009) · Zbl 1170.45006 |
| [4] | Caffarelli, L.; Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171, 3, 1903-1930 (2010) · Zbl 1204.35063 |
| [5] | Chan, C. H.; Czubak, M.; Silvestre, L., Eventual regularization of the slightly supercritical fractional Burgers equation, Discrete Contin. Dyn. Syst. Ser. A, 27, 2, 847-861 (2010) · Zbl 1194.35320 |
| [6] | Crandall, M.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, 1-67 (1992) · Zbl 0755.35015 |
| [7] | Droniou, J.; Imbert, C., Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182, 2, 299-331 (2006) · Zbl 1111.35144 |
| [8] | Droniou, J.; Gallouët, T.; Vovelle, J., Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3, 3, 499-521 (2003) · Zbl 1036.35123 |
| [9] | Imbert, C., A non-local regularization of first order Hamilton-Jacobi equations, J. Differential Equations, 211, 1, 218-246 (2005) · Zbl 1073.35059 |
| [10] | Jensen, R., The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Ration. Mech. Anal., 101, 1, 1-27 (1988) · Zbl 0708.35019 |
| [11] | Karch, G.; Woyczynski, W. A., Fractal Hamilton-Jacobi-KPZ equations, Trans. Amer. Math. Soc., 360, 5, 2423 (2008) · Zbl 1136.35012 |
| [12] | Kiselev, A.; Nazarov, F., Variation on a theme of Caffarelli and Vasseur, J. Math. Sci., 166, 1, 31-39 (2010) · Zbl 1288.35393 |
| [13] | Kiselev, Alexander; Nazarov, Fedor; Shterenberg, Roman, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ., 5, 3, 211-240 (2008) · Zbl 1186.35020 |
| [14] | Landkof, N. S., Osnovy Sovremennoi Teorii Potentsiala (1966), Nauka: Nauka Moscow |
| [15] | Lions, P.-L., Regularizing effects for first-order Hamilton-Jacobi equations, Appl. Anal., 20, 3-4, 283-307 (1985) · Zbl 0551.35014 |
| [16] | Silvestre, L., Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J., 55, 3, 1155-1174 (2006) · Zbl 1101.45004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
