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Transactions of the American Mathematical Society

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Schauder estimates for equationswith fractional derivatives


Authors: Ph. Clément, G. Gripenberg and S-O. Londen
Journal: Trans. Amer. Math. Soc. 352 (2000), 2239-2260
MSC (2000): Primary 35K99, 45K05
DOI: https://doi.org/10.1090/S0002-9947-00-02507-1
Published electronically: February 14, 2000
MathSciNet review: 1675170
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Abstract:

The equation \begin{equation*}D^\alpha_t (u-h_1) + D^\beta_x(u-h_2) =f,\quad 0< \alpha,\beta< 1, \quad t,x \geq 0,\tag{$*$ } \end{equation*}where $D^\alpha_t$ and $D^\beta_x$ are fractional derivatives of order $\alpha$ and $\beta$ is studied. It is shown that if $f=f(\underline{t}, \underline{x})$, $h_1=h_1(\underline{x})$, and $h_2=h_2(\underline{t})$ are Hölder-continuous and $f(0,0) =0$, then there is a solution such that $ D^\alpha_t u$ and $D^\beta_x u$ are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to ($*$). Finally the solution of ($*$) with $f=1$ is studied.


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  • 1. H. Amann, Linear and Quasilinear Parabolic Problems, I, Birkhäuser, Basel, 1995. MR 96g:34088
  • 2. L. Berg, Asymptotische Darstellungen und Entwicklungen, Deutscher Verlag der Wissenschaften, Berlin, 1968. MR 39:3210
  • 3. Ph. Clément and G. Da Prato, Some results on nonlinear heat equations for materials of fading memory type, J. Integral Equations App. 2 (1990), 375-391. MR 92a:45031
  • 4. B. Cockburn, G. Gripenberg, and S-O. Londen, On convergence to entropy solutions of a single conservation law, J. Differential Equations 128 (1996), 206-251. MR 98c:35107
  • 5. G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. 54 (1975), 305-387. MR 56:1129
  • 6. G. Da Prato and M. Iannelli, Existence and regularity for a class of integrodifferential equations of parabolic type, J. Math. Anal. Appl. 112 (1985), 36-55. MR 87d:45020
  • 7. G. Da Prato and E. Sinestrari, Differential operators with non dense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 285-344. MR 89f:47062
  • 8. E. Feireisl and H. Petzeltová, Singular kernels and compactness in nonlinear conservation laws, J. Differential Equations 142 (1998), 291-304. MR 99b:35132
  • 9. G. Gripenberg and S-O. Londen, Fractional derivatives and smoothing in nonlinear conservation laws, Differential Integral Equations 8 (1995), 1961-1976. MR 96f:35107
  • 10. P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pures Appl. 45 (1966), 143-206. MR 36:4362
  • 11. P. Grisvard, Équations différentielles abstraites, Ann. Sci. École Norm. Sup.(4) 2 (1969), 311-395. MR 42:5101
  • 12. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. MR 96e:47039
  • 13. H. Pollard, The representation of ${\textup e}^{-x^\lambda}$ as a Laplace integral, Bull. Amer. Math. Soc. 52 (1946), 908-910. MR 8:269a
  • 14. J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. MR 94h:45010
  • 15. E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl. 107 (1985), 16-66. MR 86g:34086
  • 16. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. MR 80i:46032a
  • 17. A. Zygmund, Trigonometric Series II, Cambridge University Press, Cambridge, 1959. MR 21:6498

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Additional Information

Ph. Clément
Affiliation: Faculty of Technical Mathematics, and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Email: clement@twi.tudelft.nl

G. Gripenberg
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
Email: gustaf.gripenberg@hut.fi

S-O. Londen
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
Email: stig-olof.londen@hut.fi

DOI: https://doi.org/10.1090/S0002-9947-00-02507-1
Keywords: Fractional derivative, maximal regularity, Schauder estimate, H\"older continuity, fundamental solution, integro-differential equation
Received by editor(s): March 20, 1997
Received by editor(s) in revised form: September 29, 1997
Published electronically: February 14, 2000
Additional Notes: The third author acknowledges the partial support of the Nederlandse organisatie voor wetenschappelijk onderzoek (NWO)
Article copyright: © Copyright 2000 American Mathematical Society

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