𝑊^{2,∞} regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension

Jakobsen, Espen

doi:10.1090/S0002-9939-04-07577-X

$W^{2,\infty }$ regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension
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Proc. Amer. Math. Soc. 132 (2004), 3203-3213 Request permission

Abstract:

In this paper we provide and analyze a nonlinear degenerate parabolic equation in one space dimension with the following smoothing property: If the initial data is only uniformly continuous, at positive times, the solution has bounded second derivatives (it belongs to $W^{2,\infty }$). We call this surprising phenomenon a $W^{2,\infty }$ regularizing effect. So far, such phenomena have only been observed in uniformly parabolic equations.

References

O. Alvarez, J.-M. Lasry, and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (9) 76 (1997), no. 3, 265–288 (English, with English and French summaries). MR 1441987, DOI 10.1016/S0021-7824(97)89952-7
Mariko Arisawa and Agnes Tourin, Regularizing effects for a class of first-order Hamilton-Jacobi equations, Nonlinear Anal. 29 (1997), no. 12, 1405–1419. MR 1484913, DOI 10.1016/S0362-546X(96)00184-8
Guy Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 17, Springer-Verlag, Paris, 1994 (French, with French summary). MR 1613876
G. Barles, Uniqueness and regularity results for first-order Hamilton-Jacobi equations, Indiana Univ. Math. J. 39 (1990), no. 2, 443–466. MR 1089047, DOI 10.1512/iumj.1990.39.39024
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
M. G. Crandall, M. Kocan, and A. Święch, $L^p$-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations 25 (2000), no. 11-12, 1997–2053. MR 1789919, DOI 10.1080/03605300008821576
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
Wendell H. Fleming and H. Mete Soner, Controlled Markov processes and viscosity solutions, Applications of Mathematics (New York), vol. 25, Springer-Verlag, New York, 1993. MR 1199811
W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J. 38 (1989), no. 2, 293–314. MR 997385, DOI 10.1512/iumj.1989.38.38015
Hitoshi Ishii, Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987), no. 2, 369–384. MR 894587, DOI 10.1215/S0012-7094-87-05521-9
E. R. Jakobsen. On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems. Math. Models Methods Appl. Sci. (M3AS) 13(5): 613-644, 2003.
Espen R. Jakobsen and Kenneth H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, J. Differential Equations 183 (2002), no. 2, 497–525. MR 1919788, DOI 10.1006/jdeq.2001.4136
Markos A. Katsoulakis, A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations, Nonlinear Anal. 24 (1995), no. 2, 147–158. MR 1312585, DOI 10.1016/0362-546X(94)00170-M
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
P.-L. Lions, Regularizing effects for first-order Hamilton-Jacobi equations, Applicable Anal. 20 (1985), no. 3-4, 283–307. MR 814955, DOI 10.1080/00036818508839575
Lihe Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math. 45 (1992), no. 1, 27–76. MR 1135923, DOI 10.1002/cpa.3160450103
Lihe Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math. 45 (1992), no. 2, 141–178. MR 1139064, DOI 10.1002/cpa.3160450202
Ning Zhu, $C^{1+\alpha }$ regularity of viscosity solutions for fully nonlinear parabolic equations, Nonlinear Anal. 38 (1999), no. 8, Ser. A: Theory Methods, 977–994. MR 1716428, DOI 10.1016/S0362-546X(98)00149-7