Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Regularizing effects for $ u\sb{t}=\Delta \varphi (u)$


Authors: Michael G. Crandall and Michel Pierre
Journal: Trans. Amer. Math. Soc. 274 (1982), 159-168
MSC: Primary 35K55
DOI: https://doi.org/10.1090/S0002-9947-1982-0670925-4
MathSciNet review: 670925
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: One expression of the fact that a nonnegative solution of the initial-value problem

$\displaystyle ({\text{IVP}})\quad \left\{ {\begin{array}{*{20}{c}} {{u_t} - \De... ...\right.\quad \begin{array}{*{20}{c}} {t > 0,x \in {R^N},} \\ {} \\ \end{array} $

where $ m > 0$, is more regular for $ t > 0$ than a rough initial datum $ {u_0}$ is the remarkable pointwise inequality $ {u_t} = \Delta {u^m} \geqslant - (N/(N(m - 1) + 2)t)u$ obtained by Aronson and Bénilan for $ t > 0$ and $ m > \max ((N - 2)/N,0)$. This inequality was used by Friedman and Caffarelli in proving that solutions of (IVP) are continuous for $ t > 0$. The main results of this paper generalize the Aronson-Bénilan inequality and show the extended inequality is valid for a much broader class of equations of the form $ {u_t} = \Delta \varphi (u)$. In particular, the results apply to the Stefan problem which is modeled by $ \varphi (r) = {(r - 1)^ + }$ and imply $ {({(u - 1)^ + })_t} \geqslant - ({(u - 1)^ + } + N/2)/t$ in this case.

References [Enhancements On Off] (What's this?)

  • [1] Donald G. Aronson and Philippe Bénilan, Régularité des solutions de l’équation des milieux poreux dans 𝑅^{𝑁}, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 2, A103–A105 (French, with English summary). MR 524760
  • [2] Ph. Bénilan, Opérateurs accrétifs et semi-groupes dans les espaces $ {L^p}(1 \leqslant p \leqslant \infty )$, Functional Analysis and Numerical Analysis Japan-France Seminar, Tokyo and Kyota, H. Fujita (Ed.), Japan Society for the Promotion of Science, 1978, pp. 15-53.
  • [3] Philippe Bénilan and Michael G. Crandall, Regularizing effects of homogeneous evolution equations, Contributions to analysis and geometry (Baltimore, Md., 1980) Johns Hopkins Univ. Press, Baltimore, Md., 1981, pp. 23–39. MR 648452
  • [4] -, The continuous dependence on $ \varphi $ of solutions of $ {u_t} - \Delta \varphi (u) = 0$, Indiana Univ. Math. J. 30 (1981), 162-177.
  • [5] Haïm Brézis and Michael G. Crandall, Uniqueness of solutions of the initial-value problem for 𝑢_{𝑡}-Δ𝜑(𝑢)=0, J. Math. Pures Appl. (9) 58 (1979), no. 2, 153–163. MR 539218
  • [6] L. A. Caffarelli and L. C. Evans, Continuity of the temperature in the two-phase Stefan problem, Arch. Rational Mech. Anal. 81 (1983), no. 3, 199–220. MR 683353, https://doi.org/10.1007/BF00250800
  • [7] John R. Cannon, Daniel B. Henry, and Daniel B. Kotlow, Classical solutions of the one-dimensional, two-phase Stefan problem, Ann. Mat. Pura Appl. (4) 107 (1975), 311–341 (1976). MR 0407456, https://doi.org/10.1007/BF02416479
  • [8] Michael Crandall and Michel Pierre, Regularizing effects for 𝑢_{𝑡}+𝐴𝜑(𝑢)=0 in 𝐿¹, J. Funct. Anal. 45 (1982), no. 2, 194–212. MR 647071, https://doi.org/10.1016/0022-1236(82)90018-0
  • [9] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Transl. Math. Monos., vol. 23, Amer. Math. Soc., Providence, R.I., 1968.
  • [10] E. S. Sabinina, On the Cauchy problem for the equation of nonstationary gas filtration in several space variables, Soviet Math. Dokl. 2 (1961), 166–169. MR 0158190
  • [11] Paul Sacks, Thesis, Univ. of Wisconsin-Madison, 1981.
  • [12] L. Véron, Coercivité et propriétés régularisantes des semi-groupes nonlinéaires dans les espaces de Banach, Publ. Math. Fac. Sci. Besançon 3 (1977).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K55

Retrieve articles in all journals with MSC: 35K55


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0670925-4
Keywords: Regularizing effect, porous media equations, strong solution, degenerate parabolic equations, Stefan problem
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society