Regularity of the free boundary for the porous medium equation

Daskalopoulos, P.¹ ; Hamilton, R.²

¹ Department of Mathematics, University of California, Irvine, California 92697-3875
² Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0001

Journal of the American Mathematical Society, Tome 11 (1998) no. 4, pp. 899-965

Résumé

We study the regularity of the free boundary for solutions of the porous medium equation $u_{t}=\Delta u^{m}$, $m >1$, on ${\mathcal {R}}^{2} \times [0,T]$, with initial data $u^{0}=u(x,0)$ nonnegative and compactly supported. We show that, under certain assumptions on the initial data $u^{0}$, the pressure $f=m u^{m-1}$ will be smooth up to the interface $\Gamma = \partial \{ u >0 \}$, when $0$, for some $T >0$. As a consequence, the free-boundary $\Gamma$ is smooth.

Détail

DOI : 10.1090/S0894-0347-98-00277-X

Bibliographie

[1] Angenent, Sigurd Analyticity of the interface of the porous media equation after the waiting time Proc. Amer. Math. Soc. 1988 329 336

[2] Aronson, D. G. Regularity propeties of flows through porous media SIAM J. Appl. Math. 1969 461 467

[3] Aronson, D. G. Regularity properties of flows through porous media: A counterexample SIAM J. Appl. Math. 1970 299 307

[4] Aronson, D. G. Regularity properties of flows through porous media: The interface Arch. Rational Mech. Anal. 1970 1 10

[5] Aronson, D. G., Caffarelli, L. A., Vã¡Zquez, Juan Luis Interfaces with a corner point in one-dimensional porous medium flow Comm. Pure Appl. Math. 1985 375 404

[6] Aronson, D. G., Vã¡Zquez, J. L. Eventual 𝐶^{∞}-regularity and concavity for flows in one-dimensional porous media Arch. Rational Mech. Anal. 1987 329 348

[7] Caffarelli, Luis A. Interior a priori estimates for solutions of fully nonlinear equations Ann. of Math. (2) 1989 189 213

[8] Caffarelli, Luis A., Friedman, Avner Regularity of the free boundary for the one-dimensional flow of gas in a porous medium Amer. J. Math. 1979 1193 1218

[9] Caffarelli, Luis A., Friedman, Avner Regularity of the free boundary of a gas flow in an 𝑛-dimensional porous medium Indiana Univ. Math. J. 1980 361 391

[10] Caffarelli, L. A., Vã¡Zquez, J. L., Wolanski, N. I. Lipschitz continuity of solutions and interfaces of the 𝑁-dimensional porous medium equation Indiana Univ. Math. J. 1987 373 401

[11] Caffarelli, Luis A., Wolanski, Noem㭠I. 𝐶^{1,𝛼} regularity of the free boundary for the 𝑁-dimensional porous media equation Comm. Pure Appl. Math. 1990 885 902

[12] Knerr, Barry F. The porous medium equation in one dimension Trans. Amer. Math. Soc. 1977 381 415

[13] Kohn, J. J., Nirenberg, L. Degenerate elliptic-parabolic equations of second order Comm. Pure Appl. Math. 1967 797 872

[14] Safonov, M. V. The classical solution of the elliptic Bellman equation Dokl. Akad. Nauk SSSR 1984 810 813

[15] Safonov, M. V. Classical solution of second-order nonlinear elliptic equations Izv. Akad. Nauk SSSR Ser. Mat. 1988

[16] Wang, Lihe On the regularity theory of fully nonlinear parabolic equations. I Comm. Pure Appl. Math. 1992 27 76

[17] Wang, Lihe On the regularity theory of fully nonlinear parabolic equations. II Comm. Pure Appl. Math. 1992 141 178