A nonlinear oblique derivative boundary value problem for the heat equation : analogy with the porous medium equation

 Access to full text

 Full (PDF)

1. [1] Aronson D.G., Bénilan P., Régularité des solutions de l'équation des milieux poreux dans RN, C. R. Acad. Sci. Paris288 (1979) 103-105. Zbl0397.35034MR524760
2. [2] Aronson D.G., Caffarelli L.A., Kamin S., How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal.14 (1983) 639-658. Zbl0542.76119MR704481
3. [3] Atkinson F.V., Peletier L.A., Similarity profiles of flows through porous media, Arch. Rational Mech. Anal.42 (1971) 369-379. Zbl0249.35043MR334666
4. [4] Atkinson F.V., Peletier L.A., Similarity solutions of the nonlinear diffusion equation, Arch. Rational Mech. Anal.54 (1974) 373-392. Zbl0293.35039MR344559
5. [5] Audounet J., Giovangigli V., Roquejoffre J.-M., A threshold phenomenon arising in the propagation of a spherical flame, Physica D121 (1998) 295-316. Zbl0938.80003MR1645427
6. [6] Caffarelli L.A., A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are C1,α, Rev. Mat. Iberoamericana3 (1987) 39-62. Zbl0676.35085
7. [7] Caffarelli L.A., Vazquez J.-L., Viscosity solutions for the porous medium equation, in: Differential Equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., 65, American Mathematical Society, Providence, RI, 1999. Zbl0929.35072MR1662747
8. [8] Caffarelli L.A., Vazquez J.-L., Wolanski N.I., Lipschitz continuity of solutions and interfaces in the N-dimensional porous medium equation, Indiana Univ. Math. J.36 (1987) 373-401. Zbl0644.35058MR891781
9. [9] Caffarelli L.A., Wolanski N.I., C1,α regularity of the free boundary for the N-dimensional porous media equation, Comm. Pure Appl. Math.43 (1990) 885-902. Zbl0728.76103
10. [10] Clément P., Gripenberg G., Londen S.-O., Hölder regularity for a linear fractional evolution equation, in: Topics in Nonlinear Analysis, H. Amann Anniversary Volume, Birkhäuser, 1999, pp. 69-82. Zbl0920.35004MR1725566
11. [11] Gorenflo R., Vessela S., Abel Integral Equations, Springer, New York, 1991. Zbl0717.45002MR1095269
12. [12] Gordeev A.V., Grechikha A.V., Kalda Y.L., Rapid penetration of a magnetic field into a plasma along an electrode, Sov. J. Plasma Phys.16 (1) (1990) 55-57. 
13. [13] Gripenberg G., Londen S.-O., Fractional derivatives and smoothing in nonlinear conservation laws, Diff. Int. Eq.8 (1995) 1961-1976. Zbl0885.45005MR1348960
14. [14] Henry D., Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981. Zbl0456.35001MR610244
15. [15] Ladyzhenskaya O.A., Ural'tceva N.N., Solonnikov V.A., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. Zbl0174.15403
16. [16] Méhats F., Thèse de doctorat de l'École polytechnique, 1997. 
17. [17] Méhats F., Roquejoffre J.-M., A nonlinear oblique derivative problem for the heat equation, Part I: Basic results, Ann. Inst. Henri Poincaré, Analyse non linéaire16 (1999) 221-253. Zbl0922.35072MR1674770
18. [18] Méhats F., Roquejoffre J.-M., A nonlinear oblique derivative problem for the Heat equation, Part II: Singular self-similar solutions, Ann. Inst. Henri Poincaré, Analyse non linéaire16 (1999) 691-724. Zbl0945.35047MR1720513
19. [19] Walter W., Differential Inequlities, Springer, Berlin, 1964. MR172076

1. Florian Mehats, Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem
2. Florian Mehats, Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem