Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity

Esteban, Juan R.; Marcati, Pierangelo

doi:10.1090/S0002-9947-1994-1214784-8

Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity
HTML articles powered by AMS MathViewer

by Juan R. Esteban and Pierangelo Marcati PDF

Trans. Amer. Math. Soc. 342 (1994), 501-521 Request permission

Abstract:

We shall consider a model problem for the fully nonlinear parabolic equation \[ {u_t} + F(x,t,u,Du,\varepsilon {D^2}u) = 0\] and we study both the approximating degenerate second order problem and the related first order equation, obtained by the limit as $\varepsilon \to 0$. The strong convergence of the gradients is provided by semiconcavity unilateral bounds and by the supremum bounds of the gradients. In this way we find solutions in the class of viscosity solutions of Crandall and Lions.

References

Luis Alvarez, Pierre-Louis Lions, and Jean-Michel Morel, Image selective smoothing and edge detection by nonlinear diffusion. II, SIAM J. Numer. Anal. 29 (1992), no. 3, 845–866. MR 1163360, DOI 10.1137/0729052
D. G. Aronson, The porous medium equation, Nonlinear diffusion problems (Montecatini Terme, 1985) Lecture Notes in Math., vol. 1224, Springer, Berlin, 1986, pp. 1–46. MR 877986, DOI 10.1007/BFb0072687
M. Bardi and L. C. Evans, On Hopf’s formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal. 8 (1984), no. 11, 1373–1381. MR 764917, DOI 10.1016/0362-546X(84)90020-8
G. Barles, A weak Bernstein method for fully nonlinear elliptic equations, Differential Integral Equations 4 (1991), no. 2, 241–262. MR 1081182
G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 557–579 (English, with French summary). MR 921827, DOI 10.1051/m2an/1987210405571
Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. MR 1100211
Michael G. Crandall, Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. 6, 419–435 (English, with French summary). MR 1035337
M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), no. 2, 487–502. MR 732102, DOI 10.1090/S0002-9947-1984-0732102-X
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
Michael G. Crandall, Pierre-Louis Lions, and Panagiotis E. Souganidis, Maximal solutions and universal bounds for some partial differential equations of evolution, Arch. Rational Mech. Anal. 105 (1989), no. 2, 163–190. MR 968459, DOI 10.1007/BF00250835
Juan R. Esteban and Juan L. Vázquez, Homogeneous diffusion in $\textbf {R}$ with power-like nonlinear diffusivity, Arch. Rational Mech. Anal. 103 (1988), no. 1, 39–80. MR 946969, DOI 10.1007/BF00292920
Juan Ramón Esteban and Juan Luis Vázquez, Régularité des solutions positives de l’équation parabolique $p$-laplacienne, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 3, 105–110 (French, with English summary). MR 1044625

Regularity of solutions of nonlinear heat equations with power-like nonlinearities in several space dimensions

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. MR 1100206
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. II, Trans. Amer. Math. Soc. 330 (1992), no. 1, 321–332. MR 1068927, DOI 10.1090/S0002-9947-1992-1068927-8
Y. Giga, S. Goto, H. Ishii, and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40 (1991), no. 2, 443–470. MR 1119185, DOI 10.1512/iumj.1991.40.40023
A. M. Il′in, A. S. Kalašnikov, and O. A. Oleĭnik, Second-order linear equations of parabolic type, Uspehi Mat. Nauk 17 (1962), no. 3 (105), 3–146 (Russian). MR 0138888
H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations 83 (1990), no. 1, 26–78. MR 1031377, DOI 10.1016/0022-0396(90)90068-Z

Linear and quasilinear equations of parabolic type

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, P. E. Souganidis, and J. L. Vázquez, The relation between the porous medium and the eikonal equations in several space dimensions, Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 275–310. MR 996819, DOI 10.4171/RMI/51
Pierangelo Marcati, Approximate solutions to conservation laws via convective parabolic equations, Comm. Partial Differential Equations 13 (1988), no. 3, 321–344. MR 917603, DOI 10.1080/03605308808820544
François Murat, L’injection du cône positif de $H^{-1}$ dans $W^{-1,\,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9) 60 (1981), no. 3, 309–322 (French, with English summary). MR 633007
Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
James Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 565–601. MR 0402274