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The pressure equation in the fast diffusion range. (English) Zbl 1073.35128

The authors study the equation \[ v_t=v\Delta v - \gamma| \nabla v| ^2\qquad \text{in } {\mathbb R}^N\times (0,\infty), \] which is an example of a fully nonlinear parabolic equation in non-divergence form. The equation is not well-understood when \(\gamma>0\). In this paper, the authors concentrate on the range \(\gamma>N/2\). They show that the Cauchy problem is well-posed with arbitrarily large initial data, understood in the sense of Borel \(p\)-trace where \(p=-\gamma<0\), and solutions are classical. The authors also point out why classical approaches do not work.

MSC:

35K65 Degenerate parabolic equations
35K15 Initial value problems for second-order parabolic equations
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[1] Adams, D. R. and Hedberg, L. I.: Function spaces and potential theory. Grundlehren Math. Wissen. 314, Springer Verlag, Berlin, 1996. · Zbl 0834.46021
[2] Angenent, S. B.: Local existence and regularity for a class of degenerate parabolic equations. Math. Ann. 280 (1988), 465-482. · Zbl 0619.35114 · doi:10.1007/BF01456337
[3] Aronson, D. G.: The Porous Medium Equation, Some problems of Non- linear Diffusion. Lectures Notes in Mathematics 1224, Springer-Verlag, New York, 1986. · Zbl 0626.76097
[4] Aronson, D. G. and Bénilan, P.: Régularité des solutions de l’équation des milieux poreux dans RN . C. R. Acad. Sci. Paris Sér. I Math. 288 (1979), 103-105. · Zbl 0397.35034
[5] Barenblatt, G. I.: On self-similar motion of compressible fluids in porous media. Prikl. Mat. Mekh. 16 (1952), 679-698 (in Russian). · Zbl 0047.19204
[6] Barenblatt, G. I.: Self-similar intermediate asymptotics for nonlinear parabolic free-boundary problems which occur in image processing. Proc. Natl. Acad. Sci. USA 98 (2001), no. 23, 12878-12881 (electronic). · Zbl 0999.35113 · doi:10.1073/pnas.241501698
[7] Barenblatt, G. I., Bertsch, M., Chertock, A. E. and Prostok- ishin, V. M.: Self-similar asymptotics for a degenerate parabolic filtration- absorption equation. Proc. Natl. Acad. Sci. USA 97 (2000), no. 18, 9844- 9848 (electronic). · Zbl 0966.35069 · doi:10.1073/pnas.97.18.9844
[8] Barenblatt, G. I. and Vázquez, J. L.: Nonlinear Diffusion and Image Enhancement, submitted. · Zbl 1064.35212 · doi:10.4171/IFB/90
[9] Bénilan, P. and Crandall, M. G.: Regularizing effects of homogeneous evolution equations. In Contribution to Analysis and Geometry (D. N. Clark et al., eds.), 23-30. John Hopkins Univ. Press, Baltimore, Md., 1981. · Zbl 0556.35067
[10] Berger, M., Gauduchon, P. and Mazet, E.: Le spectre d’une Variété Riemmanienne. Lecture Notes in Mathematics 194, Springer Verlag, Berlin, 1970. · Zbl 0223.53034
[11] Berryman, J. G. and Holland, C. J.: Stability of the separable solution for fast diffusion equation. Arch. Rat. Mech. Anal. 74 (1980), 379-388. · Zbl 0458.35046 · doi:10.1007/BF00249681
[12] Bertsch, M. and Ughi, M. Positivity properties of viscosity solutions of a degenerate parabolic equation. Nonlinear Anal. 14 (1990), 571-592. · Zbl 0702.35044 · doi:10.1016/0362-546X(90)90063-M
[13] Bertsch, M., Dal Passo, R. and Ughi, M.: Discontinuous “viscosity” solutions of a degenerate parabolic equation. Trans. Amer. Math. Soc. 320 (1990), 779-798. · Zbl 0714.35039 · doi:10.2307/2001703
[14] Bertsch, M., Dal Passo, R. and Ughi, M.: Nonuniqueness of solutions of a degenerate parabolic equation, Annali Mat. Pura Appl. 161 (1992), 57-81. 915 · Zbl 0796.35083 · doi:10.1007/BF01759632
[15] Blanchard, P., Murat, F. and Redwane, H.: Existence et unicité de la solution renormalisée d’un probl‘ eme parabolique non linéaire assez général. C. R. Acad. Sci. Paris Sér. I Math. 329, 7 (1999), 575-580. Also, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems. J. Differential Equations 177 (2001), no. 2, 331-374. · Zbl 0935.35073 · doi:10.1016/S0764-4442(00)80004-X
[16] Brezis, H. and Friedman, A.: Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. (9) 62 (1983), 73-97. · Zbl 0527.35043
[17] Caffarelli, L. A. and Cabré, X.: Fully nonlinear elliptic equations. Coll. Publ. 43, Amer. Math. Soc., Providence, 1995. · Zbl 0834.35002
[18] Caffarelli, L. A. and Vázquez, J. L.: Viscosity solutions for the porous medium equation. In Differential equations: La Pietra 1996 (Florence), 13- 26. Proc. Sympos. Pure Math. 65, Amer. Math. Soc., Providence, RI, 1999. · Zbl 0929.35072
[19] Chasseigne, E.: Thesis, Univ. Tours, France. December 2001.
[20] Chasseigne, E.: Classification of Razor Blades to the filtration equation. The sub-linear case. J. Differential Equations 187 (2003), 72-105. · Zbl 1043.35087 · doi:10.1016/S0022-0396(02)00019-0
[21] Chasseigne, E. and Vázquez, J. L.: Extended theory of fast diffusion equations in optimal classes of data. Radiation from singularities. Arch. Rat. Mech. Anal. 164 (2002), 133-187. · Zbl 1018.35048 · doi:10.1007/s00205-002-0210-0
[22] Chasseigne, E. and Vázquez, J. L.: Weak Solutions of Fast Diffusion Equations in bounded domains, submitted.
[23] Crandall, M. G., Evans, L. C. and Lions, P. L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984), 487-502. · Zbl 0543.35011 · doi:10.2307/1999247
[24] Crandall, M. G., Ishii, H. and Lions, P. L.: User’s guide to viscosity solutions for second-order partial differential equations. Bull. Amer. Math. Soc. 27 (1992), 1-67. · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[25] Crandall, M. G. and Lions, P. L.: Condition d’unicité pour les so- lutions généralisées des équations de Hamilton-Jacobi du premier ordre. (French. English summary) C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 3, 183-186. · Zbl 0469.49023
[26] Crandall, M. G. and Lions, P. L.: Viscosity solutions of Hamilton- Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1-42. · Zbl 0599.35024 · doi:10.2307/1999343
[27] Dahlberg, B. E. J., Fabes, E. and Kenig, C.: A Fatou theorem for so- lutions of the Porous Medium Equation. Proc. Amer. Math. Soc. 91 (1984), 205-212. · Zbl 0552.35047 · doi:10.2307/2044628
[28] Dal Masso, G., Murat, F., Orsina, L. and Prignet, A.: Definition and existence of renormalized solution of elliptic equations with general measure data. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 5, 481-486. · Zbl 0887.35057 · doi:10.1016/S0764-4442(97)88893-3
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