Revista Matemática Iberoamericana

The Pressure Equation in the Fast Diffusion Range

Emmanuel Chasseigne and Juan Luis Vázquez

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Abstract

We consider the following degenerate parabolic equation $$ v_{t}=v\Delta v-\gamma|\nabla v|^{2}\quad\mbox{in $\mathbb{R}^{N} \times(0,\infty)$,} $$ whose behaviour depends strongly on the parameter $\gamma$. While the range $\gamma < 0$ is well understood, qualitative and analytical novelties appear for $\gamma>0$. Thus, the standard concepts of weak or viscosity solution do not produce uniqueness. Here we show that for $\gamma>\max\{N/2,1\}$ the initial value problem is well posed in a precisely defined setting: the solutions are chosen in a class $\mathcal{W}_s$ of local weak solutions with constant support; initial data can be any nonnegative measurable function $v_{0}$ (infinite values also accepted); uniqueness is only obtained using a special concept of initial trace, the $p$-trace with $p=-\gamma < 0$, since the standard concepts of initial trace do not produce uniqueness. Here are some additional properties: the solutions turn out to be classical for $t>0$, the support is constant in time, and not all of them can be obtained by the vanishing viscosity method. We also show that singular measures are not admissible as initial data, and study the asymptotic behaviour as $t\to \infty$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 19, Number 3 (2003), 873-917.

Dates
First available in Project Euclid: 20 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1077293809

Mathematical Reviews number (MathSciNet)
MR2053567

Zentralblatt MATH identifier
1073.35128

Subjects
Primary: 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations

Keywords
pressure equation fast diffusion well-posed problem non-uniqueness measure as initial trace optimal initial data

Citation

Chasseigne, Emmanuel; Vázquez, Juan Luis. The Pressure Equation in the Fast Diffusion Range. Rev. Mat. Iberoamericana 19 (2003), no. 3, 873--917. https://projecteuclid.org/euclid.rmi/1077293809


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References

  • Adams, D. R. and Hedberg, L. I.: Function spaces and potential theory. Grundlehren Math. Wissen. 314, Springer Verlag, Berlin, 1996.
  • Angenent, S. B.: Local existence and regularity for a class of degenerate parabolic equations. Math. Ann. 280 (1988), 465-482.
  • Aronson, D. G.: The Porous Medium Equation, Some problems of Nonlinear Diffusion. Lectures Notes in Mathematics 1224, Springer-Verlag, New York, 1986.
  • Aronson, D. G. and Bénilan, P.: Régularité des solutions de l'équation des milieux poreux dans $\mathbbR^N$. C. R. Acad. Sci. Paris Sér. I Math. 288 (1979), 103-105.
  • Barenblatt, G. I.: On self-similar motion of compressible fluids in porous media. Prikl. Mat. Mekh. 16 (1952), 679-698 (in Russian).
  • 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).
  • Barenblatt, G. I., Bertsch, M., Chertock, A. E. and Prostokishin, V. M.: Self-similar asymptotics for a degenerate parabolic filtration-absorption equation. Proc. Natl. Acad. Sci. USA 97 (2000), no. 18, 9844-9848 (electronic).
  • Barenblatt, G. I. and Vázquez, J. L.: Nonlinear Diffusion and Image Enhancement, submitted.
  • 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.
  • Berger, M., Gauduchon, P. and Mazet, E.: Le spectre d'une Variété Riemmanienne. Lecture Notes in Mathematics 194, Springer Verlag, Berlin, 1970.
  • Berryman, J. G. and Holland, C. J.: Stability of the separable solution for fast diffusion equation. Arch. Rat. Mech. Anal. 74 (1980), 379-388.
  • Bertsch, M. and Ughi, M. Positivity properties of viscosity solutions of a degenerate parabolic equation. Nonlinear Anal. 14 (1990), 571-592.
  • Bertsch, M., Dal Passo, R. and Ughi, M.: Discontinuous ``viscosity" solutions of a degenerate parabolic equation. Trans. Amer. Math. Soc. 320 (1990), 779-798.
  • Bertsch, M., Dal Passo, R. and Ughi, M.: Nonuniqueness of solutions of a degenerate parabolic equation, Annali Mat. Pura Appl. 161 (1992), 57-81.
  • Blanchard, P., Murat, F. and Redwane, H.: Existence et unicité de la solution renormalisée d'un problème 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.
  • Brezis, H. and Friedman, A.: Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. (9) 62 (1983), 73-97.
  • Caffarelli, L. A. and Cabré, X.: Fully nonlinear elliptic equations. Coll. Publ. 43, Amer. Math. Soc., Providence, 1995.
  • 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.
  • Chasseigne, E.: Thesis, Univ. Tours, France. December 2001.
  • Chasseigne, E.: Classification of Razor Blades to the filtration equation. The sub-linear case. J. Differential Equations 187 (2003), 72-105.
  • 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.
  • Chasseigne, E. and Vázquez, J. L.: Weak Solutions of Fast Diffusion Equations in bounded domains, submitted.
  • 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.
  • 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.
  • Crandall, M. G. and Lions, P. L.: Condition d'unicité pour les solutions 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.
  • Crandall, M. G. and Lions, P. L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1-42.
  • Dahlberg, B. E. J., Fabes, E. and Kenig, C.: A Fatou theorem for solutions of the Porous Medium Equation. Proc. Amer. Math. Soc. 91 (1984), 205-212.
  • 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.
  • Dal Passo, R. and Luckhaus, S.: A degenerate diffusion problem not in divergence form. J. Differential Equations 69 (1987), no. 1, 1-14.
  • Esteban, J. R., Rodríguez, A. and Vázquez, J. L.: A nonlinear heat equation with singular diffusivity. Comm. Partial Differential Equations 13 (1988), 985-1039.
  • Esteban, J. R., Rodríguez, A. and Vázquez, J. L.: The maximal solution of the logarithmic fast diffusion equation in two space dimension. Adv. Differential Equations 2 (1997), 867-894.
  • Evans, L. C. and Gariepy, R. F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
  • Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
  • Friedman, A. and Kamin, S.: The asymptotic behaviour of a gas in an n-dimensional porous medium. Trans. Amer. Math. Soc. 262 (1980), 373-401.
  • Galaktionov, V. A., Peletier, L. A. and Vázquez, J. L.: Asymptotics of the fast diffusion equation with critical exponent. SIAM J. Math. Anal. 31 (2000), no 5, 1157-1174.
  • Herrero, M. A. and Pierre, M.: The Cauchy problem for $u_t=\Delta u^m$ when $0<m<1$. Trans. Amer. Math. Soc. 291 (1985), 145-158.
  • Hui, K. M.: Fatou theorem for the solutions of some nonlinear equations. J. Math. Anal. Appl. 183 (1994), 37-52.
  • Kamin, S. and Dascal, L.: Long time slow expansion of hot bubbles in gases. In Optimal Control and Partial Differential Equations. J.L. Menaldi et al. (eds), IOS Press, 2001.
  • King, J. R.: Self-similar behaviour for the equation of fast nonlinear diffusion. Phil. Trans. Roy. Soc. London A 343 (1993), 337-375.
  • King, J. R.: Exact polynomial solutions to some nonlinear diffusion equations. Phys. D 64 (1993), no. 1-3, 35-65.
  • Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural'tseva, N. N.: Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Monographs 23, Amer. Math. Soc, Providence, 1968.
  • Marcus, M. and Véron, L.: Initial trace of positive solutions of some nonlinear parabolic equations. Comm. Partial Differential Equations 24 (1999), 1445-1499.
  • Meerson, B.: On the dynamics of strong temperature disturbances in the upper atmosphere of the Earth. Phys. Fluids A 1 (1989), 887-891.
  • Meerson, B., Sasorov, P. V. and Sekimoto, K.: Logarithmically slow expansion of hot bubbles in gases. Phys. Rev. E 61 (2000), 1403-1406.
  • Peletier, M. A. and Zhang, H. F.: Self-similar solutions of fast diffusion equations that do not conserve mass. Differential Integral Equations 8 (1995), 2045-2064.
  • Rodríguez, A. and Vázquez, J. L.: Nonuniqueness of solutions of nonlinear heat equations of fast diffusion type. Ann. Inst. H. Poincaré, Anal. non Linéaire 12 (1995), 173-200.
  • Ughi, M.: A degenerate parabolic equation modelling the spread of an epidemic. Ann. Mat. Pura Appl. (4) 143 (1986), 385-400.
  • Vázquez, J. L.: Behaviour of the velocity of one-dimensional flows in porous media. Trans Amer. Mat. Soc. 286 (1984), 787-802.
  • Vázquez, J. L.: Nonexistence of solutions for nonlinear heat equations of fast-diffusion type. J. Math. Pures Appl. 71 (1992), 503-526.
  • Vázquez, J. L.: An introduction to the mathematical theory of the porous medium equation. In Shape optimization and free boundaries (Montreal, PQ, 1990), 347-389. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 380. Kluwer Acad. Publ., Dordrecht, 1992.
  • Vázquez, J. L.: Darcy's Law and the theory of shrinking solutions of fast diffusion equations. To appear in SIAM J. Math. Anal.
  • Wang, L. H.: On the regularity theory of fully nonlinear parabolic equations: I. Comm. Pure Applied Math. 45 (1992), no 1, 27-76. On the regularity theory of fully nonlinear parabolic equations: II. Comm. Pure Applied Math. 45 (1992), no 2, 141-178.