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Journal of Elliptic and Parabolic Equations

, Volume 5, Issue 2, pp 383–421 | Cite as

Existence for singular doubly nonlinear systems of porous medium type with time dependent boundary values

  • Leah SchätzlerEmail author
Article
  • 11 Downloads

Abstract

In this paper we prove the existence of variational solutions to the Cauchy–Dirichlet problem with time dependent boundary values associated with doubly nonlinear systems
$$\begin{aligned} \partial _t \big (|u|^{m-1}u\big ) - {{\,\mathrm{div}\,}}(D_\xi f(Du)) = 0 \end{aligned}$$
with \(m>1\) and a convex function f satisfying a standard p-growth condition for an exponent \(p \in (1,\infty )\). The proof relies on a nonlinear version of the method of minimizing movements.

Keywords

Porous medium equation Doubly nonlinear systems Existence Minimizing movements 

Mathematics Subject Classification

35K86 49J40 49J45 

Notes

Funding

The author has been supported by the Studienstiftung des deutschen Volkes

Compliance with ethical standards

Conflict of interest

Following the requirements of the journal, the author declares that she has no conflict of interest

Human/animal rights statement

Adding to the beauty of mathematics, this article does not contain any studies with human participants or animals.

References

  1. 1.
    Akagi, G., Stefanelli, U.: Doubly nonlinear equations as convex minimization. SIAM J. Math. Anal. 46(3), 1922–1945 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alt, H., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183(3), 311–341 (1983)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aronson, D.G.: Regularity properties of flows through porous media: the interface. Arch. Ration. Mech. Anal. 37, 1–10 (1970)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barenblatt, G.I.: On self-similar solutions of the Cauchy problem for a nonlinear parabolic equation of unsteady filtration of a gas in a porous medium (Russian). Prikl. Mat. Meh. 20, 761–763 (1956)MathSciNetGoogle Scholar
  5. 5.
    Barenblatt, G.I.: On some unsteady motions of a liquid and gas in a porous medium (Russian). Akad. Nauk SSSR. Prikl. Mat. Meh. 16, 67–78 (1952)MathSciNetGoogle Scholar
  6. 6.
    Bernis, F.: Existence results for doubly nonlinear higher order parabolic equations on unbounded domains. Math. Ann. 279(3), 373–394 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bögelein, V., Duzaar, F., Kinnunen, J., Scheven, C.: Higher integrability for doubly nonlinear parabolic systems (2018). https://arxiv.org/abs/1810.06039
  8. 8.
    Bögelein, V., Duzaar, F., Korte, R., Scheven, C.: The higher integrability of weak solutions of porous medium systems. Adv. Nonlinear Anal. 8(1), 1004–1034 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bögelein, V., Duzaar, F., Marcellini, P.: A time dependent variational approach to image restoration. SIAM J. Imaging Sci. 8(2), 968–1006 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bögelein, V., Duzaar, F., Marcellini, P.: Existence of evolutionary variational solutions via the calculus of variations. J. Differ. Equ. 256, 3912–3942 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bögelein, V., Duzaar, F., Marcellini, P.: Parabolic systems with \(p, q\)-growth: a variational approach. Arch. Ration. Mech. Anal. 210(1), 219–267 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bögelein, V., Duzaar, F., Marcellini, P., Scheven, C.: A variational approach to doubly nonlinear equations. Rend. Lincei Mat. Appl. 29, 739–772 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bögelein, V., Duzaar, F., Marcellini, P., Scheven, C.: Doubly nonlinear equations of porous medium type. Arch. Ration. Mech. Anal. 229, 503–545 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bögelein, V., Duzaar, F., Scheven, C.: The obstacle problem for parabolic minimizers. J. Evol. Equ. 17(4), 1273–1310 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Bögelein, V., Lukkari, T., Scheven, C.: The obstacle problem for the porous medium equation. Math. Ann. 363(1–2), 455–499 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dal Passo, R., Luckhaus, S.: A degenerate diffusion problem not in divergence form. J. Differ. Equ. 69(1), 1–14 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics, revised edn. Chapman and Hall/CRC, Boca Raton (2015)zbMATHGoogle Scholar
  18. 18.
    Grange, O., Mignot, F.: Sur la résolution d‘une équation et d‘une inéquation paraboliques non linéaires. (French). J. Funct. Anal. 11, 77–92 (1972)CrossRefGoogle Scholar
  19. 19.
    Ivanov, A.V.: Regularity for doubly nonlinear parabolic equations. J. Math. Sci. 83(1), 22–37 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ivanov, A.V., Mkrtychyan, P.Z.: On the existence of Hölder-continuous generalized solutions of the first boundary value problem for quasilinear doubly degenerate parabolic equations. J. Sov. Math. 62(3), 2725–2740 (1992)CrossRefGoogle Scholar
  21. 21.
    Ivanov, A.V., Mkrtychyan, P.Z., Jäger, W.: Existence and uniqueness of a regular solution of the Cauchy–Dirichlet problem for a class of doubly nonlinear parabolic equations. J. Sov. Math. 84(1), 845–855 (1997)Google Scholar
  22. 22.
    Ladyženskaja, O.A.: New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems. (Russian). Trudy Mat. Inst. Steklov. 102, 85–104 (1967)MathSciNetGoogle Scholar
  23. 23.
    Landes, R.: On the existence of weak solutions for quasilinear parabolic initial-boundary value problems. Proc. R. Soc. Edinb. Sect. A 89(3–4), 217–237 (1981)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lichnewsky, A., Temam, R.: Pseudosolutions of the time-dependent minimal surface problem. J. Differ. Equ. 30(3), 340–364 (1978)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Marcellini, P.: Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math. 51, 1–28 (1985)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Naumann, J.: Einführung in die Theorie parabolischer Variationsungleichungen. vol. 64 of Teubner-Texte zur Mathematik. BSB B.G. Teubner Verlagsgesellschaft, Leipzig (1984)Google Scholar
  27. 27.
    Showalter, R., Walkington, N.J.: Diffusion of fluid in a fissured medium with microstructure. SIAM J. Math. Anal. 22(6), 1702–1722 (1991)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Urbano, J.M.: The Method of Intrinsic Scaling. Lecture Notes in Mathematics (1930). Springer, Berlin (2008)CrossRefGoogle Scholar

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© Orthogonal Publisher and Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany

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