Skip to main content
SpringerLink
Log in

Lemmas on compensated compactness in elliptic and parabolic equations

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We study the solvability of parabolic and elliptic equations of monotone type with nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a solution, we apply a technique of passing to the limit in approximation schemes. A key element of this technique is a generalized lemma on compensated compactness. The parabolic version of this lemma is rather complicated and is proved for the first time in the present paper. The new technique applies to stationary and nonstationary problems of fast diffusion in an incompressible flow, to a parabolic equation with a p(x, t)-Laplacian and its generalization, and to a nonstationary thermistor system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. V. Zhikov, “On Passage to the Limit in Nonlinear Elliptic Equations,” Dokl. Akad. Nauk 420(3), 300–305 (2008) [Dokl. Math. 77 (3), 383–387 (2008)].

    MathSciNet  Google Scholar 

  2. V. V. Zhikov, “On the Technique for Passing to the Limit in Nonlinear Elliptic Equations,” Funkts. Anal. Prilozh. 43(2), 19–38 (2009) [Funct. Anal. Appl. 43, 96–112 (2009)].

    MathSciNet  Google Scholar 

  3. V. V. Zhikov and S. E. Pastukhova, “Parabolic Lemmas on Compensated Compactness and Their Applications,” Dokl. Akad. Nauk 431(3), 306–312 (2010) [Dokl. Math. 81 (2), 227–232 (2010)].

    MathSciNet  Google Scholar 

  4. L. Tartar, Cours Peccot, Collège de France (1977).

  5. F. Murat, “Compacité par compensation,” Ann. Sc. Norm. Super. Pisa., Cl. Sci., Ser. 4,5, 489–507 (1978).

    MathSciNet  Google Scholar 

  6. N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory (Interscience, New York, 1958; Inostrannaya Literatura, Moscow, 1962).

    Google Scholar 

  7. V. V. Zhikov and S. E. Pastukhova, “Improved Integrability of the Gradients of Solutions of Elliptic Equations with Variable Nonlinearity Exponent,” Mat. Sb. 199(12), 19–52 (2008) [Sb. Math. 199, 1751–1782 (2008)].

    MathSciNet  Google Scholar 

  8. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications (Academic Press, New York, 1980; Mir, Moscow, 1983).

    MATH  Google Scholar 

  9. J.-L. Lions, Quelques méthodes de résolution des probl`emes aux limites non linéaires (Gauthier-Villars, Paris, 1969; Mir, Moscow, 1972).

    Google Scholar 

  10. P. Billingsley, Convergence of Probability Measures (J. Wiley and Sons, New York, 1968; Nauka, Moscow, 1977).

    MATH  Google Scholar 

  11. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasi-linear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, RI, 1968), Transl. Math. Monogr. 23.

    Google Scholar 

  12. E. DiBenedetto, Degenerate Parabolic Equations (Springer, New York, 1993).

    MATH  Google Scholar 

  13. V. V. Zhikov, “To the Problem of Passage to the Limit in Divergent Nonuniformly Elliptic Equations,” Funkts. Anal. Prilozh. 35(1), 23–39 (2001) [Funct. Anal. Appl. 35, 19–33 (2001)].

    MathSciNet  Google Scholar 

  14. Yu. A. Alkhutov and V. V. Zhikov, “Existence Theorems and Qualitative Properties of Solutions to Parabolic Equations with a Variable Order of Nonlinearity,” Dokl. Akad. Nauk 430(3), 295–299 (2010) [Dokl. Math. 81 (1), 34–38 (2010)].

    MathSciNet  Google Scholar 

  15. V. V. Zhikov, “New Approach to the Solvability of Generalized Navier-Stokes Equations,” Funkts. Anal. Prilozh. 43(3), 33–53 (2009) [Funct. Anal. Appl. 43, 190–207 (2009)].

    MathSciNet  Google Scholar 

  16. J. Simon, “Compact Sets in the Space L p(0, T;B),” Ann. Mat. Pura Appl. 146, 65–96 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  17. D. E. Edmunds and J. Rákosník, “Sobolev Embeddings with Variable Exponent,” Stud. Math. 143, 267–293 (2000).

    MATH  Google Scholar 

  18. V. V. Zhikov, “On Lavrentév’s Effect,” Dokl. Akad. Nauk 345(1), 10–14 (1995) [Dokl. Math. 52 (3), 325–329 (1995)].

    MathSciNet  Google Scholar 

  19. V. V. Zhikov, “On Lavrentiev’s Phenomenon,” Russ. J. Math. Phys. 3(2), 249–269 (1995).

    MATH  MathSciNet  Google Scholar 

  20. V. V. Zhikov and S. E. Pastukhova, “On the Property of Higher Integrability for Parabolic Systems of Variable Order of Nonlinearity,” Mat. Zametki 87(2), 179–200 (2010) [Math. Notes 87, 169–188 (2010)].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vladimir State University for the Humanities, pr. Stroitelei 11, Vladimir, 600024, Russia

    V. V. Zhikov

  2. Moscow State Institute of Radio Engineering, Electronics and Automation (Technical University), pr. Vernadskogo 78, Moscow, 119454, Russia

    S. E. Pastukhova

Authors
  1. V. V. Zhikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. E. Pastukhova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Zhikov.

Additional information

Original Russian Text © V.V. Zhikov, S.E. Pastukhova, 2010, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2010, Vol. 270, pp. 110–137.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhikov, V.V., Pastukhova, S.E. Lemmas on compensated compactness in elliptic and parabolic equations. Proc. Steklov Inst. Math. 270, 104–131 (2010). https://doi.org/10.1134/S0081543810030089

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543810030089

Keywords

Search

Navigation