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Tr. Mat. Inst. Steklova, 2010, Volume 270, Pages 110–137 (Mi tm3006)  

This article is cited in 8 scientific papers (total in 8 papers)

Lemmas on compensated compactness in elliptic and parabolic equations

V. V. Zhikova, S. E. Pastukhovab

a Chair of Mathematical Analysis, Vladimir State University for the Humanities, Vladimir, Russia
b Moscow State Institute of Radio Engineering, Electronics and Automation (Technical University), Moscow, Russia

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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 270, 104–131

Bibliographic databases:

UDC: 517.956.4
Received in June 2009

Citation: V. V. Zhikov, S. E. Pastukhova, “Lemmas on compensated compactness in elliptic and parabolic equations”, Differential equations and dynamical systems, Collected papers, Tr. Mat. Inst. Steklova, 270, MAIK Nauka/Interperiodica, Moscow, 2010, 110–137; Proc. Steklov Inst. Math., 270 (2010), 104–131

Citation in format AMSBIB
\by V.~V.~Zhikov, S.~E.~Pastukhova
\paper Lemmas on compensated compactness in elliptic and parabolic equations
\inbook Differential equations and dynamical systems
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2010
\vol 270
\pages 110--137
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2010
\vol 270
\pages 104--131

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    This publication is cited in the following articles:
    1. V. V. Zhikov, S. E. Pastukhova, “Homogenization of Monotone Operators Under Conditions of Coercitivity and Growth of Variable Order”, Math. Notes, 90:1 (2011), 48–63  mathnet  crossref  crossref  mathscinet  isi
    2. S. E. Pastukhova, A. S. Khripunova, “Several versions of the compensated compactness principle”, Sb. Math., 202:9 (2011), 1387–1412  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Pastukhova S., “Zhikov's hydromechanical lemma on compensated compactness: its extension and application to generalized stationary Navier–Stokes equations”, Complex Var. Elliptic Equ., 56:7-9 (2011), 697–714  crossref  mathscinet  zmath  isi  elib  scopus
    4. V. V. Zhikov, S. E. Pastukhova, “On the Navier–Stokes equations: Existence theorems and energy equalities”, Proc. Steklov Inst. Math., 278 (2012), 67–87  mathnet  crossref  mathscinet  isi  elib  elib
    5. Zhikov V.V., Surnachev M.D., “On Existence and Uniqueness Classes for the Cauchy Problem for the Parabolic P-Laplace Equation”, Dokl. Math., 86:1 (2012), 492–496  crossref  mathscinet  zmath  isi  elib  elib  scopus
    6. Surnachev M.D. Zhikov V.V., “On Existence and Uniqueness Classes for the Cauchy Problem for Parabolic Equations of the P-Laplace Type”, Commun. Pure Appl. Anal, 12:4 (2013), 1783–1812  crossref  mathscinet  zmath  isi  elib  scopus
    7. Buhrii O.M., “on the Existence of Mild Solutions of the Initial-Boundary-Value Problems For the Petrovskii-Type Semilinear Parabolic Systems With Variable Exponents of Nonlinearity”, Ukr. Math. J., 66:4 (2014), 487–498  crossref  mathscinet  zmath  isi  elib  scopus
    8. S. E. Pastukhova, D. A. Yakubovich, “Galerkin approximations for the Dirichlet problem with the $p(x)$-Laplacian”, Sb. Math., 210:1 (2019), 145–164  mathnet  crossref  crossref  adsnasa  isi  elib
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