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

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

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
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. S. E. Pastukhova, A. S. Khripunova, “Several versions of the compensated compactness principle”, Sb. Math., 202:9 (2011), 1387–1412
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
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
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
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
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
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
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