Lemmas on compensated compactness in elliptic and parabolic equations

V. V. Zhikov ; S. E. Pastukhova

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 110-137

Résumé

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.