Pankratov L. Homogenization of nonlinear equation of the mathematical physics

The thesis is devoted to the derivation of the homogenized models for nonlinear stationary and evolutionary problems of the mathematical physics with rapidly oscillating coefficients or considered in domains with a complex microstructure by the method of mesoscopic energy characteristics or the two-scale convergence method. The First chapter is devoted to the description of the definitions and the main methods of the homogenization theory as well as the history of its development. In the Second chapter we obtain the homogenized models for the linear parabolic and nonlinear variational problems of double porosity type in the case of thin fissures. For the linear parabolic equations it is proved that in the case when the square of the opening of the fissure is of the same order as the permeability of the matrix part, the homogenized model is a model with memory. The Third chapter is devoted to the homogenization of elliptic and parabolic problems in the variable Sobolev spaces. On the periodic examples, for the elliptic and parabolic double porosity type problems it is shown that the form of the homogenized model essentially depends on the convergence rate of the growth function p to a non--disturbed growth function p0. The Fourth chapter is devoted to the derivation of the homogenized models for the Ginzburg-Landau equation in weakly connected domains. The well known Josephson phenomenon is rigorously derived as result of the homogenization process. The Fifth chapter is devoted to the homogenization of an immiscible compressible two-phase flow such as water-gas in porous media with rapidly oscillating porosity function and the global permeability tensor. In the Sixth chapter we consider nonlinear evolution problems in double porosity media and in media with traps. In the case when the square of the opening of the fissure is of the same order as the permeability coefficient in the matrix part, the homogenized reaction-diffusion model in the double porosity media is a model with memory. The obtained homogenized models for the parabolic and hyperbolic semi-linear equations are also models with memory. For these models the convergence of the global attractors of the corresponding dynamical systems is also proved.