We consider the homogeneous Dirichlet problem for the nonlinear diffusion-absorption equation with a one-sided constraint imposed on diffusion flux values. The family of approximate solutions constructed by means of Alexander Kaplan's integral penalty operator is studied. It is shown that this family converges weakly in the first-order Sobolev space to the solution of the original problem, as the small regularization parameter tends to zero. Thereafter, a property of uniform approximation of solutions is established in Hölder's spaces via systematic study of structure of the penalty operator.
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Under study are the classical three-dimensional Navier-Stokes equations of a compressible inhomogeneous viscous fluid in a smooth bounded domain endowed with no-slip conditions on the boundary of the domain and fast oscillating initial density distributions. The state equation of the medium is the state equation for a barotropic gas. We assume that the adiabatic constant is greater than 3. We give a rigorous derivation of the homogenization procedure as the frequencies of fast oscillations tend to infinity and obtain a limit effective model of the dynamics of a compressible viscous gas with fast oscillating initial data.
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