engleski

H-distributions and compensated compactness

H-measures, introduced independently by Luc Tartar and Patrick Gerard, are matrix Radon measures describing the behaviour of weak limits of quadratic quantities. They proved to be very successful tool in investigations of asymptotic limits of quadratic quan- tities. However, they turned insufficient for nonlinear problems. H-distributions were introduced by Antonić and Mitrović as an extension of H-measures to the Lp-Lq setting. heir variants have been successfully applied to problems in velocity averaging (Lazar-Mitrović 2012) and compensated compactness with variable coefficients (Mišur-Mitrović 2015). Unlike H-measures, which are nonnegative Radon measures, H- distributions are distributions in the Schwartz sense, which follows from the standard Schwartz kernel theorem. To give a precise description of H-distributions, we will introduce the notion of anisotropic distributions – distributions of different order with respect to different coordinate direc- tions. In order to show that H-distributions are anisotropic distributions of finite order with respect to every coordinate direction, we will prove a variant of Schwartz kernel theorem. In the second part of the talk, we will show a variant of compensated compactness using a variant of H-distributions. The conditions involve fractional derivatives and variable coefficients, and they represent a generalization of the known compensated compactness theory. We will apply the developed techniques to a nonlinear (degenerate) parabolic equation.

H-distributions ; compensated compactness ; commutator compactness ; kernel theorem

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