engleski

Variant H-measures and applications

H-measures have been introduced as a tool for studying quadratic terms of weakly converging $L^2$ sequences. As such, they have proved appropriate for exploring limits of energy terms. Due to special scaling of the dual variable ($\frac{; ; \xi}; ; {; ; |\xi|}; ; $) included in the definition of H-measures, they have mostly been used for hyperbolic equations and systems. The above scaling turned out to be unsuitable for the applications to equations of parabolic type. Therefore, a variant of H-measures has been developed, containing a different scaling for the dual variable, which is better suited to the equations of this type. I shall present the construction of the new variant, and its properties. Applications to the heat and Schr\" odinger equation will be compared with the results obtained by the original H-measures, where the results turned out to be insufficient, as the corresponding H-measures are supported in two points (North/South pole) of the dual space. The new variant gives measures supported on the curves, thus enabling a study of propagation phenomena. The explicit relations between macroscopic energy dissipation term and source term are obtained, as well. Furthermore, application to the nonstationary Stokes system allows us to express the term which appears by homogenisation via the variant H-measure.

H-measures; Schrodinger equation; parabolic scaling

nije evidentirano