engleski

A parabolic variant of H-measures

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 ($\xi/|\xi|$) 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. The obtained results give a good example of differences between the original H-measures and its variants. While in the first case the results have turned out to be trivial with H-measures supported in two points (North/South pole) of dual space, a new variant gives measures supported on the curves, thus enabling a study of propagation phenomena. The explicit relation between macroscopic energy dissipation term and given data is obtained. Applications to the heat and Schroedinger equation will be presented.

variant H-measures; heat equation; Schrodinger equation

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