The study on application way of the compensated compactness theory
Many important results have been got in the study of application of the compactness theory in partial differential equations. But as far as we know, there are mainly two application ways: One is the analysis of Tartar or the static structure of the Young measure; another is the analysis of Diperna of the dynamic behavior of the Young measure. The Young measure representing the weak limits theorem has been used in two ways. The final goal is to prove that the Young measures uniquely defined by the sequence of approximate solutions all are the Dirac measure. But these ways have the indirect behavior. In this report we give one direct application way to prove the convergence of the sequence of approximate solutions of the single conservation law with initial condition: \[ \partial u/\partial t+\partial f(u)/\partial x=0,\quad t>0,\quad -\infty <x<+\infty;\quad u|_{t=0}=u_ 0(x). \] In our report, we only use the weak continuous theorem of \(2\times 2\) determinant and do not introduce the mass representing the weak limits theorem.
scientific article; zbMATH DE number 4122481