Trilinear compensated compactness and nonlinear geometric optics.

*(English)*Zbl 0840.35013The paper studies the behavior of bounded families of solutions of nonlinear systems of partial differential equations. The problem deals with equations which determine the weak limits of soutions \(u^\varepsilon\) through Young measures, i.e. vague limits of the probability measures \(\delta[\lambda= u^\varepsilon(x)]\), which contain all the information on the limits of nonlinear functions \(u^\varepsilon\). In this context, the authors study the problem to derive equations which uniquely determine the Young measure of the data. Some difficulties in the weak convergence arise from “oscillations”: The equations of the profiles which asymptotically describe the oscillations depend on the structure of resonances.

In particular, when non-resonances are present, the profile equations are more decoupled. The question is to know whether the existence of resonant phases is the only obstruction for such a decoupling for general families of solutions. This paper gives a positive answer to this question, in one space dimension, either for \(3\times 3\) semilinear systems with quadratic right hand side. Next, the paper studies how the decoupling into tensor products of the Young measures is related to compensated compactness.

In particular, when non-resonances are present, the profile equations are more decoupled. The question is to know whether the existence of resonant phases is the only obstruction for such a decoupling for general families of solutions. This paper gives a positive answer to this question, in one space dimension, either for \(3\times 3\) semilinear systems with quadratic right hand side. Next, the paper studies how the decoupling into tensor products of the Young measures is related to compensated compactness.

Reviewer: M.Codegone (Torino)