Counter-examples to concentration-cancellation
This paper consider the incompressible Euler equations. In general in order to show the existence of weak solutions to Euler equations, an approximate solution sequence was constructed either by smoothing the initial data or by adding to Euler equations a small viscosity. By the property of concentration-cancellation, all the extracted \(L^2_{loc}\) weak limits still satisfy the Euler equations, through the approximate solution sequence does not converge strongly in \(L^2_{loc}\). So up to now, for the two-dimensional case, the weak limits of approximate solutions have always been observed to be solution of the Euler equations. Thus, one want to know if the property of concentration-cancellation does generalize to the space \(L^2\) of the energy estimates. It makes sense also in the less restrictive framework \(L^2_{loc}\). So the authors develop a WKB analysis and give a negative reply to the last question.
scientific article; zbMATH DE number 5375976
scientific article