Uniqueness of a solution to the Cauchy problem for the Hopf equation in the two-dimensional case
It is a very interesting paper, concerning the statistical approach of 2D turbulent flow, described by the Hopf equation. The Cauchy problem for this equation was studied by \textit{C. Foias} [Rend. Sem. Mat. Univ. Padova 48(1972), 219--348 (1973; Zbl 0283.76017)], where the existence and uniqueness were obtained, assuming that an integral related with the flow energy (containing an exponential of the solution) is finite. In the present paper, the generalized Fubini theorem for a transition measure is used to obtain the uniqueness, when the mean energy of the initial condition is finite. No additional conditions are necessary, therefore an improvement of Foias's result is obtained. The Galerkin approximation is used to solve the abstract analog of the Navier-Stokes equations in a separable Hilbert space.
scientific article; zbMATH DE number 5915269
scientific article