We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schrödinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared. We also show by an analysis of the limit hydrodynamical equaiton that, however, this WKB-type approximation breaks down before reaching the focal point: Nonlinear effects lead to the formation of singularity of the leading term of the phase function.

In A Kantorovich–type analysis for a fast iterative method for solving nonlinear equations, and Convergence and applications of Newton–type iterations, Argyros introduced a new derivative–free quadratically convergent method for solving a nonlinear equation in Banach space. In this paper, we extend this method to generalized equations in order to approximate a locally unique solution. The method uses only divided differences operators of order one. Under some Lipschitz–type conditions on the first and second order divided differences operators and Lipschitz–like property of set–valued maps, an existence–convergence theorem and a radius of convergence are obtained. Our method has the following advantages: we extend the applicability of this method than all the previous ones, and we do not need to evaluate any Fréchet derivative. We provide also an improvement on the radius of convergence for our algorithm, under some center–condition and less computational cost. Numerical examples are also provided.

In this paper, we establish a blow up criterion for strong solutions of the full compressible Navier-Stokes equations just in terms of the gradient of the velocity. It shows that the gradient of the velocity alone dominates the global existence of strong solutions.