In this paper, the three-dimensional compressible Navier-Stokes equations are considered on a periodic domain. We propose a semi-discrete numerical scheme and derive a priori bounds that ensures that the resulting system of ordinary differential equations is solvable for any h > 0. An a posteriori examination that density remain uniformly bounded away from 0 will establish that a subsequence of...

The main aim of this paper is a geometrical approach to simultaneous solutions of the abstract weak Dirichlet problem. We answer partially a question from the paper 2] where a similar problem was discussed from a potential-theoretical point of view for the case of function spaces consisting of harmonic functions.

New exact solutions are obtained for several nonlinear physical equations, namely the Navier–Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schrödinger equation. The solution methods make use of the symmetry group of the system in situations when the standard Lie method of symmetry reduction is not applicable.

We describe recent results obtained by G. Barles, P. Cardaliaguet, R. Monneau and the author in [9, 7]. They are concerned with nonlocal Eikonal equations arising in the study of the dynamics of dislocation lines in crystals. These equations are nonlocal but also non monotone. We use a notion of weak solution to provide solutions for all time. Then, we discuss the link between these weak soluti...

We study the nonlinear stochastic heat equation driven by spacetime white noise in the case that the initial datum u0 is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We prove instead that it is possible to establish the existence and uniqueness of a weak solution with values in a suitable function space. Our approach is based on a...

In this work we give su cient conditions for the nite convergence of descent algorithms for solving variational inequalities involving generalized monotone mappings.

We show that the Willmore ow for smoothly immersed initial surfaces has eternal solutions in a weak sense, which are smooth except at nitely many times. Furthermore we study the set of singularities of such a weak solution.

We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of Lx estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate L 4 n−1 t Lx for n = 2, 3 even for data in inhomogeneous Sobolev spaces.

We prove a weak version of the dynamic programming principle for standard stochastic control problems and mixed control-stopping problems, which avoids the technical difficulties related to the measurable selection argument. In the Markov case, our result is tailor-maid for the derivation of the dynamic programming equation in the sense of viscosity solutions.