We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing that in general the transition flow is only of class $C^3$.

This paper is the third of a series where the convergence analysis of SPH method for multidimensional conservation laws is analyzed. In this paper, two original numerical models for the treatment of boundary conditions are elaborated. To take into account nonlinear effects in agreement with Bardos, LeRoux and Nedelec boundary conditions, the state at the boundary is computed by solving appropriate Riemann problems. The first numerical model is developed around the idea of boundary forces in surrounding walls, recently initiated in Simulating Gravity Currents with SPH. III : Boundary Forces by Monaghan in his simulation of gravity currents. The second one extends the well-known approach of ghost particles for plane boundaries to the case of general curved boundaries. The convergence analysis in $L_loc^p ( p< \infty)$ is achieved thanks to the uniqueness result of measure-valued solutions recently established in Ben Moussa B., Szepessy A., Scalar Conservation Laws with Boundary Conditions and Rough Data Measure Solutions for $L^\infty$ initial and boundary data.

In this paper, we establish the space-time estimates in the Besov spaces of the solution to the Navier-Stokes equations in $\bold R^n , n\geq 3$. As an application, we improve some known results about the regularity criterion of weak solutions and the blow-up criterion of smooth solutions. Our main tools are the frequency localization and the Littlewood-Paley decomposition.