Self-similar asymptotics of solutions to the Navier-Stokes system in two dimensional exterior domain

We consider the 2D incompressible Navier-Stokes equations with Dirichlet boundary condition in the exterior of one obstacle. Assuming that the circulation at infinity of the velocity is sufficiently small, we prove that the large time behavior of the corresponding solution to the initial-boundary value problem is described by the Lamb-Oseen vortex. The later is the well-known explicit self-similar solution to the Navier-Stokes system in the whole space $\R^2$.Comment: 13 page

Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics

Geophysical fluids all exhibit a common feature: their aspect ratio (depth to horizontal width) is very small. This leads to an asymptotic model widely used in meteorology, oceanography, and limnology, namely the hydrostatic approximation of the time-dependent incompressible Navier–Stokes equations. It relies on the hypothesis that pressure increases linearly in the vertical direction. In the following, we prove a convergence and existence theorem for this model by means of anisotropic estimates and a new time-compactness criterium.Fonds Franco-Espagnol D.R.E.I.FMinisterio de Educación y Cienci

Asymptotics of solutions to the Navier-Stokes system in exterior domains

We consider the incompressible Navier-Stokes equations with the Dirichlet boundary condition in an exterior domain of $\mathbb{R}^n$ with $n\geq2$. We compare the long-time behaviour of solutions to this initial-boundary value problem with the long-time behaviour of solutions of the analogous Cauchy problem in the whole space $\mathbb{R}^n$. We find that the long-time asymptotics of solutions to both problems coincide either in the case of small initial data in the weak $L^{n}$-space or for a certain class of large initial data

The FENE dumbbell polymer model: existence and uniqueness of solutions for the momentum balance equation

We consider the FENE dumbbell polymer model which is the coupling of the incompressible Navier-Stokes equations with the corresponding Fokker-Planck-Smoluchowski di ffusion equation. We show global well-posedness in the case of a 2D bounded domain. We assume in the general case that the initial velocity is sufficiently small and the initial probability density is sufficiently close to the equilibrium solution; moreover an additional condition on the coeffcients is imposed. In the corotational case, we only assume that the initial probability density is sufficiently close to the equilibrium solution