Tag - Navier-Stokes equation

In this talk I will discuss some recent results on Euler and Navier-Stokes equations concerning the construction of quasi-periodic solutions and the problem of the invscid limit for the Navier-Stokes equation. I will discuss the following two results:

1) Construction of quasi-periodic solutions for the Euler equation with a time quasi-periodic external force, bifurcating from a constant, diophantine velocity field;

2) I shall discuss the inviscid limit problem from the perspective of KAM theory, namely I shall prove the existence of quasi-periodic solutions of the Navier Stokes equation converging to the one of the Euler equation constructed in 1).

The main difficulty is that this is a singular limit problem. We overcome this difficulty by implementing a normal form methods which allow to prove sharp estimates (global in time) with respect to the viscosity parameter.

In this talk, we are concerned with the propagation of the regularity for the non-isentropic Navier-Stokes system in a domain with boundaries, uniformly in the Mach number and the Reynolds number, in a general setting of ill-prepared data. This is an essential step towards the study of the low Mach number and the inviscid limit for strong solutions. The main obstacle to estabilish such uniform regularity estimates lie in the existence of fast oscillating acoustic waves (due to the ill-prepared assumption) and the interactions of two kinds of boundary layers (due to the large time temperature variation and the non-vanishing thermal conductivity). To start, I will first explain the ideas to propagate the high regularity for the isentropic Navier-Stokes system uniformly only in Mach number, which was a joint work with Professor N. Masmoudi and F. Rousset.

Since seminal papers published in the middle of the sixties, low-order staggered schemes for incompressible flow computations have received a considerable attention. The staggered discretization is a space structured or unstructured discretization where the scalar unknowns (the pressure) are located at the cell centers while the vector unknowns (the velocity) are located at the cells faces. This discretization is essentially motivated by the fact that it combines a low computational cost with the so-called inf-sup or LBB stability condition, which prevents from the odd-even decoupling of the pressure. For several years, an important effort has been dedicated to the extension of staggered numerical schemes for the approximation of compressible flows. In this talk, I will focus on the barotropic compressible Navier-Stokes equations. I will prove that, as the Mach number tends to zero, the solution of the implicit staggered scheme for these equations converges towards the solution of the standard staggered scheme for the incompressible Navier-Stokes equations. In particular, the numerical density tends towards a constant as the Mach number tends to zero. Such a result follows from a similar analysis to that of Lions and Masmoudi (1998) at the continuous level for weak solutions of the barotropic compressible Navier-Stokes equations. It extends to other time discretizations such as the so-called pressure-correction scheme (adapted to compressible models). These numerical schemes are used in practice in industrial codes such as CALIF3S, a code developed by the IRSN (French Institut for Radioprotection and Nuclear Safety) for the simulation of deflagrations.

In this talk we will introduce the recent results about the non-linear stability of a shear layer profile for Navier-Stokes equations near a boundary. This question plays a major role in the study of the inviscid limit of Navier-Stokes equations in a bounded domain as the viscosity goes to 0. We mainly study the effect of cubic interactions on the growth of the linear instability here. In the case of the exponential profile and Blasius profile we obtain that the non-linearity tames the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude O(ν^1/4) only, forming small rolls in the critical layer near the boundary.