Naturally the question of inviscid limit arises. Does the solution u of the CGL equation (1.1) tend to (in an appropriate space norm) the solution v of the NLS equation (1.2) as the parameters a and b tend to 0? What is the convergence rate? The answers are not immediate especially when the initial data for these equations are not smooth. Because of its importance in both mathematical theory an...

We describe the method for finding the non-Gaussian tails of the probability distribution function ~PDF! for solutions of a stochastic differential equation, such as the convection equation for a passive scalar, the random driven Navier-Stokes equation, etc. The existence of such tails is generally regarded as a manifestation of the intermittency phenomenon. Our formalism is based on the WKB ap...

A random perturbation of a deterministic Navier-Stokes equation is considered in the form of an SPDE with Wick type nonlinearity. The nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term u∇u. This perturbation is unbiased in that the expectation of a solution of the perturbed/quantized equation solves the determinis...

A stochastic Navier–Stokes equation with space-time Gaussian white noise is considered, having as infinitesimal invariant measure a Gaussian measure µν whose covariance is given in terms of the enstrophy. Pathwise uniqueness for µν-a.e. initial velocity is proven for solutions having µν as invariant measure. 1. Introduction. We are interested in the stochastic Navier–Stokes equation with a spac...

We describe how Borel summability of divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for nonanalytic initial data, the present approach generates an integral equation applicable to much more general data. We apply these concepts to the 3-D Navier-Stokes system and show how the integral equation a...

In [3, 4, 5], we have developed a new tool called tquasi solutions which approximate in some sense the compressible Navier-Stokes equation. In particular it allows to obtain global strong solution for the compressible Navier-Stokes equations with large initial data on the irrotational part of the velocity (large in the sense that the smallness assumption is subcritical in terms of scaling, it t...

We study a liquid droplet moving inside rarefied gas. In other words, we consider two-phase flow with and gas phases an interface between the two which deforms respect to time space. The phase is modelled by Bhatanager–Gross–Krook (BGK) model of Boltzmann equation. incompressible Navier–Stokes equations. Interface boundary conditions for are presented. BGK solved semi-Lagrangian scheme meshfree...