Attractors for the inhomogeneous incompressible Navier–Stokes flows

This paper is devoted to the evolution of Lions’s weak solutions to the inhomogeneous Navier–Stokes equations. After proving that the kinetic energy is eventually bounded, we obtain a weakly compact global attractor that all Lions’s weak solutions approach as time tends to infinity. Furthermore, the existence of attracting sets in strong topology is established for short trajectories satisfying an additional compactness condition on the density.     

Space-time asymptotics of the two dimensional Navier–Stokes flow in the whole plane

We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in $L^1({\mathbb{R}}^2)\cap L_{\sigma }^2({\mathbb{R}}^2)$. Moreover, we characterize the necessary and sufficient condition for the rapid energy decay ${6u(t)6}_2=o(t^{?1})$ as $t\to \infty$ motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow $u(t)$ lies in the Hardy space $H^1({\mathbb{R}}^2)$ for $t>0$, we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay ${6u(t)6}_2=o(t^{?\frac{3}{2}})$ as $t\to \infty$ with cyclic symmetry introduced by Brandolese [2].     

High-order accurate iterative solution of the Navier–Stokes equations for incompressible flows

An iterative solution scheme for the incompressible Navier–Stokes equations is presented. It is split into inner and outer iteration cycles, such that the momentum and continuity equations are satisfied within prescribed accuracy. The spatial discretization is based on high-order finite differences which makes it well suited for massively parallel computers. This is demonstrated in a scaling test. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the stationary Navier–Stokes flows around a rotating body

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body $ \mathcal{K} = \mathbb{R}^3 \, \backslash \, {\Omega}$ which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U, ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier–Stokes equations on the exterior domain Ω for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying ${\nabla u, p \in L_{3/2, \infty} (\Omega)}$ and ${u \in L_3, \infty (\Omega)}$ under the smallness condition on ${|U| + |\omega| + ||F||_{L_{3/2, \infty} (\Omega)}}$ . Then the uniqueness is shown for solutions (u, p) satisfying ${\nabla u, p \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}$ and ${u \in L_{3, \infty} (\Omega) \cap L_{q*, r} (\Omega)}$ provided that 3/2 <? q <? 3 and ${{F \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}}$ . Here L _q,r (Ω) denotes the well-known Lorentz space and q* =? 3q /(3 ? q) is the Sobolev exponent to q.     

Algorithm for solving the Navier–Stokes equations for the modeling of creeping flows

We study a coupled algorithm for solving the two-dimensional Navier–Stokes equations in the stream function–vorticity variables. The algorithm is based on a finite-difference scheme in which the inertial terms in the vortex transport equation are taken from the lower time layer and the dissipative terms, from the upper time layer. In the linear approximation, we study the stability of this algorithm and use test computations to show its advantages when modeling flows at moderate Reynolds numbers.     

Approximations of stochastic Navier–Stokes equations

In this paper we show that solutions of two-dimensional stochastic Navier–Stokes equations driven by Brownian motion can be approximated by stochastic Navier–Stokes equations forced by pure jump noise/random kicks.     

A first Integral of Navier­Stokes equations for two-dimensional flows

As wellknown, Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity. Even in case of non-vanishing vorticity, a first integral from Euler's equations is obtained by using the so called Clebsch transformation [1] for inviscid flows. In contrast to this, a generalisation of this procedure towards viscous flows has not been established so far. In the present paper a first integral of Navier-Stokes equations is constructed in the case of two-dimensional flow by making use of an alternative representation of the fields in terms of complex coordinates and introducing a potential representation for the pressure. The associated boundary conditions are also considered. The first integral is a suitable tool for the development of new analytical methods and numerical codes in fluid dynamics. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains

Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to the incompressible Navier–Stokes equation. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.     

Blow-up of Compressible Navier–Stokes–Korteweg Equations

Based on the results of Xin (Commun. Pure Appl. Math. 51(3):229–240, 1998), Zhang and Tan (Acta Math. Sin. Engl. Ser. 28(3):645–652, 2012), we show the blow-up phenomena of smooth solutions to the non-isothermal compressible Navier–Stokes–Korteweg equations in arbitrary dimensions, under the assumption that the initial density has compact support. Here the coefficients are generalized to a more general case which depends on density and temperature. Our work extends the previous corresponding results.