A blowup criterion for nonhomogeneous incompressible Navier–Stokes–Landau–Lifshitz system in 2-D

In this paper, we investigate nonhomogeneous incompressible Navier–Stokes–Landau–Lifshitz system in two-dimensional (2-D). This system consists of Navier–Stokes equations coupled with Landau–Lifshitz–Gilbert equation, an evolutionary equation for the magnetization vector. We establish a blowup criterion for the 2-D incompressible Navier–Stokes–Landau–Lifshitz system with finite positive initial density.     

An Application of BMO-type Space to Chemotaxis-fluid Equations

Acta Mathematica Sinica, English Series - We consider a Keller–Segel model coupled to the incompressible Navier–Stokes system in 3-dimensional case. We prove that the system has a...     

Random attractor for the stochastic Cahn–Hilliard–Navier–Stokes system with small additive noise

The objective of this paper is to study the asymptotic behavior of solutions, in terms of the upper semi-continuous property of random attractor, of the Cahn–Hilliard–Navier–Stokes system with small additive noise. We prove the existence of a random attractor for the Cahn–Hilliard–Navier–Stokes system with small additive noise. Furthermore, we consider the stability of global attractor and prove the random attractor of the Cahn–Hilliard–Navier–Stokes system with small additive noise will convergent to the global attractor of the unperturbed Cahn–Hilliard–Navier–Stokes system when the parameter of the perturbation ε tends to zero.     

Global well-posedness and twist-wave solutions for the inertial Qian–Sheng model of liquid crystals

We consider the inertial Qian–Sheng model of liquid crystals which couples a hyperbolic-type equation involving a second-order material derivative with a forced incompressible Navier–Stokes system. We study the energy law and prove a global well-posedness result. We further provide an example of twist-wave solutions, that is solutions of the coupled system for which the flow vanishes for all times.     

The inviscid limit of the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition

In this paper, we study the asymptotic behavior for the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition in a half space of ${\mathbb{R}^3}$ when the vertical viscosity goes to zero. Firstly, by multi-scale analysis, we formally deduce an asymptotic expansion of the solution to the problem with respect to the vertical viscosity, which shows that the boundary layer appears in the tangential velocity field and satisfies a nonlinear parabolic–elliptic coupled system. Also from the expansion, it is observed that away from the boundary the solution of the anisotropic Navier–Stokes equations formally converges to a solution of a degenerate incompressible Navier–Stokes equation. Secondly, we study the well-posedness of the problems for the boundary layer equations and then rigorously justify the asymptotic expansion by using the energy method. We obtain the convergence results of the vanishing vertical viscosity limit, that is, the solution to the incompressible anisotropic Navier–Stokes equations tends to the solution to degenerate incompressible Navier–Stokes equations away from the boundary, while near the boundary, it tends to the boundary layer profile, in both the energy space and the L ^∞ space.     

Non‐homogeneous Navier–Stokes systems with order‐parameter‐dependent stresses

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order‐parameter c. Moreover, an extra stress depending on c and ?c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two‐phase flow of viscous incompressible fluids. The so‐called density‐dependent Navier–Stokes system is also a special case of our system. We prove short‐time existence of strong solution in L^q‐Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.     

Global existence of weak solutions to a diffuse interface model for magnetic fluids

This article is devoted to the derivation and analysis of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier–Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn–Hilliard equations. We show global in time existence of weak solutions to the system using the time discretization method.     

Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

In this paper, we are concerned with the system of the non‐isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time‐decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large‐time behavior is based on the linearized analysis of the non‐isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time‐decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866‐891] for the linearized non‐isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.     

Homogenization of a Coupled System with Periodic Oscillating Coefficients

We study the homogenization of a coupled system with periodic oscillating coefficients in bounded non-homogeneous media. The system couples the Navier–Stokes and a classical parabolic diffusive equation. To do that, we introduce a generalized compensate compactness result and a suitable class of test function to this problem. By passing the limit, we obtain the homogenized model of this problem.     

On an incompressible model in radiation hydrodynamics

We consider a simplified model arising in radiation hydrodynamics based on the incompressible Navier–Stokes–Fourier system describing a macroscopic fluid motion coupled to a transport equation modeling the propagation of radiative intensity. We establish global‐in‐time existence for the associated initial‐boundary value problem in the framework of weak solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence and zero‐electron‐mass limit of strong solutions to the stationary compressible Navier‐Stokes‐Poisson equation with large external force

In this study, we consider a viscous compressible model of plasma and semiconductors, which is expressed as a compressible Navier‐Stokes‐Poisson equation. We prove that there exists a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces in bounded domain, provided that the ratio of the electron/ions mass is appropriately small. Moreover, the zero‐electron‐mass limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into 4 parts, a system of stationary incompressible Navier‐Stokes equations with large forces, a system of stationary compressible Navier‐Stokes equations with small forces, coupled with 2 Poisson equations. Based on the known results about linear incompressible Navier‐Stokes equation, linear compressible Navier‐Stokes, linear transport, and Poisson equations, we try to establish uniform in the ratio of the electron/ions mass a priori estimates. Further, using Schauder fixed point theorem, we can show the existence of a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces. At the same time, from the uniform a priori estimates, we present the zero‐electron‐mass limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier‐Stokes‐Poisson equations.     

A boundary layer problem in nonflat domains with measurable viscous coefficients

We study the convergence of weak solutions of the Navier–Stokes equations with vanishing measurable viscous coefficients in domains with nonflat boundaries. Sufficient anisotropic conditions on the vanishing rates of the viscous coefficients are found to prove the convergence of Leray–Hopf weak solutions of the Navier–Stokes equations to solutions of the corresponding Euler equations. As the domains are not flat, we apply a change of variables to flatten the domains. We then construct explicit boundary layers for the system of Navier–Stokes equations in the upper-half space with measurable viscous coefficients. The result is new even when the viscous coefficients are constant, and it recovers the classical results when domains are flat and with constant viscous coefficients.     

Well-posedness of a Debye type system endowed with a full wave equation

We prove well-posedness for a transport-diffusion problem coupled with a wave equation for the potential. We assume that the initial data are small. A bilinear form in the spirit of Kato’s proof for the Navier–Stokes equations is used, coupled with suitable estimates in Chemin–Lerner spaces. In the one dimensional case, we get well-posedness for arbitrarily large initial data by using Gagliardo–Nirenberg inequalities.     

Finite-dimensional global attractor for a system modeling the 2<Emphasis Type="Italic">D</Emphasis> nematic liquid crystal flow

We consider a 2D system that models the nematic liquid crystal flow through the Navier–Stokes equations suitably coupled with a transport-reaction-diffusion equation for the averaged molecular orientations. This system has been proposed as a reasonable approximation of the well-known Ericksen–Leslie system. Taking advantage of previous well-posedness results and proving suitable dissipative estimates, here we show that the system endowed with periodic boundary conditions is a dissipative dynamical system with a smooth global attractor of finite fractal dimension.     

Blow-up criteria for the three dimensional nonlinear dissipative system modeling electro-hydrodynamics

In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier–Stokes equations and the classical Poisson–Nernst–Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover, some Prodi–Serrin type blow-up criteria are also established.     

Asymptotic behavior toward the combination of contact discontinuity with rarefaction waves for 1-D compressible viscous gas with radiation

This paper is concerned with the large-time behavior toward the combination of two rarefaction waves and viscous contact wave for the Cauchy problem to a one-dimensional Navier–Stokes–Poisson coupled system, modeling the dynamics of a viscous gas in the presence of radiation. We show that the composite wave with small strength is asymptotically stable under partially large initial perturbations. The proofs are based on the more refined energy estimates to control the possible growth of the perturbations induced by two different waves and large data.     

On the low Mach number limit of compressible flows in exterior moving domains

We study the incompressible limit of solutions to the compressible barotropic Navier–Stokes system in the exterior of a bounded domain undergoing a simple translation. The problem is reformulated using a change of coordinates to fixed exterior domain. Using the spectral analysis of the wave propagator, the dispersion of acoustic waves is proved by means of the RAGE theorem. The solution to the incompressible Navier–Stokes equations is identified as a limit.     

On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain by allowing for a degenerate mobility. The model has been developed by Abels, Garcke and Grün for fluids with different densities and leads to a solenoidal velocity field. It is given by a non-homogeneous Navier–Stokes system with a modified convective term coupled to a Cahn–Hilliard system, such that an energy estimate is fulfilled which follows from the fact that the model is thermodynamically consistent.     

Optimal decay rate of the two-fluid incompressible Navier–Stokes–Fourier–Poisson system with Ohm’s law

This paper is to investigate the optimal $L^p$ ($p\ge 2$) time decay rate of global solutions to the two-fluid incompressible Navier–Stokes–Fourier–Poisson system with Ohm’s law. The result shows that the time decay rate of this system achieves the same as that of the incompressible Navier–Stokes equation, in other words, the coupling of the self-consistent Poisson equation does not change the decay rate of the incompressible Navier–Stokes equation. This phenomenon is interesting, compared to the compressible Navier–Stokes equation, whose time decay rate decreases when it is coupled with the self-consistent Poisson equation.