Global existence of weak solution to Navier–Stokes equations with large external force and general pressure

We study the 3‐D compressible Navier–Stokes equations with an external potential force and a general pressure. We prove the global‐in‐time existence of weak solutions with small‐energy initial data and with densities being positive and essentially bounded. No smallness assumption is made on the external force. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of a weak solution to the Navier–Stokes equation in a general time‐varying domain by the Rothe method

We assume that Ω^t is a domain in ?³, arbitrarily (but continuously) varying for 0?t?T. We impose no conditions on smoothness or shape of Ω^t. We prove the global in time existence of a weak solution of the Navier–Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition in Q_{[0, T)} := {( x , t);0?t?T, x ∈Ω^t}. The solution satisfies the energy‐type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall. Copyright © 2008 John Wiley & Sons, Ltd.     

Strong solutions to the equations of a ferrofluid flow model

We study the differential system introduced by M.I. Shliomis to describe the motion of a ferrofluid driven by an external magnetic field. The system is a combination of the Navier-Stokes equations, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of strong solutions to the system.     

Global weak solution to the flow of liquid crystals system

In this paper, we study a simplified system for the flow of nematic liquid crystals in a bounded domain in the three‐dimensional space. We derive the basic energy law which enables us to prove the global existence of the weak solutions under the condition that the initial density belongs to L^γ(Ω) for any $\gamma >\frac{3}{2}$. Especially, we also obtain that the weak solutions satisfy the energy inequality in integral or differential form. Copyright © 2009 John Wiley & Sons, Ltd.     

Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this paper, we consider the equations of Magnetohydrodynamics with Coulomb force which is of hyperbolic–parabolic–elliptic mixed type. By constructing the approximate solutions to the modified system with an artificial pressure term added, global existence of finite energy weak solutions is established via the weak convergence method. More careful argument has been paid to overcome the new difficulty arising from the Poisson term of Coulomb force in two dimensions when the adiabatic exponent is close to one. We also investigate the large-time behavior of such weak solutions after discussing the regularity and uniqueness of solutions to the stationary problem.     

Global large solutions to incompressible Navier‐Stokes equations with gravity

In this per, we consider a special class of initial data for the three‐dimensional incompressible Navier–Stokes equations with gravity. We show that, under such conditions, the incompressible Navier‐Stokes equations with gravity are globally well posed, and the velocity minus gravity term has finite energy. The important features of the initial data is that the velocity fields minus gravity term are almost parallel to the corresponding vorticity fields in a very large space domain. Copyright © 2014 John Wiley & Sons, Ltd.     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

Sufficient conditions for the regularity of the solutions of the Navier–Stokes equations

In this paper we find sufficient conditions, involving only the pressure, that ensure the regularity of weak solutions to the Navier–Stokes equations. Conditions involving only the pressure were previously obtained in [7,4]. Following a remark in this last reference we improve, in particular, Kaniel's result [7]. Our condition can be seen at the light of the heuristic idea that the pressure behaves similarly to the modulus squared of the velocity. Copyright © 1999 John Wiley & Sons, Ltd.     

On the domain dependence of solutions to the Navier–Stokes equations of a two‐dimensional compressible flow

We consider the Navier–Stokes equations for compressible, barotropic flow in two space dimensions, with pressure satisfying p(?)=a?log^d(?) for large ?, here d>1 and a>0. After introducing useful tools from the theory of Orlicz spaces, we prove a compactness result for the solution set of the equations with respect to the variation of the underlying bounded spatial domain. Especially, we get a general existence theorem for the system in question with no restrictions on smoothness of the bounded spatial domain. Copyright © 2009 John Wiley & Sons, Ltd.     

Moment estimates for weak solutions to the Navier–Stokes equations in half‐space

We establish the moment estimates for a class of global weak solutions to the Navier–Stokes equations in the half‐space. Copyright © 2009 John Wiley & Sons, Ltd.     

Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

A remark on the decay of solutions to the 3‐D Navier–Stokes equations

In this paper we derive a decay rate of the L²‐norm of the solution to the 3‐D Navier–Stokes equations. Although the result which is proved by Fourier splitting method is well known, our method is new, concise and direct. Moreover, it turns out that the new method established here has a wide application on other equations. Copyright © 2007 John Wiley & Sons, Ltd.     

Global existence of weak solutions for the anisotropic compressible Stokes system

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Sequential stability of weak solutions in compressible self‐gravitating fluids and stationary problem

In this paper, we prove the sequential stability of weak solutions over time, in relation to the Navier–Stokes system of compressible self‐gravitating fluids in a three‐dimensional domain. As a byproduct, we show that there exists at least one non‐negative solution to the stationary problem in any bounded domain with a given mass for the adiabatic constant γ > 3 ∕ 2. In particular, for the spherically symmetric case, these conclusions still hold for γ > 4 ∕ 3 or γ = 4 ∕ 3 with a small mass. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of global weak solutions for a phase–field model of a vesicle moving into a viscous incompressible fluid

We consider in this article a model of vesicle moving into a viscous incompressible fluid. The vesicle is described through a phase–field equation and through a transport equation modeling the local incompressibility of its membrane. The equations for the fluid are the classical Navier–Stokes equations with a force resulting from the presence of the vesicle. Our main result states the existence of weak solutions for the corresponding system. The proof is based on compactness/monotonicity arguments. Copyright © 2013 John Wiley & Sons, Ltd.     

Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ₀, u ₀ satisfy the compatibility condition for some radially symmetric g ∈ L². The initial density ρ₀ needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Remarks on regularity criteria for axially symmetric weak solutions to the Navier–Stokes equations

We examine the conditional regularity of the solutions to the Navier–Stokes equations in the entire three‐dimensional space under the assumption that the data are axially symmetric. We show that if a radial or angular component of velocity satisfies a weighted Serrin condition, then the solution is regular. Copyright © 2011 John Wiley & Sons, Ltd.     

Global weak solutions to the Vlasov–Poisson–Fokker–Planck–Navier–Stokes system

We consider the compressible Vlasov–Poisson–Fokker–Planck–Navier–Stokes system in a three-dimensional bounded domain with nonhomogeneous Dirichlet boundary conditions. The system describes the evolution of charged particles ensemble dispersed in an isentropic fluid. For the adiabatic coefficient , we establish the global existence of weak solutions to this system with arbitrary large initial and boundary data.