On the asymptotic stability of steady solutions of the Navier–Stokes equations in unbounded domains

We consider the problem of the asymptotic behaviour in the L²‐norm of solutions of the Navier–Stokes equations. We consider perturbations to the rest state and to stationary motions. In both cases we study the initial‐boundary value problem in unbounded domains with non‐compact boundary. In particular, we deal with domains with varying and possibly divergent exits to infinity and aperture domains. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonlinear artificial boundary conditions with pointwise error estimates for the exterior three dimensional Navier–Stokes problem

On a three–dimensional exterior domain Ω we consider the Dirichlet problem for the stationary Navier–Stokes system. We construct an approximation problem on the domain Ω_R, which is the intersection of Ω with a sufficiently large ball, while we create nonlinear, but local artificial boundary conditions on the truncation boundary. We prove existence and uniqueness of the solutions to the approximating problem together with asymptotically precise pointwise error estimates as R tends to infinity.     

Asymptotic properties of the steady fall of a body in viscous fluids

The paper deals with the properties of the steady fall of a body in linear fluids (Oseen and Stokes cases) and in nonlinear fluids (Navier–Stokes fluids) in L^q structure. Copyright © 2004 John Wiley & Sons, Ltd.     

A remark on free boundary problem of 1‐D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we prove the existence and uniqueness of the weak solution of the one‐dimensional compressible Navier–Stokes equations with density‐dependent viscosity µ(ρ)=ρ^θ with θ∈(0, γ?2], γ>1. The initial data are a perturbation of a corresponding steady solution and continuously contact with vacuum on the free boundary. The obtained results apply for the one‐dimensional Siant–Venant model of shallow water and generalize ones in (Arch. Rational Mech. Anal. 2006; 182: 223–253). Copyright © 2009 John Wiley & Sons, Ltd.     

The coupling of boundary integral and finite element methods for the nonstationary exterior flow

In this article, we represent a new numerical method for solving the nonstationary Navier–Stokes equations in an unbounded domain. The technique consists of coupling the boundary integral and the finite element method. The variational formulation and the well-posedness of the coupling method are obtained. The convergence and optimal error estimates for the approximate solution are provided. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 549–565, 1998     

A regularizing effect of radiation in the equations of fluid dynamics

We investigate the properties of a class of variational solutions to the equations of fluid dynamics when radiation effects are taken into account. The main aim is to prove weak sequential stability of the solution set under certain hypotheses imposed on the pressure, viscosity, and heat conductivity. Copyright © 2004 John Wiley & Sons, Ltd.     

Energy stable numerical scheme for the viscous Cahn–Hilliard–Navier–Stokes equations with moving contact line

In the present article we study the numerics of the viscous Cahn–Hilliard–Navier–Stokes model, endowed with dynamic boundary conditions which allow us to take into account the interaction between the fluids interface and the moving walls of the physical domain. In what follows, we propose an energy stable temporal scheme for the problem and we prove the stability and the unconditional solvability of the discretization proposed. We also propose a fully discrete scheme for which we prove the stability and the unconditional solvability. Numerical simulations are presented to illustrate the theoretical results.     

Estimates for the asymptotic expansion of a viscous fluid satisfying Navier's law on a rugous boundary

In a previous paper, we have studied the asymptotic behavior of a viscous fluid satisfying Navier's law on a periodic rugous boundary of period ε and amplitude δ _ε , with δ _ε / ε tending to zero. In the critical size, δ _ε ～ ε ^3/2, in order to obtain a strong approximation of the velocity and the pressure it is necessary to consider a boundary layer term in the corresponding ansatz. The purpose of this paper is to estimate the approximation given by this ansatz. Copyright © 2011 John Wiley & Sons, Ltd.     

Strong solutions for the nonhomogeneous Navier–Stokes equations in unbounded domains

We show the existence of strong solutions for the nonhomogeneous Navier–Stokes equations in three‐dimensional domains with boundary uniformly of class C³. Under suitable assumptions, uniqueness is also proved. Copyright © 2009 John Wiley & Sons, Ltd.     

Asymptotic behavior of exact solutions for the Cauchy problem to the 3D cylindrically symmetric Navier–Stokes equations

In this paper, we establish exact solutions of the Cauchy problem for the 3D cylindrically symmetric incompressible Navier–Stokes equations and further study the global existence and asymptotic behavior of solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Global C∞‐solutions to 1D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries. The initial density ρ₀∈W^1,2n is bounded below away from zero and the initial velocity u₀∈L²ⁿ. The viscosity coefficient µ is proportional to ρ^θ with 0<θ?1, where ρis the density. The existence and uniqueness of global solutions in Hⁱ([0,1])(i = 1,2,4) have been established in (J. Math. Phys. 2009; 50 :023101; Meth. Appl. Anal. 2005; 12 :239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7 :373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries when the initial data ρ₀ and u₀ are smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.     

Stabilized finite element methods for a blood flow model of arteriosclerosis

In this article, a blood flow model of arteriosclerosis, which is governed by the incompressible Navier–Stokes equations with nonlinear slip boundary conditions, is constructed and analyzed. By means of suitable numerical integration approximation for the nonlinear boundary term in this model, a discrete variational inequality for the model based on stabilized finite elements is proposed. Optimal order error estimates are obtained. Finally, numerical examples are shown to demonstrate the validity of the theoretical analysis and the efficiency of the presented methods. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2063–2079, 2015     

Nonstationary flow for the Navier–Stokes equations in a cylindrical pipe

In cylindrical domain, we consider the nonstationary flow with prescribed inflow and outflow, modelled with Navier–Stokes equations under the slip boundary conditions. Using smallness of some derivatives of inflow function, external force and initial velocity of the flow, but with no smallness restrictions on the inflow, initial velocity neither force, we prove existence of solutions in . Copyright © 2012 John Wiley & Sons, Ltd.     

On non‐stationary viscous incompressible flow through a cascade of profiles

The paper deals with theoretical analysis of non‐stationary incompressible flow through a cascade of profiles. The initial‐boundary value problem for the Navier–Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 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.     

Analysis of nonlocal model of compressible fluid in 1‐D

We study a nonlocal modification of the compressible Navier–Stokes equations in mono‐dimensional case with a boundary condition characteristic for the free boundaries problem. From the formal point of view, our system is an intermediate between the Euler and Navier–Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient, we obtain the local and global existence of solutions. Particularly, we show the uniform in time bound on the density of fluid. Copyright © 2010 John Wiley & Sons, Ltd.     

A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for steady compressible viscous flow

In this paper, the authors consider the Navier–Stokes equations for steady compressible viscous flow in three-dimensional cylindrical domain. A differential inequality for appropriate energy associated with the solutions of the Navier–Stokes isentropic flow in semi-infinite pipe is derived, from which the authors show a Phragmén–Lindelöf alternative result, i.e. the solutions for steady compressible viscous N–S flow problem either grow or decay exponentially as the distance from the entry section tends to infinity. In the decay case, the authors indicate how to bound explicitly the total energy in terms of data.     

Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition

In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition. We establish the local well‐posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established, and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition. Copyright © 2017 John Wiley & Sons, Ltd.     

Global attractor for the Navier–Stokes equations with the heat convection in a cylindrical pipe

Global existence of regular solutions to the Navier–Stokes equations coupled with the heat convection in a cylindrical pipe has already been shown. In this paper, we prove the existence of the global attractor to the equations and convergence of their solutions to a stationary one. Copyright © 2012 John Wiley & Sons, Ltd.