The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel

In this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three-dimensional Navier–Stokes equations in the cylinder while the asymptotic expansion method is used to solving the linearly elastic shell equations. Specifically, in order to discrete the Navier–Stokes equations, the dimensional splitting strategy is constructed under the cylinder coordinate system. The spectral method is adopted along the rotation direction while the finite element method is used along the other directions. By using the above strategy, we get a series of two-dimensional-three-components (2D-3C) fluid problems. By introduce the S-coordinate system in E³ and employ the thickness of blood vessel wall as the expanding parameter, the asymptotic expansion method can be established to approximate the solution of the 3D elastic problem. The interface contact conditions can be treated exactly based on the knowledge of tensor analysis. Finally, numerical test shows that our method is reasonable.     

On some properties of the Navier–Stokes system of equations

We discuss the initial and boundary value problems for the system of dimensionless Navier–Stokes equations describing the dynamics of a viscous incompressible fluid using the method of characteristics and the geometric method developed by the authors. Some properties of the formulation of these problems are considered. We study the effect of the Reynolds number on the flow of a viscous fluid near the surface of a body.     

A mixed problem for the steady Navier–Stokes equations

We consider a mixed boundary problem for the Navier–Stokes equations in a bounded Lipschitz two-dimensional domain: we assign a Dirichlet condition on the curve portion of the boundary and a slip zero condition on its straight portion. We prove that the problem has a solution provided the boundary datum and the body force belong to a Lebesgue’s space and to the Hardy space respectively.     

Liouville theorems for the Navier–Stokes equations and applications

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in R ⁿ × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].     

On a model for the Navier–Stokes equations using magnetization variables

It is known that in a classical setting, the Navier–Stokes equations can be reformulated in terms of so-called magnetization variables w that satisfy

(1)${?}_{t}w+(\mathbb{P}w?\mathrm{?})w+{(\mathrm{?}\mathbb{P}w)}^{?}w?\mathrm{\Delta }w=0,$

and relate to the velocity u via a Leray projection $u=\mathbb{P}w$. We will prove the equivalence of these formulations in the setting of weak solutions that are also in $L^{\infty }(0,T;H^{1/2})\cap L^2(0,T;H^{3/2})$ on the 3-dimensional torus.Our main focus is the proof of global well-posedness in $H^{1/2}$ for a new variant of (1), where $\mathbb{P}w$ is replaced by w in the second nonlinear term:

(2)${?}_{t}w+(\mathbb{P}w?\mathrm{?})w+\frac{1}{2}\mathrm{?}|w{|}^2?\mathrm{\Delta }w=0.$

This is based on a maximum principle, analogous to a similar property of the Burgers equations.     

Navier–Stokes equations,the algebraic aspect

Theoretical and Mathematical Physics - We present an analysis of the Navier–Stokes equations in the framework of an algebraic approach to systems of partial differential equations (the formal...     

A logarithmically improved regularity criterion for the Navier–Stokes equations

In this note we prove a logarithmically improved regularity criterion in terms of the Besov space norm for the Navier–Stokes equations. The result shows that if a mild solution u satisfies ${\int_{0}^{T}\frac{\|u (t,\cdot)\|_{{\dot{B}}_{\infty,\infty}^{-r}}^{\frac{2}{1-r}}}{1+\ln(e+\| u(t,\cdot)\|_{H^{s}})}\text{d}t < \infty}$ for some 0?≤ r?<?1 and ${s\geq\frac{n}{2}-1}$ , then u is regular at t?=?T.     

A convergent FEM-DG method for the compressible Navier–Stokes equations

This paper presents a new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix–Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax–Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.     

A steady Navier–Stokes model for compressible fluid with partially strong solutions

In this Note, we prove the existence of a partially strong solution to the steady Navier–Stokes equations for viscous barotropic compressible fluids, in a bounded simply connected domain of ${\mathbb{R}}^3$ with the prescribed generalized impermeability conditions ${\mathbf{curl}}^k\mathbf{u}?\mathbf{n}=0$, $k=0,1,2$ on the boundary. We call the solution “partially strong” because only the divergence-free part of the velocity field and the associated effective pressure have regularity typical for strong solution, while the density and the gradient part of the velocity have regularity typical for weak solution.     

A fluid–structure interaction model with interior damping and delay in the structure

A coupled system of partial differential equations modeling the interaction of a fluid and a structure with delay in the feedback is studied. The model describes the dynamics of an elastic body immersed in a fluid that is contained in a vessel, whose boundary is made of a solid wall. The fluid component is modeled by the linearized Navier-Stokes equation, while the solid component is given by the wave equation neglecting transverse elastic force. Spectral properties and exponential or strong stability of the interaction model under appropriate conditions on the damping factor, delay factor and the delay parameter are established using a generalized Lax-Milgram method.     

Unique solvability for the density-dependent Navier–Stokes equations

In this paper we consider the incompressible Navier–Stokes equations with a density-dependent viscosity in a bounded domain $\Omega$ of ${\mathbf{R}}^n(n=2,3)$. We prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition. This condition is also necessary for a very general initial data. Moreover, we provide a blow-up criterion for the regularity of the strong solution. For these results, the initial density need not be strictly positive. It may vanish in an open subset of $\Omega$.     

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.