Blowup of classical solutions to the pressureless Euler/isentropic Navier‐Stokes equations

In this paper, we will show the blowup of classical solutions to the Cauchy problem for the pressureless Euler/isentropic Navier‐Stokes equations in arbitrary dimensions under some restrictions on the initial data. Compared with the degenerate viscosities appeared in the recent work, we consider the constant viscosities, but we can remove the condition that the adiabatic exponent has a upper bound, which was a key constraint in the proof of the blow‐up result is based on the construction of some new differential inequalities.     

Stability of a colocated finite volume scheme for the Navier‐Stokes equations

The aim of this article is to describe a colocated finite volume approximation of the incompressible Navier‐Stokes equation and study its stability. One of the advantages of colocated finite volume space discretizations over staggered space discretizations is that all the variables share the same location; hence, the possibility to more easily use complex geometries and hierarchical decompositions of the unknowns. The time discretization used in the scheme studied here is a projection method. First, we give the full discretization of the incompressible Navier‐Stokes equations, then, we state the stability result and prove it following the methods of Marion and Temam. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Semi‐invariant solutions of the Navier‐Stokes equations

We present new exact solutions and reduced differential systems of the Navier‐Stokes equations of incompressible viscous fluid flow. We apply the method of semi‐invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier‐Stokes equations are, in fact, semi‐invariant and that the reduced differential systems we derive using semi‐invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi‐invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier‐Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier‐Stokes equations or find their approximations, and physical interpretation.     

Pullback attractors of 2D incompressible Navier‐Stokes‐Voight equations with delay

In this paper, the 2D Navier‐Stokes‐Voight equations with 3 delays in is considered. By using the Faedo‐Galerkin method, Lions‐Aubin lemma, and Arzelà‐Ascoli theorem, we establish the global well‐posedness of solutions and the existence of pullback attractors in H¹.     

The Crank‐Nicolson/Adams‐Bashforth scheme for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u₀ with div u₀ = 0, i.e., the time step τ satisfies: τ ≤ C₀ if u₀ ∈ H¹ ∩ L^∞; τ |log h| ≤ C₀ if u₀ ∈ H¹ for the mesh size h and some positive constant C₀. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012     

A Liouville theorem for the compressible Navier‐Stokes equations

This paper is concerned with 3‐dimensional steady compressible Navier‐Stokes equations. A Liouville‐type theorem is proved when some suitable conditions are satisfied.     

A remark on the Navier‐Stokes equations with the Coriolis force

In this short paper, the initial value problem for the Navier‐Stokes equations with the Coriolis force is investigated. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large‐scale phenomena. We prove that existence of uniform global large solutions to the Navier‐Stokes equations with the Coriolis force for a class of special initial data. The results obtained in this paper are different from the previous 2 types of results.     

Study of the mixed finite volume method for Stokes and Navier‐Stokes equations

We present finite volume schemes for Stokes and Navier‐Stokes equations. These schemes are based on the mixed finite volume introduced in (Droniou and Eymard, Numer Math 105 (2006), 35‐71), and can be applied to any type of grid (without “orthogonality” assumptions as for classical finite volume methods) and in any space dimension. We present numerical results on some irregular grids, and we prove, for both Stokes and Navier‐Stokes equations, the convergence of the scheme toward a solution of the continuous problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H¹‐norm and L²‐norm of the velocity and the L²‐norm of the pressure are derived. Moreover, on the basis of the optimal L²‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.     

Global stability for the fractional Navier‐Stokes equations in the Fourier‐Herz space

We consider global stability for the fractional incompressible Navier‐Stokes equations in a 3‐D critical Fourier‐Herz space. By introducing a weighted norm space and using Fourier localization technique, the stability of mild solutions with small initial perturbation is established. With the Friedrichs method, the stability of weak solutions is proved under arbitrary large initial perturbation.     

Discretization of the Navier‐Stokes equations with slip boundary condition

We propose and analyze a two‐level method of discretizing the nonlinear Navier‐Stokes equations with slip boundary condition. The slip boundary condition is appropriate for problems that involve free boundaries, flows past chemically reacting walls, and other examples where the usual no‐slip condition u = 0 is not valid. The two‐level algorithm consists of solving a small nonlinear system of equations on the coarse mesh and then using that solution to solve a larger linear system on the fine mesh. The two‐level method exploits the quadratic nonlinearity in the Navier‐Stokes equations. Our error estimates show that it has optimal order accuracy, provided that the best approximation to the true solution in the velocity and pressure spaces is bounded above by the data. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 26–42, 2001     

Semi‐implicit schemes for transient Navier–Stokes equations and eddy viscosity models

This study presents two computational schemes for the numerical approximation of solutions to eddy viscosity models as well as transient Navier–Stokes equations. The eddy viscosity model is one example of a class of Large Eddy Simulation models, which are used to simulate turbulent flow. The first approximation scheme is a first order single step method that treats the nonlinear term using a semi‐implicit discretization. The second scheme employs a two step approach that applies a Crank–Nicolson method for the nonlinear term while also retaining the semi‐implicit treatment used in the first scheme. A finite element approximation is used in the spatial discretization of the partial differential equations. The convergence analysis for both schemes is discussed in detail, and numerical results are given for two test problems one of which is the two dimensional flow around a cylinder. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Existence of global weak solutions for Navier‐Stokes‐Poisson equations with quantum effect and convergence to incompressible Navier‐Stokes equations

In this paper, we consider a three dimensional quantum Navier‐Stokes‐Poisson equations. Existence of global weak solutions is obtained, and convergence toward the classical solution of the incompressible Navier‐Stokes equation is rigorously proven for well prepared initial data. Furthermore, the associated convergence rates are also obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

A two‐level method in space and time for the Navier‐Stokes equations

A two‐level method in space and time for the time‐dependent Navier‐Stokes equations is considered in this article. The approximate solution u_M∈H_M is decomposed into the large eddy component v∈H_m(m < M) and the small eddy component w∈H. We obtain the large eddy component v by solving a standard Galerkin equation in a coarse‐level subspace H_m with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H with a time step length pk, where p is a positive integer. The analysis shows that our two‐level scheme has long‐time stability and can reach the same accuracy as the standard Galerkin method in fine‐level subspace H_M for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Least‐squares mixed finite element methods for the incompressible Navier‐Stokes equations

Least‐squares mixed finite element schemes are formulated to solve the evolutionary Navier‐Stokes equations and the convergence is analyzed. We recast the Navier‐Stokes equations as a first‐order system by introducing a vorticity flux variable, and show that a least‐squares principle based on L² norms applied to this system yields optimal discretization error estimates. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 441–453, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10015     

Global regularity criterion for the Navier‐Stokes equations based on the direction of vorticity

We study the regularity criterion for the Navier‐Stokes equations and show that the (β₁,β₂,β₃)‐Hölder continuity assumption in (x₁,x₂,x₃) on the direction of the vorticity ensures the regularity of the solution. This may be viewed as an extension of many previous results, since some of the β_i can be arbitrarily small.     

Shape reconstruction for the time‐dependent Navier‐Stokes flow

This article deals with the shape reconstruction of a bounded domain with a viscous incompressible fluid driven by the time‐dependent Navier‐Stokes equations. For the approximate solution of the ill‐posed and nonlinear problem we propose a regularized Newton method. A theoretical foundation for the Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of the domain derivative. Numerical examples indicate the feasibility of our method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Orthogonal spline collocation methods for the stream function‐vorticity formulation of the Navier–Stokes equations

An orthogonal spline collocation (OSC) spatial discretization is proposed for the solution of the fully coupled stream function‐vorticity formulation of the Navier–Stokes equations in two dimensions. For the time‐stepping, a three‐level leapfrog scheme is employed. This method is algebraically linear, and, at each time step, gives rise to a system of linear equations of the form arising in the OSC approximation of the biharmonic Dirichlet problem and can be solved by a fast direct method. Error estimates in the H^l–norm in space, l = 1,2, are derived for the semi‐discrete method and the fully‐discrete leapfrog scheme which is also shown to be second order accurate in time. Numerical results are presented which confirm the theoretical analysis and exhibit superconvergence phenomena, which provide superconvergent approximations to the components of the velocity. © John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008