Analysis of the grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable finite elements

This paper studies inf-sup stable finite element discretizations of the evolutionary Navier–Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order \(\mathcal O(h^{2})\) in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-in-time case and the fully discrete scheme with the backward Euler method as time integrator are analyzed.     

Navier-Stokes equations in 3D thin domains with Navier friction boundary condition

In this article we study the 3D Navier-Stokes equations with Navier friction boundary condition in thin domains. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. We generalize the techniques developed to study the 3D Navier-Stokes equations in thin domains, see [G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503-568; G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains II: Global regularity of spatially periodic conditions, in: Nonlinear Partial Differential Equations and Their Application, College de France Seminar, vol. XI, Longman, Harlow, 1994, pp. 205-247; R. Temam, M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996) 499-546; R. Temam, M. Ziane, Navier-Stokes equations in thin spherical shells, in: Optimization Methods in Partial Differential Equations, in: Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1996, pp. 281-314], to the Navier friction boundary condition by introducing a new average operator Mε in the thin direction according to the spectral decomposition of the Stokes operator Aε. Our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the Stokes operator Aε with Navier friction boundary condition.     

For the stationary compressible viscous Navier-Stokes equations with no-slip condition on a convex polygon

Our concern is with existence and regularity of the stationary compressible viscous Navier-Stokes equations with no-slip condition on convex polygonal domains. Note that [u,p]=[0,c], c a constant, is the eigenpair for the singular value λ=1 of the Stokes problem on the convex sector. It is shown that, except the pair [0,c], the leading order of the corner singularities for the nonlinear equations is the same as that of the Stokes problem. We split the leading corner singularity from the solution and show an increased regularity for the remainder. As a consequence the pressure solution changes the sign at the convex corner and its derivatives blow up.     

Spectral approximation of the periodic-nonperiodic Navier-Stokes equations

Summary In order to approximate the Navier-Stokes equations with periodic boundary conditions in two directions and a no-slip boundary condition in the third direction by spectral methods, we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure. The compatibility between these two spaces is checked via an infsup condition. We analyze a spectral and a collocation pseudo-spectral method for the Stokes problem and a collocation pseudo-spectral method for the Navier-Stokes equations. We derive error bounds of spectral type, i.e. which behave likeM ^– whereM depends on the number of degrees of freedom of the method and represents the regularity of the data.     

A remark on regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with the free boundary. The viscosity coefficient μ is proportional to ρθ with θ>0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H¹([0,1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.     

Asymptotics of Stationary Navier Stokes Equations in Higher Dimensions

We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.     

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69 (1990), 237–254.     

A Two-Grid Finite Element Method for Time-Dependent Incompressible Navier-Stokes Equations with Non-Smooth Initial Data

We analyze here, a two-grid finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size $H$ and solving a Stokes problem on a fine grid of size $h, h <相似文献   

Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

The Incompressible Navier-Stokes Equations in Vacuum

We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier-Stokes equations supplemented with H¹ initial velocity and only bounded nonnegative density. In contrast to all the previous works on those topics, we do not require regularity or a positive lower bound for the initial density or compatibility conditions for the initial velocity and still obtain unique solutions. Those solutions are global in the two-dimensional case for general data, and in the three-dimensional case if the velocity satisfies a suitable scaling-invariant smallness condition. As a straightforward application, we provide a complete answer to Lions' question in his 1996 book Mathematical Topics in Fluid Mechanics, vol. 1, Incompressible Models, concerning the evolution of a drop of incompressible viscous fluid in the vacuum. © 2018 Wiley Periodicals, Inc.     

Strong solutions of the Navier-Stokes equations in aperture domains

We consider the nonstationary Navier-Stokes equations in an aperture domain Ω⊂R³ consisting of two halfspaces separated by a wall, but connected by a hole in this wall. In this special domain one has to impose an auxiliary condition to single out a unique solution. This can be done by prescribing either the flux through the hole or the pressure drop between the two halfspaces. We construct suitable Stokes operators for both of the auxiliary conditions and show that they generate holomorphic semigroups. Then we prove the existence and uniqueness of solutions as well as a maximal regularity estimate for the Stokes equations subject to one of the auxiliary conditions. For the corresponding Navier-Stokes equations we prove existence and uniqueness of local in time solutions.

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Sunto In questo lavoro consideriamo le equazioni di Navier-Stokes non stazionarie in un dominio con un’apertura, che consiste di due semispazi separati da una parete, ma collegati da un’apertura in quest’ultima. In questo dominio particolare è necessario imporre, per avere un’unica soluzione, una opportuna condizione ausiliaria. Questo può essere fatto sia assegnando il flusso attraverso l’apertura sia prescrivendo il salto di pressione tra i due semispazi. Qui costruiamo degli operatori di Stokes opportuni per ambedue i tipi di condizioni ausiliarie e mostriamo come essi generino semigruppi olomorfi. Dimostriamo, quindi, esistenza e unicità di soluzioni, assieme ad una stima di massima regolarità per le equazioni di Stokes soggette ad una delle condizioni ausiliarie. Per le corrispondenti equazioni di Navier-Stokes, dimostriamo esistenza e unicità di soluzioni locali nel tempo.

     

Time stepping procedures for the non-stationary Stokes equations

We consider first and second-order implicit time stepping procedures for the non-stationary Stokes equations in bounded domains of ?³. Using energy estimates we prove the optimal convergence properties in the Sobolev spaces H^m(G)(m = 0, 1, 2) uniformly in time, provided that the Stokes solution has a certain degree of regularity. Here in the case of the second-order scheme (method of Crank–Nicholson) the Stokes solution has to satisfy a non-local compatibility condition at the initial time t = O, which can be satisfied by a special initial construction.     

The Strong Solution of a Class of Generalized Navier-Stokes Equations

We study initial boundary value (lBV) problem for a class of generalized Navier-Stokes equations in L^q([0, T); L^p(Ω)). Our main tools are regularity of analytic semigroup by Stokes operator and space-time estimates. As an application we can obtain some classical results of the Navier-Stokes equations such as global classical solution of 2-dimensional Navier-Stokes equation etc.     

Global strong solutions of Navier-Stokes equations with interface boundary in three-dimensional thin domains

In this article, we study the spectrum of the Stokes operator in a 3D two layer domain with interface, obtain the asymptotic estimates on the spectrum of the Stokes operator as thickness ε goes to zero. Based on the spectral decomposition of the Stokes operator, a new average-like operator is introduced and applied to the study of Navier-Stokes equation in the two layer thin domains under interface boundary condition. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. This article is a continuation of our study on the Stokes operator under Navier friction boundary condition. Due to the viscosity distinction between the two layers, the Stokes operator displays radically different spectral structure from that under Navier friction boundary condition, then causes great difficulty to the analysis.     

A note on the regularity criteria for the Navier-Stokes equations

We consider the regularity problem for 3D Navier-Stokes equations in a bounded domain with smooth boundary. A new sufficient condition which guarantees the regularity of weak solutions on the quotient ∇p/(1+|u|^δ₁+|∇u|^δ₂) for the Navier-Stokes equations is established.     

On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations with vacuum

This paper concerns the Cauchy problem of the barotropic compressible Navier–Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier–Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.     

Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of(x, t) in the region R3×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.     

Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.     

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.