An L2 finite element approximation for the incompressible Navier–Stokes equations

This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least-squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L²-approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second-order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L²-approximation to the stationary incompressible Navier–Stokes equations. The validity of L²-approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.     

On stationary solutions of the Vlasov-Poisson equations

We study the existence of stationary solutions of the Vlasov-Poisson equations for x ∈ ℝ ^N . We prove the absence of stationary solutions represented by Maxwell-Boltzmann distributions.     

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.     

An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

We analyze a finite-element approximation of the stationary incompressible Navier–Stokes equations in primitive variables. This approximation is based on the nonconforming P₁/P₀ element pair of Crouzeix/Raviart and a special upwind discretization of the convective term. An optimal error estimate in a discrete H¹-norm for the velocity and in the L²-norm for the pressure is proved. Some numerical results are presented. © 1996 John Wiley & Sons, Inc.     

An exterior boundary value problem in Minkowski space

In three‐dimensional Lorentz–Minkowski space ??³, we consider a spacelike plane Π and a round disc Ω over Π. In this article we seek the shapes of unbounded surfaces whose boundary is ? Ω and its mean curvature is a linear function of the distance to Π. These surfaces, called stationary surfaces, are solutions of a variational problem and governed by the Young–Laplace equation. In this sense, they generalize the surfaces with constant mean curvature in ??³. We shall describe all axially symmetric unbounded stationary surfaces with special attention in the case that the surface is asymptotic to Π at the infinity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and exponential stability in Lr‐spaces of stationary Navier–Stokes flows with prescribed flux in infinite cylindrical domains

We prove existence, uniqueness and exponential stability of stationary Navier–Stokes flows with prescribed flux in an unbounded cylinder of ?ⁿ,n?3, with several exits to infinity provided the total flux and external force are sufficiently small. The proofs are based on analytic semigroup theory, perturbation theory and L^r ? L^q‐estimates of a perturbation of the Stokes operator in L^q‐spaces. Copyright © 2006 John Wiley & Sons, Ltd.     

Global solutions of the Laplace equation with a nonlinear dynamical boundary condition

We study the boundedness and a priori bounds of global solutions of the problem Δu=0 in Ω×(0, T), (∂u/∂t) + (∂u/∂ν) = h(u) on ∂Ω×(0, T), where Ω is a bounded domain in ℝ^N, ν is the outer normal on ∂Ω and h is a superlinear function. As an application of our results we show the existence of sign-changing stationary solutions. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A remark on the pressure for the Navier–Stokes flows in 2‐D straight channel with an obstacle

Let T=?×(‐1,1) and &ℴ??² be a smoothly bounded open set, closure of which is contained in T. We consider the stationary Navier–Stokes flows in . In general, the pressure is determined up to a constant. Since Ω has two extremities, we want to know if we can choose the constant same. We study the behaviour of the pressure at the infinity in Ω and give a relation between the velocity and the pressure difference. Copyright © 2004 John Wiley & Sons, Ltd.     

Convergence of FFT‐based homogenization for strongly heterogeneous media

The FFT‐based homogenization method of Moulinec–Suquet has recently attracted attention because of its wide range of applicability and short computational time. In this article, we deduce an optimal a priori error estimate for the homogenization method of Moulinec–Suquet, which can be interpreted as a spectral collocation method. Such methods are well‐known to converge for sufficiently smooth coefficients. We extend this result to rough coefficients. More precisely, we prove convergence of the fields involved for Riemann‐integrable coercive coefficients without the need for an a priori regularization. We show that our L² estimates are optimal and extend to mildly nonlinear situations and L^p estimates for p in the vicinity of 2. The results carry over to the case of scalar elliptic and curl ? curl‐type equations, encountered, for instance, in stationary electromagnetism. Copyright © 2014 John Wiley & Sons, Ltd.     

A stability of the steady flow of compressible viscous fluid with respect to initial disturbance (v∞ ≠ 0)

We consider a compressible viscous fluid with the velocity at infinity equal to a strictly non‐zero constant vector in ?³. Under the assumptions on the smallness of the external force and velocity at infinity, Novotny–Padula (Math. Ann. 1997; 308 :439– 489) proved the existence and uniqueness of steady flow in the class of functions possessing some pointwise decay. In this paper, we study stability of the steady flow with respect to the initial disturbance. We proved that if H³‐norm of the initial disturbance is small enough, then the solution to the non‐stationary problem exists uniquely and globally in time, which satisfies a uniform estimate on prescribed velocity at infinity and converges to the steady flow in L_q‐norm for any number q? 2. Copyright © 2006 John Wiley & Sons, Ltd.     

Decay Estimates for Solutions of Linear Elasticity for Anisotropic Media

We analyse the time decay of solutions to the Cauchy problem for the linear hyperbolic system of elasticity for anisotropic media. As an example, we will consider media with hexagonal symmetry. First we derive decay estimates for special initial data using the method of stationary phase in several variables and degenerate phase function based on the Malgrange preparation theorem. Asymptotic expansions are given to prove the sharpness of the weaker time decay found for zinc and beryl than in the isotropic case. A method using Besov spaces leads to ℒ︁^p–ℒ︁^q-estimates.     

A Duality Method in Prediction Theory of Multivariate Stationary Sequences

Let W be an integrable positive Hermitian q × q–matrix valued function on the dual group of a discrete abelian group G such that W^–1 is integrable. Generalizing results of T. Nakazi [N] and of A. G. Miamee and M. Pourahmadi [MiP] for q = 1 we establish a correspondence between trigonometric approximation problems in L²(W) and certain approximation problems in L²(W^–1). The result is applied to prediction problems for q–variate stationary processes over G , inparticular, to the case G = ℤ.     

On the Nonexistence of Global Weak Solutions to the Navier–Stokes–Poisson Equations in ℝ N

In this paper we prove nonexistence of stationary weak solutions to the Euler–Poisson equations and the Navier–Stokes–Poisson equations in ? ^N , N ≥ 2, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler–Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier–Stokes–Poisson equations in ? ^N this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case.     

Spectral Analysis for Stationary Solutions of the Cahn–Hilliard Equation in ℝ d

We consider the spectrum associated with three types of bounded stationary solutions for the Cahn–Hilliard equation on ? ^d , d ≥ 2: radial solutions, saddle solutions (only for d = 2), and planar periodic solutions. In particular, we establish spectral instability for each type of solution. The important case of multiply periodic solutions does not fit into the framework of our approach, and we do not consider it here.     

On Two Iterative Least-Squares Finite Element Schemes for the Incompressible Navier–Stokes Problem

This paper is mainly devoted to a comparative study of two iterative least-squares finite element schemes for solving the stationary incompressible Navier–Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, we recast the Navier–Stokes problem into a first-order quasilinear velocity–vorticity–pressure system. Two Picard-type iterative least-squares finite element schemes are proposed to approximate the solution to the nonlinear first-order problem. In each iteration, we adopt the usual L ² least-squares scheme or a weighted L ² least-squares scheme to solve the corresponding Oseen problem and provide error estimates. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that the iterative L ² least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L ² least-squares scheme seems to be better than the iterative L ² least-squares scheme. Numerical simulations of the two-dimensional driven cavity flow are presented to demonstrate the effectiveness of the iterative least-squares finite element approach.     

Hyperuniform and rigid stable matchings

We study a stable partial matching τ of the d‐dimensional lattice with a stationary determinantal point process Ψ on R^d with intensity α>1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψ^τ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψ^τ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process Ψ, whose so‐called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behavior. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Furthermore, if Ψ is a Poisson process then Ψ^τ has an exponentially decreasing truncated pair correlation function.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

A weighted Lq‐approach to Oseen flow around a rotating body

We study time-periodic Oseen flows past a rotating body in ℝ³ proving weighted a priori estimates in L^q-spaces using Muckenhoupt weights. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional terms (ω ∧ x) ⋅ ∇ u and −ω ∧ u in the equation of momentum where ω denotes the angular velocity. Due to the asymmetry of Oseen flow and to describe its wake we use anisotropic Muckenhoupt weights, a weighted theory of Littlewood–Paley decomposition and of maximal operators as well as one-sided univariate weights, one-sided maximal operators and a new version of Jones' factorization theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

Stationary solution of the Navier-Stokes equations in a 2d bounded domain for incompressible flow with discontinuous density

We consider the boundary value problem for the stationary Navier-Stokes equations describing an inhomogeneous incompressible fluid in a two dimensional bounded domain. We show the existence of a weak solution with boundary values for the density prescribed in L^￥L^{\infty}.