Navier-Stokes Equations In Exterior Domains Research Articles

The Boundary Layer Equations and a Dimensional Split Method for Navier-Stokes Equations in Exterior Domain of a Spheroid and Ellipsoid

In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.

Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data

We investigate the nonstationary Navier-Stokes equations for an exterior domain Ω ⊂ R 3 in a solution class L s ( 0 , T ; L q ( Ω ) ) of very low regularity in space and time, satisfying Serrin's condition 2 s + 3 q = 1 but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by u | ∂ Ω = g ∈ L s ( 0 , T ; W - 1 / q , q ( ∂ Ω ) ) , and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition k = ÷ u ∈ L s ( 0 , T ; L r ( Ω ) ) , where 1 3 + 1 q = 1 r .

Stability of periodic solutions of Navier-Stokes equations in a three-dimensional exterior domain

We study the problem of stability of periodic solutions of Navier-Stokes equations in an exterior domain with a given constant velocity vector at infinity. We study spectral properties of the monodromy operator, using its connection with a certain Wiener-Hopf operator with an operator-valued symbol.

Decay estimates for strong solutions of the Navier-Stokes equations in exterior domains

Based on theL p−Lq-estimates for the Stokes semigroup on exterior domains, proven by Iwashita and refined by Maremonti-Solonnikov and Shibata, the precise time-asymptotic behaviour of solutions of the Navier-Stokes equations is derived. The estimates are improved, if some additional information on the initial value is available. The results apply also to weak solutions in three and four dimensions.

Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains

For an arbitrary domain Ω ⊂ ℝn, n=2,3, Ω ≠ ℝn, we prove the existence of weak periodic solutions to the Navier-Stokes equations and of regular solutions if the data are small or satisfy certain symmetry conditions. We also show that the periodic regular solutions are stable. Bibliography: 38 titles.

On a larger class of stable solutions to the Navier-Stokes equations in exterior domains

where w = w(x) = (w1(x), · · · , wn(x)) and π = π(x) denote the unknown velocity vector and the unknown pressure of the fluid at point x ∈ Ω, respectively, while F = F (x) = (F i j (x))i,j=1,··· ,n is the given tensor with div F = ( ∑n j=1 ∂F 1 j ∂xj , · · · ,∑nj=1 ∂F n j ∂xj ) denoting the external force. In our previous paper [23], we established the theory of existence, uniqueness and regularity for solutions to (N − S0) in the class w ∈ Ln,∞(Ω),

Global estimates in weighted spaces of weak solutions of the Navier-Stokes equations in exterior domains

which describes the unknown velocity u = (ul, u2, u3) and pressure p of some fluid for x ~ ~ and all times t > 0, while the external force f and the initial velocity Uo are given. It is well known that (1.1) has global weak solutions, but it is an open problem whether a weak solution is unique, globally strong or even classical. Therefore there is a great interest to obtain additional global properties of weak solutions besides the fact that the energy norms II u tl2,oo and II Vu 112.2 are finite; here II " IIq,s denotes the norm in the space /z ~ (L q) = L ~ (0, oo; L q (Q)), 1 -<_ q, s =< ~. In this paper we start - under suitable assumptions on f and u o - with a weak solution

Weighted energy inequalities for the navier-stokes equations in exterior domains

Consider the nonstationary Navier-Stokes equations on an exterior domain in IR3. Under suitable assumptions on the initial value and the external force we prove the existence of a global, suitable weak solution satisfying a weighted energy inequality with weight functions |x|α,0≤α≤1. This weak solution is constructed using the Yosida approximation procedure, Lg-theory and localized energy inequalities as int,roduced by Caffarelli, Kohn and Nirenberg [3].

Solutions of the stationary and nonstationary Navier-Stokes equations in exterior domains

It is shown that a nonstationary exterior Navier-Stokes flow tends to a small stationary flow in L 2 like t −3/4 as t→∞

Asymptotic behaviour inL r for weak solutions of the Navier-Stokes equations in exterior domains

Asymptotic behaviour inL r for weak solutions of the Navier-Stokes equations in exterior domains

Density properties for solenoidal vector fields, with applications to the Navier-Stokes equations in exterior domains

Density properties for solenoidal vector fields, with applications to the Navier-Stokes equations in exterior domains

Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains

We apply an abstract perturbation theorem to derive global in time L q estimates for the Cauchy problem and L q − L s estimates for the nonstationary Stokes equations in exterior domains. This will be applied to obtain various new global in time estimates for weak solutions of the Navier-Stokes equations.

On energy inequality, smoothness and large time behavior inL 2 for weak solutions of the Navier-Stokes equations in exterior domains

Soit Ω un domaine exterieur de R n (n=3,4) a frontiere lisse ∂Ω. Sur le cylindre d'espace temps Ω×[0,∞) on considere le probleme de Navier-Stokes non stationnaire: u t −Δu+u•⊇u+⊇p=f dans Ω×(0,∞), ⊇•u=0 dans Ω×(0,∞), u=0 sur ∂Ω×(0,∞), u(x,0)=a(x) avec des donnees prescrites f et a

Partial regularity of a generalized solution to the Navier-Stokes equations in exterior domain

In this note, we prove two regularity theorems for solutions to the Navier-Stokes equations of an I.B.V.P. in exterior domains. Namely, we prove that the setS of the singular points of a solution, if not empty, has at most 1-Hausdorff measureH 1(S)=0. Moreover, the setS is enclosed in a sphere of rayR for anyt>0. These results are obtained as corollaries to the partial regularity results furnished in [2].

Monotonic decreasing and asymptotic behavior of the kinetic energy for weak solutions of the Navier-Stokes equations in exterior domains

Existence of weak solutions of the three-dimensional Navier-Stokes problem was first proved by J. Leray in the case of the Cauchy problem, cf. Leray (1934). As is well known, these solutions are important in that they are the only solutions which, so far, are known to exist for all times, without restriction on the data. Unfortunately, however, the question of whether they are classical, in an ordinary sense, is still open, even though partial conclusions regarding regularity are avilable: Leray (1934), Scheffer (1976, 1980), Caffarelli, Kohn, & Nirenberg (1982). Subsequently, E. Hopf (1951), using a different technique, constructed weak solutions for a general initial-boundary value problem. However, hopf’s solution (even for the Cauchy problem) has weaker properties than Leray’s solution. Among others, we refer to the “energy inequality” for the velocity field u(x, t), i.e. $$ ntimits_mega {{u^2}} (x,t)dx - ntimits_mega {{u^2}(x,s)sx} eqq - 2ntimits_s^t {ntimits_mega {abla u:abla udxdau } } $$ (I) for s = 0, for almost all s > 0 and for all t ≧ s.

Navier-Stokes equations in exterior domains

We prove existence and regularity of solutions to the nonstationary Navier-Stokes equations in exterior domains.