Navier-Stokes Equations In Unbounded Domains Research Articles

Uniqueness of Solutions on the Whole Time Axis to the Navier-Stokes Equations in Unbounded Domains

We consider the uniqueness of bounded continuous L3, ∞-solutions on the whole time axis to the Navier-Stokes equations in 3-dimensional unbounded domains. Here, Lp, q denotes the scale of Lorentz spaces. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution in BC(ℝ; L3, ∞) within the class of solutions which have sufficiently small L∞(L3, ∞)-norm. In this paper, we discuss another type of uniqueness theorem for solutions in BC(ℝ; L3, ∞) using a smallness condition for one solution and a precompact range condition for the other one. The proof is based on the method of dual equations.

Existence of solutions on the whole time axis to the Navier-Stokes equations with precompact range in L 3

We present an existence theorem of mild solutions on the whole time axis to the Navier-Stokes equations in unbounded domains $${\Omega\subset \mathbb{R}^3}$$ having precompact range in $${L^3(\Omega)}$$ if the external force is small and has precompact range in some function space. In our forthcoming paper [7] we proved the uniqueness of such solutions.

Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains

We present a uniqueness theorem for backward asymptotically almost periodic solutions to the Navier-Stokes equations in $3$-dimensional unbounded domains. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution in $BC(-\infty,T;L^{3}_w)$ within the class of solutions which have sufficiently small $L^{\infty}( L^{3}_w)$-norm. In this paper, we show that a small backward asymptotically almost periodic solution in $BC(-\infty,T;L^{3}_w\cap L^{6,2})$ is unique within the class of all backward asymptotically almost periodic solutions in $BC(-\infty,T;L^{3}_w\cap L^{6,2})$.

Navier-stokes equations on unbounded domains with rough initial data

We consider the Navier-Stokes equations in unbounded domains Ω ⊆ ℝ n of uniform C 1,1-type. We construct mild solutions for initial values in certain extrapolation spaces associated to the Stokes operator on these domains. Here we rely on recent results due to Farwig, Kozono and Sohr, the fact that the Stokes operator has a bounded H ∞-calculus on such domains, and use a general form of Kato’s method. We also obtain information on the corresponding pressure term.

A simple proof of global solvability of 2-D Navier-Stokes equations in unbounded domains

In this paper we provide an elementary proof of the classical result of J.L. Lions and G. Prodi on the global unique solvability of two-dimensional Navier-Stokes equations that avoids compact embedding and strong convergence. The method applies to unbounded domains without special treatment. The essential idea is to utilize the local monotonicity of the sum of the Stokes operator and the inertia term. This method was first discovered in the context of stochastic Navier-Stokes equations by J.L. Menaldi and S.S. Sritharan.

A Coupled Legendre-Laguerre Spectral-Element Method for the Navier-Stokes Equations in Unbounded Domains

A coupled Legendre-Laguerre spectral element method is proposed for the Stokes and Navier-Stokes equations in unbounded domains. The method combines advantages of the high accuracy of the Laguerre-spectral method for unbounded domains and the geometric flexibility of the spectral-element method. Rigorous stability and error analysis for the Stokes problem is carried out. Numerical results indicate that the proposed method is very effective for some realistic flow problems in unbounded domains, such as flows passing a circular cylinder.

Strong solutions for the nonhomogeneous Navier-Stokes equations in unbounded domains

We show the existence of strong solutions for the nonhomogeneous Navier–Stokes equations in three-dimensional domains with boundary uniformly of class C3. Under suitable assumptions, uniqueness is also proved. Copyright © 2009 John Wiley & Sons, Ltd.

Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains

We investigate the flow of a magneto-micropolar fluid in an arbitrary unbounded domain on which the Poincare inequality holds. Assuming homogeneous boundary conditions and the external fields to be almost periodic in time we prove the existence of the uniform attractor by using the energy method [10] which we generalize to nonautonomous systems. We consider the problem in an abstract setting that allows to include also other hydrodynamical models. In particular, we extend the result of R. Rosa [12] from autonomous to nonautonomous Navier-Stokes equations in unbounded domains.

On stability of periodic solutions of the Navier-Stokes equations in unbounded domains

On stability of periodic solutions of the Navier-Stokes equations in unbounded domains

Periodic solutions of the Navier-Stokes equations in unbounded domains

We shall construct a periodic strong solution of the Navier-Stokes equations for the prescribed external force in unbounded domains.

Decay of the L?-norm of solutions of Navier-Stokes equations in unbounded domains

We show that for nn⩽ 4 the L∞-norm of weak solutions of the Navier-Stokes equations on ℝn with generalized energy inequality decays like\(\parallel u(t, \cdot )\parallel _\infty = O(t^{ - ({{n + 1)} \mathord{\left/ {\vphantom {{n + 1)} 2}} \right. \kern-\nulldelimiterspace} 2}} ),if(1 + | \cdot |)|u(0, \cdot )| \in L_1 \) and $$\int_{\mathbb{R}^n } {u(0,x)} dx = 0$$ .

L2Decay for the Compressible Navier-Stokes Equations in Unbounded Domains

(1993). L2 Decay for the Compressible Navier-Stokes Equations in Unbounded Domains. Communications in Partial Differential Equations: Vol. 18, No. 9-10, pp. 1445-1476.

Some results on the asymptotic behavior of Hopf weak solutions to the Navier-Stokes equations in unbounded domains

Some results on the asymptotic behavior of Hopf weak solutions to the Navier-Stokes equations in unbounded domains

Epochs of regularity for weak solutions of the Navier-Stokes equations in unbounded domains

On demontre l'existence de solutions faibles possedant les epoques de propriete de regularite dans des domaines non bornes autres que l'espace R 3 tout entier