Stokes Equations In Unbounded Domains Research Articles

On uniqueness of mild L3,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{3,\\infty }$$\\end{document}-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains

This paper is concerned with the uniqueness of bounded continuous L3,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{3,\\infty }$$\\end{document}-solutions on the whole time axis R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R$$\\end{document} or the half-line (-∞,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-\\infty ,T)$$\\end{document} to the Navier–Stokes equations in 3-dimensional unbounded domains. When Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is an unbounded domain, it is known that a small solution in BC(R;L3,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$BC(\\mathbb R;L^{3,\\infty })$$\\end{document} is unique within the class of solutions which have sufficiently small L∞(R;L3,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }(\\mathbb R; L^{3,\\infty })$$\\end{document}-norm; i.e., if two solutions u and v exist for the same force f, bothu and v are small, then the two solutions coincide. There is another type of uniqueness theorem. Farwig et al. (Commun Partial Differ Equ 40:1884–1904, 2015) showed that if two solutions u and v exist for the same force f, u is small and if v has a precompact range R(v):={v(t);-∞<t<T}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {R}}(v):=\\{v(t); -\\infty<t<T\\}$$\\end{document} in L3,∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{3,\\infty },$$\\end{document} then the two solutions coincide. However, there exist many solutions which do not have precompact range. In this paper, instead of the precompact range condition, by assuming some decay property of v(x, t) with respect to the spatial variable x near t=-∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=-\\infty ,$$\\end{document} we show a modified version of the above-mentioned uniqueness theorem. As a by-product, in the half-spaceR+3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R^3_+,$$\\end{document} we obtain a non-existence result of backward self-similar L3,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{3,\\infty }$$\\end{document}-solutions sufficiently close to some homogeneous function Q(x/|x|)/|x| in a certain sense.

A remark on the uniqueness of Kozono–Nakao’s mild $$L^3$$-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains

A remark on the uniqueness of Kozono–Nakao’s mild $$L^3$$-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains

Pullback Attractors in $$V_g$$ for Non-autonomous 2D g-Navier–Stokes Equations in Unbounded Domains

Pullback Attractors in $$V_g$$ for Non-autonomous 2D g-Navier–Stokes Equations in Unbounded Domains

Brezis–Gallouet–Wainger Type Inequalities and Blow-Up Criteria for Navier–Stokes Equations in Unbounded Domains

We shall find the weakest norm that satisfies the Brezis–Gallouet–Wainger type inequality, under some conditions. As an application of the Brezis–Gallouet–Wainger type inequality, we shall establish Beale–Kato–Majda type blow-up criteria of smooth solutions to the 3-D Navier–Stokes equations in unbounded domains.

Very weak solutions of the stationary Stokes equations in unbounded domains of half space type

We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as $\mathbb R^n_+$, bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of $\mathbb R^n_+$, and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous Sobolev spaces. In addition to very weak solutions we also construct corresponding pressure functions in negative homogeneous Sobolev spaces.

The existence of energy solutions to 2-dimensional non-Lipschitz stochastic Navier–Stokes equations in unbounded domains

In this paper we consider the existence and uniqueness of weak energy solutions to a stochastic 2-dimensional non-Lipschitz Navier–Stokes equation perturbed by the cylindrical Wiener process W ( t ) in a bounded or unbounded domain D with the smooth boundary ∂ D or D = R 2 : d X ( t ) = [ − ν A X ( t ) + B ( X ( t ) ) ] d t + f ( t , X ( t ) ) d t + g ( t , X ( t ) ) d W ( t ) , where A is the Stokes operator and f, g satisfy the non-Lipschitz condition.

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

We present a uniqueness theorem for almost periodic-in-time solutions to the Navier–Stokes equations in 3-dimensional unbounded domains. Thus far, uniqueness of almost periodic-in-time solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, is known only for a small almost periodic-in-time solution in \({BC(\mathbb {R};L^{3}_w)}\) within the class of solutions that have sufficiently small \({L^{\infty}(L^{3}_w)}\) -norm. In this paper, we show that a small almost periodic-in-time solution in \({BC(\mathbb {R};L^{3}_w\cap L^{6,2})}\) is unique within the class of all almost periodic-in-time solutions in \({BC(\mathbb {R};L^{3}_w\cap L^{6,2})}\) . The proof of the present uniqueness theorem is based on the method of dual equations.

On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains

We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far, results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, have only found that a small time-periodic Ln-solution is unique within the class of solutions which have sufficiently small L∞(Ln)-norm. In this paper, we show that a small time-periodic Ln-solution is unique within the class of all time-periodic Ln-solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-Ln space. The proof of the present uniqueness theorem is based on the method of dual equations.

On partial regularity of suitable weak solutions to the Navier–Stokes equations in unbounded domains

Consider the nonstationary Navier–Stokes equations in Ω × (0, T), where Ω is a general unbounded domain with non-compact boundary in R3. We prove the regularity of suitable weak solutions for large |x|. It should be noted that our result also holds near the boundary. Our result extends the previous ones by Caffarelli–Kohn–Nirenberg in R3 and Sohr-von Wahl in exterior domains to general domains.

On the asymptotic stability of steady solutions of the Navier–Stokes equations in unbounded domains

AbstractWe 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 &amp; Sons, Ltd.

Difference potentials for the Navier–Stokes equations in unbounded domains

We develop a numerical method for the Navier–Stokes equations over unbounded domains. From the analytic methods used to show existence and uniqueness, we obtain their discrete counterparts which allows us to establish a problem-adapted numerical solver based on finite differences for functions with low regularity.

Finite fractal dimension of pullback attractors for non-autonomous 2D Navier–Stokes equations in some unbounded domains

We study the asymptotic behaviour of non-autonomous 2D Navier–Stokes equations in unbounded domains for which a Poincaré inequality holds. In particular, we give sufficient conditions for their pullback attractor to have finite fractal dimension. The existence of pullback attractors in this framework comes from the existence of bounded absorbing sets of pullback asymptotically compact processes [T. Caraballo, G. Łukaszewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal. 64 (3) (2006) 484–498]. We show that, under suitable conditions, the method of Lyapunov exponents in [P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1984) [5]] for the dimension of attractors can be developed in this new context.

A spectral/B-spline method for the Navier–Stokes equations in unbounded domains

The numerical method presented in this paper aims at solving the incompressible Navier–Stokes equations in unbounded domains. The problem is formulated in cylindrical coordinates and the method is based on a Galerkin approximation scheme that makes use of vector expansions that exactly satisfy the continuity constraint. More specifically, the divergence-free basis vector functions are constructed with Fourier expansions in the θ and z directions while mapped B-splines are used in the semi-infinite radial direction. Special care has been taken to account for the particular analytical behaviors at both end points r=0 and r→∞. A modal reduction algorithm has also been implemented in the azimuthal direction, allowing for a relaxation of the CFL constraint on the timestep size and a possibly significant reduction of the number of DOF. The time marching is carried out using a mixed quasi-third order scheme. Besides the advantages of a divergence-free formulation and a quasi-spectral convergence, the local character of the B-splines allows for a great flexibility in node positioning while keeping narrow bandwidth matrices. Numerical tests show that the present method compares advantageously with other similar methodologies using purely global expansions.