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Distribution of energy and convergence to equilibria in extended dissipative systems. (English) Zbl 1338.35045
Summary: We are interested in understanding the dynamics of dissipative partial differential equations on unbounded spatial domains. We consider systems for which the energy density $$e\geq 0$$ satisfies an evolution law of the form $$\partial_te=\operatorname{div}_xf-d$$, where $$-f$$ is the energy flux and $$d\geq 0$$ the energy dissipation rate. We also suppose that $$| f|^2\leq b(e)d$$ for some nonnegative function $$b$$. Under these assumptions we establish simple and universal bounds on the time-integrated energy flux, which in turn allow us to estimate the amount of energy that is dissipated in a given domain over a long interval of time. In low space dimensions $$N\leq 2$$, we deduce that any relatively compact trajectory converges on average to the set of equilibria, in a sense that we quantify precisely. As an application, we consider the incompressible Navier-Stokes equation in the infinite cylinder $$\mathbb R\times\mathbb T$$, and for solutions that are merely bounded we prove that the vorticity converges uniformly to zero on large subdomains, if we disregard a small subset of the time interval.

MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35Q30 Navier-Stokes equations
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 [1] Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. Courier Dover Publications, New York (1964) · Zbl 0171.38503 [2] Afendikov, A; Mielke, A, Dynamical properties of spatially non-decaying 2D Navier-Stokes flows with Kolmogorov forcing in an infinite strip, J. Math. Fluid Mech., 7, s51-s67, (2005) · Zbl 1062.35057 [3] Allen, S; Cahn, J, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metal., 27, 1085-1095, (1979) [4] Aranson, I; Kramer, L, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74, 99-143, (2002) · Zbl 1205.35299 [5] Aronson, D; Weinberger, H, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30, 33-76, (1978) · Zbl 0407.92014 [6] Arrieta, J; Rodriguez-Bernal, A; Cholewa, J; Dlotko, T, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14, 253-293, (2004) · Zbl 1058.35076 [7] Babin, A; Vishik, M, Attractors of partial differential equations in an unbounded domain, Proc. R. Soc. Edinburgh, 116A, 221-243, (1990) · Zbl 0721.35029 [8] Carr, J; Pego, R, Metastable patterns in solutions of $$u_t=ϵ ^2u_{xx}-f(u)$$, Commun. Pure Appl. Math., 42, 523-576, (1989) · Zbl 0685.35054 [9] Collet, P, Thermodynamic limit of the Ginzburg-Landau equations, Nonlinearity, 7, 1175-1190, (1994) · Zbl 0803.35066 [10] Collet, P; Eckmann, J-P, Space-time behaviour in problems of hydrodynamic type: a case study, Nonlinearity, 5, 1265-1302, (1992) · Zbl 0757.35059 [11] Conway, J; Sloane, N, Sphere packings, lattices and groups, No. 290, (1988), New York · Zbl 0634.52002 [12] Eckmann, J-P; Rougemont, J, Coarsening by Ginzburg-Landau dynamics, Commun. Math. Phys., 199, 441-470, (1998) · Zbl 1057.35508 [13] Ei, S-I, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dyn. Differ. Equ., 14, 85-137, (2002) · Zbl 1007.35039 [14] Feireisl, E, Bounded, locally compact global attractors for semilinear damped wave equations on $${\mathbb{R}}^n$$, J. Diff. Integral Equ., 9, 1147-1156, (1996) · Zbl 0858.35084 [15] Gallay, Th; Slijepčević, S, Energy flow in formally gradient partial differential equations on unbounded domains, J. Dyn. Differ. Equ., 13, 757-789, (2001) · Zbl 1003.35085 [16] Gallay, Th., Slijepčević, S.: Energy bounds for the two-dimensional Navier-Stokes equations in an infinite cylinder, to appear. Commun. Partial Diff. Equ. (2014) · Zbl 0685.35054 [17] Giga, Y; Matsui, S; Sawada, O, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech., 3, 302-315, (2001) · Zbl 0992.35066 [18] Guo, B; Ding, S, Landau-Lifshitz equations, No. 1, (2008), Hackensack · Zbl 1158.35096 [19] Hale, J.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25. AMS, Providence (1988) · Zbl 0642.58013 [20] Henry, D, Geometric theory of semilinear parabolic equations, No. 840, (1981), Berlin · Zbl 0456.35001 [21] Massatt, P, Limiting behavior for strongly damped nonlinear wave equations, J. Differ. Equ., 48, 334-349, (1983) · Zbl 0561.35049 [22] Mielke, A, The Ginzburg-Landau equation in its role as a modulation equation, No. 2, (2002), Amsterdam · Zbl 1041.37037 [23] Mielke, A; Schneider, G, Attractors for modulation equations on unbounded domains: existence and comparison, Nonlinearity, 8, 743-768, (1995) · Zbl 0833.35016 [24] Mischaikow, K; Morita, Y, Dynamics on the global attractor of a gradient flow arising from the Ginzburg-Landau equation, Jpn. J. Ind. Appl. Math., 11, 185-202, (1994) · Zbl 0807.58029 [25] Pata, V; Zelik, S, Smooth attractors for strongly damped wave equations, Nonlinearity, 19, 1495-1506, (2006) · Zbl 1113.35023 [26] Rougemont, J, Dynamics of kinks in the Ginzburg-Landau equation: approach to metastable shape and collapse of embedded pair of kinks, Nonlinearity, 12, 539-554, (1999) · Zbl 0984.35148 [27] Sawada, O; Taniuchi, Y, A remark on $$L^∞$$ solutions to the 2-D Navier-Stokes equations, J. Math. Fluid Mech., 9, 533-542, (2007) · Zbl 1132.35437 [28] Zelik, S, Infinite energy solutions for damped Navier-Stokes equations in $${\mathbb{R}}^2$$, J. Math. Fluid Mech., 15, 717-745, (2013) · Zbl 1293.35221
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