zbMATH — the first resource for mathematics

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.

35B40 Asymptotic behavior of solutions to PDEs
35Q30 Navier-Stokes equations
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.