Abstract
We are interested in understanding the dynamics of dissipative partial differential equations on unbounded spatial domains. We consider systems for which the energy density \(e \ge 0\) satisfies an evolution law of the form \(\partial _t e = \mathrm{div}_x f - d\), where \(-f\) is the energy flux and \(d \ge 0\) the energy dissipation rate. We also suppose that \(|f|^2 \le b(e)d\) 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 \(N \le 2\), 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.
Similar content being viewed by others
References
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)
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(suppl. 1), S51–S67 (2005)
Allen, S., Cahn, J.: A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metal. 27, 1085–1095 (1979)
Aranson, I., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)
Aronson, D., Weinberger, H.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)
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)
Babin, A., Vishik, M.: Attractors of partial differential equations in an unbounded domain. Proc. R. Soc. Edinburgh 116A, 221–243 (1990)
Carr, J., Pego, R.: Metastable patterns in solutions of \(u_t=\epsilon ^2u_{xx}-f(u)\). Commun. Pure Appl. Math. 42, 523–576 (1989)
Collet, P.: Thermodynamic limit of the Ginzburg–Landau equations. Nonlinearity 7, 1175–1190 (1994)
Collet, P., Eckmann, J.-P.: Space-time behaviour in problems of hydrodynamic type: a case study. Nonlinearity 5, 1265–1302 (1992)
Conway, J., Sloane, N.: Sphere packings, lattices and groups. Grundlehren der Mathematischen Wissenschaften, vol. 290. Springer, New York (1988)
Eckmann, J.-P., Rougemont, J.: Coarsening by Ginzburg–Landau dynamics. Commun. Math. Phys. 199, 441–470 (1998)
Ei, S.-I.: The motion of weakly interacting pulses in reaction-diffusion systems. J. Dyn. Differ. Equ. 14, 85–137 (2002)
Feireisl, E.: Bounded, locally compact global attractors for semilinear damped wave equations on \({\mathbb{R}}^n\). J. Diff. Integral Equ. 9, 1147–1156 (1996)
Gallay, Th, Slijepčević, S.: Energy flow in formally gradient partial differential equations on unbounded domains. J. Dyn. Differ. Equ. 13, 757–789 (2001)
Gallay, Th., Slijepčević, S.: Energy bounds for the two-dimensional Navier–Stokes equations in an infinite cylinder, to appear. Commun. Partial Diff. Equ. (2014)
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)
Guo, B., Ding, S.: Landau–Lifshitz equations. Frontiers of Research with the Chinese Academy of Sciences, vol. 1. World Scientific, Hackensack (2008)
Hale, J.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25. AMS, Providence (1988)
Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)
Massatt, P.: Limiting behavior for strongly damped nonlinear wave equations. J. Differ. Equ. 48, 334–349 (1983)
Mielke, A.: The Ginzburg–Landau equation in its role as a modulation equation. Handbook of Dynamical Systems, vol. 2. North-Holland, Amsterdam (2002)
Mielke, A., Schneider, G.: Attractors for modulation equations on unbounded domains: existence and comparison. Nonlinearity 8, 743–768 (1995)
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)
Pata, V., Zelik, S.: Smooth attractors for strongly damped wave equations. Nonlinearity 19, 1495–1506 (2006)
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)
Sawada, O., Taniuchi, Y.: A remark on \(L^\infty \) solutions to the 2-D Navier–Stokes equations. J. Math. Fluid Mech. 9, 533–542 (2007)
Zelik, S.: Infinite energy solutions for damped Navier–Stokes equations in \({\mathbb{R}}^2\). J. Math. Fluid Mech. 15, 717–745 (2013)
Acknowledgments
Part of this work was done when the second author visited Institut Fourier at University of Grenoble, whose hospitality is gratefully acknowledged. The authors thank P. Poláčik and A. Scheel for fruitful discussions. Th.G. was supported in part by the ANR projet PREFERED of the French Ministry of Research, and S.S. by the grant No 037-0372791-2803 of the Croatian Ministry of Science.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated with deep respect to the memory of Klaus Kirchgässner.
Rights and permissions
About this article
Cite this article
Gallay, T., Slijepčević, S. Distribution of Energy and Convergence to Equilibria in Extended Dissipative Systems. J Dyn Diff Equat 27, 653–682 (2015). https://doi.org/10.1007/s10884-014-9376-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-014-9376-z