Long-time limit for a class of quadratic infinite-dimensional dynamical systems inspired by models of viscoelastic fluids

Guy Katriel; Raz Kupferman; Edriss S. Titi

doi:10.1016/j.jde.2007.11.014

Long-time limit for a class of quadratic infinite-dimensional dynamical systems inspired by models of viscoelastic fluids

Guy Katriel, Raz Kupferman, Edriss S. Titi^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study a class of quadratic, infinite-dimensional dynamical systems, inspired by models for viscoelastic fluids. We prove that these equations define a semi-flow on the cone of positive, essentially bounded functions. As time tends to infinity, the solutions tend to an equilibrium manifold in the L²-norm. Convergence to a particular function on the equilibrium manifold is only proved under additional assumptions. We discuss several possible generalizations.

Original language	English
Pages (from-to)	2771-2784
Number of pages	14
Journal	Journal of Differential Equations
Volume	245
Issue number	10
DOIs	https://doi.org/10.1016/j.jde.2007.11.014
State	Published - 15 Nov 2008

Bibliographical note

Funding Information:
We are grateful to Raanan Fattal for discussions that motivated this present work. G.K. was partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). R.K. was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, and by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the US Department of Energy under Contract DE-AC03-76-SF00098. The work of E.S.T. was supported in part by the NSF grant No. DMS-0504619, the ISF grant No. 120/6, and the BSF grant No. 2004271.

Keywords

Equilibrium manifold
Global attractor
Quadratic differential systems
Viscoelastic toy model

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10.1016/j.jde.2007.11.014

Cite this

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abstract = "We study a class of quadratic, infinite-dimensional dynamical systems, inspired by models for viscoelastic fluids. We prove that these equations define a semi-flow on the cone of positive, essentially bounded functions. As time tends to infinity, the solutions tend to an equilibrium manifold in the L2-norm. Convergence to a particular function on the equilibrium manifold is only proved under additional assumptions. We discuss several possible generalizations.",

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Long-time limit for a class of quadratic infinite-dimensional dynamical systems inspired by models of viscoelastic fluids

Abstract

Bibliographical note

Keywords

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