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.
Bibliographical noteFunding 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.
- Equilibrium manifold
- Global attractor
- Quadratic differential systems
- Viscoelastic toy model