On relation between attractors for single and multivalued semiflows for a certain class of PDEs
This article is about dissipative PDE, which admit weak solutions that are \textit{not} necessarily unique, and a global attractor. Such problems give rise to multivalued flows (e.g. generalized semiflows [\textit{J. M. Ball}, J. Nonlinear Sci. 7, No. 5, 475--502 (1997; Zbl 0903.58020); erratum ibid. 8, No. 2, 233 (1998); Corrected version appears in: Mechanics: from theory to computation. Essays in honor of Juan-Carlos Simo. Papers invited by Journal of Nonlinear Science editors. New York, NY: Springer. 447--474 (2000; Zbl 0958.35101)] or trajectory attractors [\textit{V. V. Chepyzhov} and \textit{M. I. Vishik}, C. R. Acad. Sci., Paris, Sér. I 321, No. 10, 1309--1314 (1995; Zbl 0843.35038); ibid. 321, No. 2, 153--158 (1995; Zbl 0837.35059)]). For the class of problems in which these weak solutions may be bootstrapped into existing (unique) strong solutions, it is shown that the global attractor for the strong solutions coincides with global attractors for weak solutions. One illuminating aspect to this is that the finite-dimensionality of the global attractor, when this is present for the unique strong solutions, carries over for the case of the weak solutions. Two applications are given.
scientific article; zbMATH DE number 7039946
scientific article