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A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Univ Sao Paulo, Inst Ciencias Matemat & Comp, BR-13560970 Sao Carlos, SP, Brazil..
Univ Laval, Dept Math & Stat, Quebec City, PQ G1V 0A6, Canada..
Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA..
2017 (English)In: SIAM Journal on Applied Dynamical Systems, ISSN 1536-0040, E-ISSN 1536-0040, Vol. 16, no 2, 1070-1088 p.Article in journal (Refereed) Published
Abstract [en]

We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits, and invariant manifolds attached to fixed points or periodic orbits. The core of the study is writing down the invariance condition as a zero of an operator. These operators are in general not continuous, so one needs to smooth them by means of preconditioners before classical fixed point theorems can be applied. We develop in detail all the aspects of how to work with these objects: how to precondition the equations, how to work with the nonlinear terms, which function spaces can be useful, and how to work with them in a computationally rigorous way. In two companion papers, we present two different implementations of the tools developed in this paper to study periodic orbits.

Place, publisher, year, edition, pages
SIAM PUBLICATIONS , 2017. Vol. 16, no 2, 1070-1088 p.
Keyword [en]
evolution equation, periodic orbits, contraction mapping, invariant manifolds, rigorous computations, interval analysis
National Category
Physical Sciences Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-330073DOI: 10.1137/16M1073777ISI: 000404777600010OAI: oai:DiVA.org:uu-330073DiVA: diva2:1150702
Available from: 2017-10-19 Created: 2017-10-19 Last updated: 2017-10-19Bibliographically approved

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Figueras, Jordi-Lluis

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