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A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations. (English) Zbl 1370.65046

Summary: We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear partial differential equations. 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.

MSC:

65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
37C27 Periodic orbits of vector fields and flows
65G40 General methods in interval analysis
65F08 Preconditioners for iterative methods

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References:

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