Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations

Berg, Jan Bouwe van den; Jaquette, Jonathan; James, J. D. Mireles

doi:10.1007/s10884-022-10146-1

Cited by 4 publications

(1 citation statement)

References 68 publications

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“…While we cannot for the moment prove such claims because 'parameterizing' the infinite dimensional stable manifold of the unstable steady states is highly non-trivial, we are aware of some preliminary work in this direction [47]. That said, computer-assisted proofs of connecting orbits from saddle points to asymptotically stable steady states in parabolic PDEs are starting to appear [48,49].…”

Section: Connections Between Steady Statesmentioning

confidence: 99%

Microscopic patterns in the 2D phase-field-crystal model

2022

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Using the recently developed theory of rigorously validated numerics, we address the phase-field-crystal model at the microscopic (atomistic) level. We show the existence of critical points and local minimizers associated with ‘classical’ candidates, grain boundaries, and localized patterns. We further address the dynamical relationships between the observed patterns for fixed parameters and across parameter space, then formulate several conjectures on the dynamical connections (or orbits) between steady states.

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Section: Connections Between Steady Statesmentioning

confidence: 99%

Microscopic patterns in the 2D phase-field-crystal model

2022

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Rigorous numerics for nonlinear heat equations in the complex plane of time

et al. 2022

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In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

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