On the basicity of trigonometric systems with a linear phase in a weighted Sobolev space

Muradov, Togrul; Salmanov, Valid F.

doi:10.1134/s1064562414060301

Cited by 1 publication

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“…where the boundary conditions in the second and third equations are considered in the sense of the trace operator. The first identity follows as a special case from the results in [15,22], the second identity has been established in [21,Lemma 3.2], and also the third identity can be verified as in [21,Lemma 3.2]. For the sake of simplicity we further define H 0 = L 2 (Ω).…”

Section: Evelyn Sander and Thomas Wannermentioning

confidence: 86%

Equilibrium validation in models for pattern formation based on Sobolev embeddings

Sander¹,

Wanner²

2021

Discrete &Amp; Continuous Dynamical Systems - B

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In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.

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