Validated bounds on embedding constants for Sobolev space Banach algebras

Wanner, Thomas

doi:10.1002/mma.5294

“…More precisely, we begin by discussing the necessary function spaces in Subsection 3.1, which form the foundation for our spectral approach based on Fourier cosine series. The following Subsection 3.2 recalls a number of rigorous Sobolev embedding results that originated in [38], and results which allow us to replace the standard Sobolev norms with more computationally appropriate ones. Subsection 3.3 is devoted to the required finite-dimensional approximation spaces and associated projection operators, which are used in our computerassisted proofs.…”

Section: Rigorously Verified Microstructuresmentioning

confidence: 99%

Equilibrium Validation for Triblock Copolymers via Inverse Norm Bounds for Fourth-Order Elliptic Operators

Rizzi¹,

Sander²,

Wanner³

2022

Preprint

Self Cite

0

View full text Add to dashboard Cite

Block copolymers play an important role in materials sciences and have found widespread use in many applications. From a mathematical perspective, they are governed by a nonlinear fourth-order partial differential equation which is a suitable gradient of the Ohta-Kawasaki energy. While the equilibrium states associated with this equation are of central importance for the description of the dynamics of block copolymers, their mathematical study remains challenging. In the current paper, we develop computerassisted proof methods which can be used to study equilibrium solutions in block copolymers consisting of more than two monomer chains, with a focus on triblock copolymers. This is achieved by establishing a computer-assisted proof technique for bounding the norm of the inverses of certain fourth-order elliptic operators, in combination with an application of a constructive version of the implicit function theorem. While these results are only applied to the triblock copolymer case, we demonstrate that the obtained norm estimates can also be directly used in other contexts such as the rigorous verification of bifurcation points, or pseudo-arclength continuation in fourth-order parabolic problems.

show abstract

“…The values of C m and C b in dimensions 1, 2, and 3 were established in [38] using rigorous computational techniques. The values of C m can be obtained by adapting the approach in this paper, as outlined in the next lemma.…”

Section: 3mentioning

confidence: 99%

“…In order to complete the proof one only has to find a rigorous upper bound on the second factor in the last line of the above estimate. For this, one can first use the proof of [38,Corollary 3.3] to establish the tail bound…”

Section: 3mentioning

confidence: 99%

Equilibrium validation in models for pattern formation based on Sobolev embeddings

Sander¹,

Wanner²

2021

Discrete &Amp; Continuous Dynamical Systems - B

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…The values of C m and C b in dimensions 1, 2, and 3 were established in [35] using rigorous computational techniques. The values of C m can be obtained by adapting the approach in this paper, as outlined in the next lemma.…”

Section: 3mentioning

confidence: 99%

“…In order to complete the proof one only has to find a rigorous upper bound on the second factor in the last line of the above estimate. For this, one can first use the proof of [35,Corollary 3.3] to establish the tail bound…”

Section: 3mentioning

confidence: 99%

Equilibrium Validation in Models for Pattern Formation Based on Sobolev Embeddings

Sander,

Wanner

2020

Preprint

Self Cite

0

View full text Add to dashboard Cite

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.

show abstract

Validated bounds on embedding constants for Sobolev space Banach algebras

Cited by 4 publications

References 28 publications

Equilibrium Validation for Triblock Copolymers via Inverse Norm Bounds for Fourth-Order Elliptic Operators

Equilibrium Validation for Triblock Copolymers via Inverse Norm Bounds for Fourth-Order Elliptic Operators

Equilibrium validation in models for pattern formation based on Sobolev embeddings

Equilibrium Validation in Models for Pattern Formation Based on Sobolev Embeddings

Contact Info

Product

Resources

About