Nonlinear Stability of Periodic Roll Solutions in the Real Ginzburg–Landau Equation Against Cubm -Perturbations
(2023) In Communications in Mathematical Physics 400(1). p.277-314- Abstract
The real Ginzburg–Landau equation arises as a universal amplitude equation for the description of pattern-forming systems exhibiting a Turing bifurcation. It possesses spatially periodic roll solutions which are known to be stable against localized perturbations. It is the purpose of this paper to prove their stability against bounded perturbations, which are not necessarily localized. Since all state-of-the-art techniques rely on localization or periodicity properties of perturbations, we develop a new method, which employs pure L∞-estimates only. By fully exploiting the smoothing properties of the semigroup generated by the linearization, we are able to close the nonlinear iteration despite the slower decay rates. To show... (More)
The real Ginzburg–Landau equation arises as a universal amplitude equation for the description of pattern-forming systems exhibiting a Turing bifurcation. It possesses spatially periodic roll solutions which are known to be stable against localized perturbations. It is the purpose of this paper to prove their stability against bounded perturbations, which are not necessarily localized. Since all state-of-the-art techniques rely on localization or periodicity properties of perturbations, we develop a new method, which employs pure L∞-estimates only. By fully exploiting the smoothing properties of the semigroup generated by the linearization, we are able to close the nonlinear iteration despite the slower decay rates. To show the wider relevance of our method, we also apply it to the amplitude equation as it appears for pattern-forming systems with an additional conservation law.
(Less)
- author
- Hilder, Bastian LU ; de Rijk, Björn and Schneider, Guido
- organization
- publishing date
- 2023
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Communications in Mathematical Physics
- volume
- 400
- issue
- 1
- pages
- 277 - 314
- publisher
- Springer
- external identifiers
-
- scopus:85147367266
- ISSN
- 0010-3616
- DOI
- 10.1007/s00220-022-04619-z
- language
- English
- LU publication?
- yes
- id
- cbc734a3-3f9f-4c03-becf-b3be43053042
- date added to LUP
- 2023-02-24 12:52:31
- date last changed
- 2023-10-26 14:50:36
@article{cbc734a3-3f9f-4c03-becf-b3be43053042,
abstract = {{<p>The real Ginzburg–Landau equation arises as a universal amplitude equation for the description of pattern-forming systems exhibiting a Turing bifurcation. It possesses spatially periodic roll solutions which are known to be stable against localized perturbations. It is the purpose of this paper to prove their stability against bounded perturbations, which are not necessarily localized. Since all state-of-the-art techniques rely on localization or periodicity properties of perturbations, we develop a new method, which employs pure L<sup>∞</sup>-estimates only. By fully exploiting the smoothing properties of the semigroup generated by the linearization, we are able to close the nonlinear iteration despite the slower decay rates. To show the wider relevance of our method, we also apply it to the amplitude equation as it appears for pattern-forming systems with an additional conservation law.</p>}},
author = {{Hilder, Bastian and de Rijk, Björn and Schneider, Guido}},
issn = {{0010-3616}},
language = {{eng}},
number = {{1}},
pages = {{277--314}},
publisher = {{Springer}},
series = {{Communications in Mathematical Physics}},
title = {{Nonlinear Stability of Periodic Roll Solutions in the Real Ginzburg–Landau Equation Against Cubm -Perturbations}},
url = {{http://dx.doi.org/10.1007/s00220-022-04619-z}},
doi = {{10.1007/s00220-022-04619-z}},
volume = {{400}},
year = {{2023}},
}