scholarly journals Asymptotic stability of stationary solutions for magnetohydrodynamic equations

2017 ◽  
Vol 37 (6) ◽  
pp. 3435-3465
Author(s):  
Zhong Tan ◽  
◽  
Leilei Tong
Keyword(s):  
Asymptotic Stability ◽  
Stationary Solutions ◽  
Magnetohydrodynamic Equations

scholarly journals The mathematics of asymptotic stability in the Kuramoto model

2018 ◽  
Vol 474 (2220) ◽  
pp. 20180467 ◽  
Author(s):  
Helge Dietert ◽  
Bastien Fernandez
Keyword(s):  
Asymptotic Stability ◽  
Coupled Oscillators ◽  
Continuum Limit ◽  
Heterogeneous Systems ◽  
Stationary Solutions ◽  
Kuramoto Model ◽  
The Kuramoto Model ◽  
The Continuum

Now a standard in Nonlinear Sciences, the Kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators. While its basic phenomenology has been sketched in early works, the corresponding rigorous validation has long remained problematic and was achieved only recently. This paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the Kuramoto model, and provides insights into the principal arguments of proofs. This review is complemented with additional original results, various examples, and possible extensions to some variations of the model in the literature.

Asymptotic stability of stationary Navier–Stokes flow in Besov spaces

2021 ◽  
pp. 1-22
Author(s):  
Jayson Cunanan ◽  
Takahiro Okabe ◽  
Yohei Tsutsui
Keyword(s):  
Asymptotic Stability ◽  
Stokes Flow ◽  
Besov Spaces ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Main Ingredient ◽  
Navier Stokes Equations ◽  
Whole Space

We discuss the asymptotic stability of stationary solutions to the incompressible Navier–Stokes equations on the whole space in Besov spaces. A critical estimate for the semigroup generated by the Laplacian with a perturbation is the main ingredient of the argument.

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