Self-Similar Solutions to the Nonstationary Navier-Stokes Equations

Jia, Hao; Šverák, Vladimír; Tsai, Tai‐Peng

doi:10.1007/978-3-319-10151-4_9-1

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The Landau-lifshitz-gilbert Equation: Self-similar Solutions1

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2018

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Cited by 2 publications

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“…Examples of equations for which self-similar solutions have been studied include, among others, the Navier-Stokes equation, semilinear parabolic equations, and geometric flows such as Yang-Mills, mean curvature flow and harmonic map flow. We refer to [37,48,36,51,5] and the references therein for more details.…”

Section: The Landau-lifshitz-gilbert Equation: Self-similar Solutionsmentioning

confidence: 99%

Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation

Gutiérrez

Laire

2015

Nonlinearity

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The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere S 2 , at an exponential rate.In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

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Section: The Landau-lifshitz-gilbert Equation: Self-similar Solutionsmentioning

confidence: 99%