In this paper we use formal asymptotic arguments to understand the stability proper- ties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. We also analyze both the harmonic map heatflow and Schrodinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic. Solutions permitted to deviate from radial symmetry remain global for all time but may, for suitable initial data, approach arbitrarily close to blowup. A careful asymptotic analysis of solutions near blowup shows that finite-time blowup corresponds to a saddle fixed point in a low dimensional dynamical system. Radial symmetry precludes motion anywhere but on the stable manifold towards blowup. A similar scenario emerges in the equivariant setting: blowup is unstable. To be more precise, blowup is co-dimension one both within the equivariant symmetry class and in the unrestricted class of initial data. The value of the parameter in the Landau-Lifshitz-Gilbert equation plays a very subdued role in the analysis of equivariant blowup, leading to identical blowup rates and spatial scales for all parameter values. One notable exception is the angle between solution in inner scale (which bubbles off) and outer scale (which remains), which does depend on parameter values. Analyzing near-blowup solutions, we find that in the inner scale these solution quickly rotate over an angle π. As a consequence, for the blowup solution it is natural to consider a continuation scenario after blowup where one immediately re-attaches a sphere (thus restoring the energy lost in blowup), yet rotated over an angle π. This continuation is natural since it leads to continuous dependence on initial data.
Compressible Navier-Stokes-Landau-Lifshitz-Gilbert system: derivations and well-posedness
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Finance and Mathematics, Huainan Normal University（淮南师范学院金融与数学学院）；School of Mathematics, Hunan University（湖南大学数学学院）；School of Mathematics and Physics, China University of Geosciences（中国地质大学数理学院）
A decoupled, stable, and second-order integrator for the Landau--Lifshitz--Gilbert equation with magnetoelastic effects
Institute of Information Theory and Automation, Czech Academy of Sciences（自动化与信息理论研究所，捷克科学院）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Department of Mathematics, University of Bologna（博洛尼亚大学数学系）
Blowup dynamics for equivariant critical Landau--Lifshitz flow
二维Landau-Lifshitz方程的有限时间爆破解的存在性是一个长期存在的问题，至少从2001年开始就存在于文献中（E，Mathematics Unlimited-2001 and Beyond，Springer，Berlin，P.410，2001）。在等变类中更精确的描述在（van den Berg和Williams，European J. Appl. Math.，24（6），912- 948，2013）。本文考虑Landau-Lifshitz方程$$ \partial_tu=\mathfrak{a}_1u\timesΔu-\mathfrak{a}_2u\times(u\timesΔu),\quad x\in\mathbb{R}^2, $$的爆破动力学，其中$u\in\mathbb{S}^2$，$\mathfrak{a}_1+i\mathfrak{a}_2\in\mathbb{C}$ with $\mathfrak{a}_2\geq0$和$\mathfrak{a}_1+\mathfrak{a}_2=1$ .我们证明了在最低能量稳态附近存在1-等变Krieger-Schlag-Tataru型爆破解。更精确地说，我们证明了，对于任何$ν>1$，存在一个1-等变有限时间爆破解的形式$$ u(x,t)=ϕ(λ(t)x)+ζ(x,t),\quad λ(t)=t^{-1/2-ν}, $$…
