Valentin Schwinte, Laurent Thomann
We study coupled systems of nonlinear lowest Landau level equations, for which we prove global existence results with polynomial bounds on the possible growth of Sobolev norms of the solutions. We also exhibit explicit unbounded trajectories which show that these bounds are optimal.
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We consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping.
We prove the existence of time periodic solution to the 3D Ginzburg-Landau equation in weighted Sobolev spaces. We consider the cubic Ginzburg-Landau equation with an external force $g$ satisfying the oddness condition $g(-x,t)=-g(x,t)$. The existence of the periodic solution is proved for small time-periodic external force. The stability of the time periodic solution is also considered.
The Kuramoto model is a mean-field model for the synchronisation behaviour of oscillators, which exhibits Landau damping. In a recent work, the nonlinear stability of a class of spatially inhomogeneous stationary states was shown under the assumption of analytic regularity. This paper proves the nonlinear Landau damping under the assumption of Sobolev regularity. The weaker regularity required the construction of a different more robust bootstrap argument, which focuses on the nonlinear Volterra equation of the order parameter.
A Ginzburg-Landau approximation theorem for quasilinear pattern-forming systems in uniformly local Sobolev spaces
该研究在一致局部Sobolev空间中建立逼近理论，证明Ginzburg-Landau方程可预测近首次失稳时拟线性或完全非线性斑图形成系统的动力学，并将该理论应用于三类具体系统。
A global solvability result of the Cauchy problem of the two-species Vlasov-Maxwell-Landau system near a given global Maxwellian is established by employing an approach different than that of [5]. Compared with that of [5], the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker. Our analysis does not rely on the decay of the corresponding linearized system and the Duhamel principle and thus it can be used to treat the one-species Vlasov-Maxwell-Landau system for the case of $γ>-3$ and the one-species Vlasov-Maxwell-Boltzmann system for the case of …
We study dynamical properties of the cubic lowest Landau level equation, which is used in the modeling of fast rotating Bose-Einstein condensates. We obtain bounds on the decay of general stationary solutions. We then provide a classification of stationary waves with a finite number of zeros. Finally, we are able to establish which of these stationary waves are stable, through a variational analysis.
An Optimal Minimization Problem In The Lowest Landau Level And Related Questions
本文解决了与玻色-爱因斯坦凝聚中三次最低朗道能级方程相关的极小化问题，给出了高斯函数为唯一全局极小值点的最优条件，并进一步给出了第二类特殊埃尔米特函数为全局极小值点的条件，推广了已有结果。
We consider a coupled system of nonlinear Lowest Landau Level equations. We first show the existence of multi-solitons with an exponentially localised error term in space, and then we prove a uniqueness result. We also show a long time stability result of the sum of traveling waves having all the same speed, under the condition that they are localised far away enough from each other. Finally, we observe that these multi-solitons provide examples of dynamics for the linear Schr{ö}dinger equation with harmonic potential perturbed by a time-dependent potential.
On the stability of the Abrikosov lattice in the Lowest Landau Level
研究最低朗道能级方程组在单双周期域上的情况，先探讨适定性与稳态解存在性，再研究晶格解线性稳定性，得出阿布里科索夫晶格稳定、矩形晶格不稳定的结论。
In this paper, we study Landau damping in the weakly collisional limit of a Vlasov-Fokker-Planck equation with nonlinear collisions in the phase-space $(x,v) \in \mathbb T_x^n \times \mathbb R^n_v$. The goal is four-fold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collision-driven nonlinear instabilities dominate. W…
We give an example of a linear, time-dependent, Schr{ö}dinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent potential.
We are concerned with the uniqueness of weak solution to the spatially homogeneous Landau equation with Coulomb interactions under the assumption that the solution is bounded in the space $L^\infty(0,T,L^p(\R^3))$ for some $p>3/2$. The proof uses a weighted Poincaré-Sobolev inequality recently introduced in \cite{GG18}.
We prove the Lp,q-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.
The analytic Gelfand-Shilov smoothing effect of the Landau equation with hard potential
本文在整体平衡态扰动框架下研究硬势非齐次 Landau 方程 Cauchy 问题，证明 Sobolev 初值对应的解在正时刻产生解析 Gelfand-Shilov 光滑化效应。
In this paper, we consider the linearized Vlasov-Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter $\eps$ in front of the collision operator which will tend to $0$. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker-Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter $\eps$ as it goes to $0$.
Global existence of weak solutions for Landau-Lifshitz equation with helical derivatives
本文研究了Landau-Lifshitz方程含螺旋导数的右手边界值问题，通过引入适应于螺旋导数的Sobolev空间并建立与右手边界条件兼容的能量估计，证明了在有无阻尼项情况下该问题弱解的全局存在性。
The double phase Dirichlet problem when the lowest exponent is equal to 1
本文证明了最低指数等于1时双相 Dirichlet 问题的存在性与唯一性，其解为有界变差函数且属于加权 Sobolev 空间，通过中间问题解的极限求得，并涉及广义 Orlicz-Sobolev 空间性质的研究。
We prove that the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations on $\mathbb{T}_x \times \mathbb{R}_v$ cannot, in general, be extended to high Sobolev spaces in the case of gravitational interactions. This is done by showing in every Sobolev space, there exists background distributions such that one can construct arbitrarily small perturbations that exhibit arbitrarily many isolated nonlinear oscillations in the density. These oscillations are known as plasma echoes in the physics community. For the case of electrostatic interactions, we demon…
The Lowest Landau Level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of Bose-Einstein Condensates (BEC) in two-dimensional isotropic harmonic traps in the limit of weak interactions. Building on recent developments in the field of spatially confined extended Hamiltonian systems, we find a fully nonlinear solution of this equation representing periodically modulated precession of a single vortex. Motions of this type have been previously seen in numerical simulations and experiments at moderately weak coupling. Our work provides the first controlled analy…
Algebraic rates of stability for front-type modulated waves in Ginzburg Landau equations
该研究针对复金兹堡-朗道方程中双稳区域的波前型调制波，运用指数三分理论分析空间模，证明了代数衰减初始扰动下带渐近相位的代数速率收敛结果。
This note is devoted to the study of the well-posedness of the Cauchy problem for the Landau Hamiltonian wave equation in the plane, with nonzero constant magnetic field. We show the well-posedness in suitably defined Sobolev spaces taking into account the spectral properties of the operator.
We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various hypoelliptic kinetic equations, we establish for linear Landau-type operators optimal global hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is exactly related to the anisotropy of the diffusion.
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