具有径向非谐约束势的玻尔兹曼方程的平衡态与线性稳定性

中文摘要 AI 辅助

我们研究了在整个空间中，当$p>2$时$\Phi(x)=|x|^p/p$以及$1<p<2$时$\Phi(x)=\langle x\rangle^p/p$的径向非谐约束势下的玻尔兹曼方程。首先对所有正的有限质量和能量且熵不变（即熵产生为零）的解进行分类。对于$p>2$，非线性平衡流形由质量、温度和角动量的三个分量参数化；对于$1<p<2$，可积性排除了旋转平衡，仅质量和能量作为平衡参数。接着确定方程在任意平衡态附近线性化后的五维平稳空间，并构建由守恒矩确定的显式投影。经归一化后，碰撞项具有特定形式，其微观强制性在空间无穷远处退化，通过远场权重转移估计进行补偿。减去平稳投影后，相应半群解在指数加权$L^2$空间中代数收敛，收敛速率由增长指数$p$和两个权重之间的差距决定，对于$1<p<2$，二维非线性平衡流形与五维线性平稳空间的不匹配对携带非零角动量的扰动的非线性渐近吸引产生了条件性阻碍。

英文摘要

We study the Boltzmann equation in the whole space under the radial anharmonic confining potentials $Φ(x)=|x|^p/p$ for $p>2$ and $Φ(x)=\langle x\rangle^p/p$ for $1<p<2$. We first classify all positive finite-mass-and-energy entropy-invariant, equivalently zero-entropy-production, solutions. For $p>2$, the nonlinear equilibrium manifold is parametrized by mass, temperature, and the three components of angular momentum; for $1<p<2$, integrability excludes rotating equilibria and only mass and energy remain as equilibrium parameters. We then identify the five-dimensional stationary space of the equation linearized about an arbitrary equilibrium and construct an explicit projection determined by the conserved moments. After normalization, the collision term takes the form $C_Me^{-\widetildeΦ(x)}\mathsf L$, so its microscopic coercivity degenerates at spatial infinity. A far-field weight-transfer estimate compensates for this degeneracy. After subtracting the stationary projection, the corresponding semigroup solution converges algebraically in exponentially weighted $L^2$ spaces. The rate is governed by the growth exponent $p$ and the gap between the two weights, up to an arbitrarily small loss. For $1<p<2$, the mismatch between the two-dimensional nonlinear equilibrium manifold and the five-dimensional linear stationary space yields a conditional obstruction to nonlinear asymptotic attraction for perturbations carrying nonzero angular momentum.

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61pages, no figure