AI 中文总结
本文证明 Vlasov--Poisson 方程的解映射在 $L_x^pL_v^\infty$ 空间中于零初值处不连续，通过构造格点上的窄空间包反例，展示其质量与初值范数趋于零但解范数至少为一，同时相空间 $L^q$ 范数一致收敛到零。
AI 中文摘要
我们证明，对于每个 $1\leq p<\infty$，在 $\mathbb{R}^{d} \times \mathbb{R}^{d}$（其中 $d \le 3$）中，Vlasov--Poisson 方程的解映射在零初值处关于 $X_p=L_x^pL_v^\infty$ 是不连续的，并且对于吸引和排斥相互作用均成立。对于每个 $T>0$，解在 $L^\infty([0,T];X_p)$ 中的轨迹在零处不连续，并且对于每个足够小的固定 $t>0$，取值于 $X_p$ 的固定时间映射在零处也不连续。反例是光滑的、非负的、以 1 为界，并且支撑在相空间的公共紧子集中。它们的质量和初始 $X_p$ 范数趋于零，而它们在观测时刻的解范数至少为 1。构造将窄的空间包放置在格点上。自由输运允许不同的速度在每个空间点选择一个包，而场的小性估计确保非线性特征接近自由输运的特征。对于相同的族，对于每个 $1\leq q<\infty$，相空间 $L^q$ 范数随时间一致地收敛到零。
英文摘要
We prove that solution maps of the Vlasov--Poisson equation are discontinuous at the zero initial datum in $X_p=L_x^pL_v^\infty$ for every $1\leq p<\infty$, in $\mathbb{R}^{d} \times \mathbb{R}^{d}$ with $d \le 3$, and for both attractive and repulsive interactions. For every $T>0$, the trajectory of the solution in $L^\infty([0,T];X_p)$ is discontinuous at zero, and for every sufficiently small fixed $t>0$, the fixed-time map with values in $X_p$ is also discontinuous at zero. The counterexamples are smooth, nonnegative, bounded by one, and supported in a common compact subset of phase space. Their mass and initial $X_p$ norm tend to zero, whereas their solution norm at the observation time is at least one. The construction places narrow spatial packets on a lattice. Free transport allows a different velocity to select a packet at each spatial point, while a smallness estimate on the field ensures that the nonlinear characteristics are close to that of the free transport. For the same families, the phase-space $L^q$ norms converge to zero uniformly in time for every $1\leq q<\infty$.
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