论文标题
从鲍尔茨曼到Yudovich类以下的2D不可压缩的Euler方程的涡度收敛
Vorticity convergence from Boltzmann to 2D incompressible Euler equations below Yudovich class
论文作者
论文摘要
对以宏观尺度表现出奇异性的介观系统进行多尺度分析是一项挑战。在本文中，我们研究了Boltzmann方程的流体动力学极限$$ \ MATHRM {st} \ partial_t f + v \ cdot \ cdot \ nabla_x f = \ frac {1} {\ mathrm {knrm {kn}} q（f，f，f，f，f，f） $ \ partial_t u + u \ cdot \ nabla_x u + \ nabla_x p = 0，\ text {div} u = 0。$$，我们通过所谓的动力学涡流获得了奇异性的微观描述，并了解其在巨型sidularity的遗传中的行为。由于我们的新分析，我们肯定地解决了流体动力学限制的一个开放问题，该问题朝着2D不可压缩的Euler方程的拉格朗日解决方案解决，其涡度无限制（$ω\ in L^\ Mathfrak {p} $ in L^\ Mathfrak {p} $，对于任何固定的$ 1 \ leq \ leq \ mathfrak {p Mathfrak {p} <\ infty $）。此外，我们证明了动力学涡度朝着欧拉方程拉格朗日溶液的涡度的融合。特别是，当涡度中适度地吹动$ l^\ mathfrak {p} $作为$ \ mathfrak {p} \ rightarrow \ infty $（本地化的Yudovich类）时，我们获得了收敛速度。
It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations $$\mathrm{St} \partial_t F + v\cdot \nabla_x F = \frac{1}{\mathrm{Kn}} Q(F ,F ) $$ toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded $$\partial_t u + u \cdot \nabla_x u + \nabla_x p = 0,\text{div }u =0.$$ We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of the hydrodynamic limit toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded ($ω\in L^\mathfrak{p}$ for any fixed $1 \leq \mathfrak{p} < \infty$). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in $L^\mathfrak{p}$ as $\mathfrak{p} \rightarrow \infty$ (localized Yudovich class).
