三维球对称可压缩Navier-Stokes方程中具有任意大初值的强解全局存在与爆破的尖锐符号判据
Sharp sign criterion for global existence and blow-up of strong solutions with arbitrarily large initial data in the 3D spherically symmetric compressible Navier-Stokes equations
- State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）
- School of Mathematics, Jilin University（吉林大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对三维球对称可压缩Navier-Stokes方程，本文基于有效速度初始符号建立了强解全局存在与有限时间爆破的尖锐判据，并首次将全局存在性推广到超临界α>1情形。
AI中文摘要：
我们针对实心球上的三维球对称可压缩Navier-Stokes方程，建立了强解在时间上全局存在与有限时间爆破的尖锐判据，该判据仅基于有效速度的初始符号。粘性系数假定满足Bresch-Desjardins结构 μ=ρ^α，λ=(α-1)ρ^α，其中有效速度由 v=u+αρ^{α-2}ρ_r 给出。此前，高维强解的全局存在性结果仅限于 α≤1 的情形，其中端点 α=1 对应于粘性Saint-Venant（浅水）系统。在本文中，我们将全局存在性理论推广到超过该阈值的超临界区域 α>1。具体而言，只要球边界上的初始有效速度非负，我们就证明对于任意大的初值，强解在时间上全局存在。主要困难在于推导密度的均匀下界，这需要处理球心处的奇异估计。为了克服这一困难，我们利用球心附近的若干新量，并采用精细的最大值原理，首次建立了球心处速度的Lipschitz连续性。随后，我们为适当构造的函数推导出Dini-Gronwall型不等式，从而封闭密度下界估计。反之，我们可以构造一族初值，使得边界上的初始有效速度为负，相应的强解在有限时间内爆破，且在爆破时刻真空恰好出现在边界上。我们的结果建立了全局正则性与奇性形成之间的清晰二分法，该二分法完全由有效速度的初始符号决定。
英文摘要：
We establish a sharp criterion for global-in-time existence versus finite-time blow-up of strong solutions to the 3D spherically symmetric compressible Navier-Stokes equations on a solid ball, based solely on the initial sign of effective velocity. The viscosity coefficients are assumed to satisfy the Bresch-Desjardins structure $μ=ρ^α$, $λ=(α-1)ρ^α$, with the effective velocity given by $v=u+αρ^{α-2}ρ_r$. Previously, global existence results for strong solutions in higher dimensions were restricted to the case $α\le 1$ with the endpoint $α=1$ corresponding to the viscous Saint-Venant (shallow water) system. In this paper, we extend the global existence theory beyond this threshold to the supercritical regime $α>1$. Specifically, whenever the initial effective velocity is nonnegative on the boundary of the ball, we prove the global-in-time existence of strong solutions for arbitrarily large initial data. The main difficulty lies in deriving a uniform lower bound for the density, which requires handling singular estimates at the center of the ball. To overcome this, we exploit several novel quantities near the center and employ a refined maximum principle to establish, for the first time, the Lipschitz continuity of the velocity at the center. Subsequently, we derive Dini-Gronwall-type inequalities for suitably constructed functions, which close the density lower bound estimate. Conversely, we can construct a family of initial data for which the initial effective velocity is negative on the boundary, such that the corresponding strong solutions blow up in finite time, with vacuum appearing exactly on the boundary at the blow-up time. Our results establish a clean dichotomy between global regularity and singularity formation, governed purely by the initial sign of the effective velocity.
