全局正则性：$\mathbb{R}^3$ 中具有信号消耗的趋化-斯托克斯系统

中文摘要 AI 辅助

2005年，Tuval等人提出了一个描述好氧细菌、氧气与不可压缩流体相互作用的连续介质模型。对于相应的三维趋化-消耗系统，即使在流体方程中省略非线性对流项之后，大数据的全局经典正则性仍是一个长期悬而未决的问题。我们回答了这个问题，并证明了在$\mathbb{R}^3$中，具有线性细胞扩散和双线性信号消耗的趋化-斯托克斯柯西问题的全局经典可解性，在以下数据假设下成立：初始密度和信号是光滑、有界且非负的，具有有限的密度质量和二阶矩以及有限的信号Fisher能量；无散度初始速度属于$H^4$。不施加小性假设，并允许正的常数信号背景。解在每个有限时间区间上保持有界。主要的新工具是在密度归一化下一致的局部熵估计，即使由此产生的消耗系数变得任意大。信号的终端表示分离了两种互补机制：消耗排除了在正信号点处的归一化密度集中，而局部加权估计改善了在零信号点处的密度可积性。这些机制共同产生了尺度不变密度量的衰减，并允许在不单独对信号梯度施加小性假设的情况下应用局部$\varepsilon$-正则性准则。

英文摘要

In 2005, Tuval et al. introduced a continuum model for the interaction of aerobic bacteria, oxygen, and an incompressible fluid. For the associated three-dimensional chemotaxis-consumption systems, global classical regularity for large data has remained a longstanding question, even after omitting the nonlinear convection term from the fluid equation. We answer this question and prove global classical solvability of the chemotaxis--Stokes Cauchy problem in \(\mathbb{R}^3\), with linear cell diffusion and bilinear signal consumption, under the following data assumptions. The initial density and signal are smooth, bounded, and nonnegative, with finite density mass and second moment and finite signal Fisher energy; the divergence-free initial velocity belongs to \(H^4\). No smallness assumption is imposed, and a positive constant signal background is allowed. The solution remains bounded on every finite time interval. The main new ingredient is a localized entropy estimate uniform under density normalization, even when the resulting consumption coefficient becomes arbitrarily large. A terminal representation of the signal separates two complementary mechanisms: consumption excludes normalized density concentration at positive-signal points, while a localized weighted estimate improves density integrability at zero-signal points. Together, these mechanisms yield decay of a scale-invariant density quantity and allow a local $\varepsilon$-regularity criterion to be applied without a separate smallness assumption on the signal gradient.

发表机构

• School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）

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