arXiv 2408.095142024-08math.AP

不可压缩两相流中具有趋化性和主动输运的Navier-Stokes-Cahn-Hilliard系统全局弱解的正则性传播

Regularity Propagation of Global Weak Solutions to a Navier-Stokes-Cahn-Hilliard System for Incompressible Two-phase Flows with Chemotaxis and Active Transport

School of Mathematics, Hangzhou Normal University（杭州师范大学数学学院）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）

Jingning He, Hao Wu

备注：: Major revision in Section 3.2 has been made. Typos fixed, references updated

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AI总结：

该研究分析了一个包含趋化、主动输运和长程相互作用的不可压缩两相流扩散界面模型，证明了全局弱解随时间逐步正则化并最终稳定到平衡态，揭示了各机制对正则性传播的影响。

摘要：

我们分析了一个描述不可压缩粘性两相流动力学的扩散界面模型，该模型包含了趋化性、主动输运以及Oono型长程相互作用等机制。该演化系统耦合了体积平均流体速度$\bm{v}$的Navier-Stokes方程、相场变量$φ$的对流Cahn-Hilliard方程，以及化学物质密度$σ$的平流-扩散方程。对于三维空间中具有物理相关奇异势的初边值问题，我们证明了每个全局弱解$(\bm{v}, φ, σ)$随时间表现出正则性的传播。具体而言，在任意正时间之后，相场变量$φ$转变为强解，而化学密度$σ$仅部分正则化。随后，速度场$\bm{v}$在足够长的时间后变得正则，这进而促进了化学密度$σ$的进一步正则化，从而又提升了$φ$的空间正则性。此外，我们证明了当$t\to +\infty$时，每个全局弱解都稳定到单一平衡态。我们的分析揭示了趋化性、主动输运和长程相互作用在不同时间阶段对正则性传播的影响。证明依赖于几个关键点，包括具有Leray型速度场$\bm{v}$的对流Cahn-Hilliard-扩散系统的一个新的正则性结果、$φ$在长时间下的严格分离性质，以及分别针对全系统和给定速度下$(φ, σ)$子系统的两个条件唯一性结果。

英文摘要：

We analyze a diffuse interface model that describes the dynamics of incompressible viscous two-phase flows, incorporating mechanisms such as chemotaxis, active transport, and long-range interactions of Oono's type. The evolution system couples the Navier--Stokes equations for the volume-averaged fluid velocity $\bm{v}$, a convective Cahn--Hilliard equation for the phase-field variable $φ$, and an advection-diffusion equation for the density of a chemical substance $σ$. For the initial boundary value problem with a physically relevant singular potential in three dimensions, we demonstrate that every global weak solution $(\bm{v}, φ, σ)$ exhibits a propagation of regularity over time. Specifically, after an arbitrary positive time, the phase-field variable $φ$ transitions into a strong solution, whereas the chemical density $σ$ only partially regularizes. Subsequently, the velocity field $\bm{v}$ becomes regular after a sufficiently large time, followed by a further regularization of the chemical density $σ$, which in turn enhances the spatial regularity of $φ$. Furthermore, we show that every global weak solution stabilizes towards a single equilibrium as $t\to +\infty$. Our analysis uncovers the influence of chemotaxis, active transport, and long-range interactions on the propagation of regularity at different stages of time. The proof relies on several key points, including a novel regularity result for a convective Cahn--Hilliard--diffusion system with a velocity field $\bm{v}$ of Leray type, the strict separation property of $φ$ for large times, as well as two conditional uniqueness results pertaining to the full system and its subsystem for $(φ, σ)$ with a given velocity, respectively.

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