Global weak solutions and eventual smoothness in a 3D two-competing-species chemotaxis-Navier-Stokes system with two consumed signals
Global weak solutions and eventual smoothness in a 3D two-competing-species chemotaxis-Navier-Stokes system with two consumed signals
复制标题
具有两个消耗信号的 3D 两种竞争物种趋化-纳维-斯托克斯系统中的全局弱解和最终平滑度
DOI：
10.1002/mma.6154
复制
发表时间：
2020
影响因子：
2.9
通讯作者：
Willie Robert
中科院分区：
文献类型：
--
作者：
Zheng Pan;Willie Robert
本文研究了具有两个不同消耗信号的两竞争种群趋化性Navier Stokes系统（n1）t+u·n1=d1Δn1−χ1·（n1 c）+μ 1 n1（1−n1− a1 n2），在Ω×（0，∞）中，ct+u·c=d2Δc−α 1cn 2，在Ω×（0，∞）中，（n2）t+u·n2=d3Δn2−χ2·（n2 v）+μ 2n 2（1− a2 n1 −n2），在Ω×（0，∞）中，vt+u·v=d4Δv−α 2 vn 1，在Ω×（0，∞）中，ut+（u·）u=Δu+ P+（β 1 n1 +β 2n 2）n = 0，在Ω×（0，∞）中，n·u=0，在Ω×（0，∞）中，在零Neumann边界条件下，在齐次Dirichlet边界条件下，其中参数（）和（）为正数。该系统描述了两种竞争物种的进化，它们在液体周围环境中对两种不同的化学信号进行反应。最近，在以前的工作中已经推导出了二维情况下上述系统经典解的有界性和稳定性。然而，据我们所知，由于Navier-Stokes系统的困难，上述系统的解的适定性问题在三维环境中仍然是开放的。本文的目的是构造整体弱解，并证明经过一段时间的等待，这些弱解最终变得光滑。
This paper deals with a two‐competition‐species chemotaxis‐Navier‐Stokes system with two different consumed signals (n1)t+u·∇n1=d1Δn1−χ1∇·(n1∇c)+μ1n1(1−n1−a1n2),inΩ×(0,∞),ct+u·∇c=d2Δc−α1cn2,inΩ×(0,∞),(n2)t+u·∇n2=d3Δn2−χ2∇·(n2∇v)+μ2n2(1−a2n1−n2),inΩ×(0,∞),vt+u·∇v=d4Δv−α2vn1,inΩ×(0,∞),ut+(u·∇)u=Δu+∇P+(β1n1+β2n2)∇ϕ,inΩ×(0,∞),∇·u=0,inΩ×(0,∞), in a smooth bounded domainunder zero Neumann boundary conditions for, and homogeneous Dirichlet boundary condition for, where the parameters() and() are positive. This system describes the evolution of two‐competing species which react on two different chemical signals in a liquid surrounding environment. Recently, the boundedness and stabilization of classical solutions to the above system under two‐dimensional case have been derived in the previous works. However, to the best of our knowledge, the well‐posedness problem of solutions for the above system is still open in the three dimensional setting, because of the difficulties in the Navier‐Stokes system. The aim of this paper is to construct global weak solutions and show that after some waiting time, these weak solutions become eventually smooth.