论文标题
具有奇异灵敏度和逻辑源的二维趋化性的渐近型
Asymptotic profile of a two-dimensional chemotaxis--Navier--Stokes system with singular sensitivity and logistic source
论文作者
论文摘要
趋化性 - 纳维尔 - stokes system \ begin {equination*} \ label {0.1} \ left \ {\ begin {array} {ll} n_t+u \ cdot \ nabla n = \ triangle n-χ\ nabla \ cdotp \ left（\ displaystyle \ frac \ frac n {c} \ nabla c \ nabla c \ right）+n（r-μn）， C_T+U \ CDOT \ Nabla C = \三角形C-NC， u_t+（u \ cdot \ nabla）u =ΔU+\ nabla p+n \ nabla ϕ， \ nabla \ cdot u = 0， \ end {array} \ right。 \ end {equation*}在一个有限的平滑域$ω\ subset \ mathbb {r}^2 $中考虑，其中$ ϕ \ in w^{1，\ infty}（ω）$，$χ> 0 $，$χ> 0 $，$ r \ in \ MATHBB {r} $和$ umuty> 0 $ 0 $> 0 $。结果表明，存在一个值$μ_*（ω，χ，r）\ geq 0 $，以便每当$μ>μ_*（ω，χ，r）$时，系统的全球经典解决方案都相对于$ x \ incom yinch yinch y clibly。此外，对于情况，对于$ r> 0 $，$（n，c，\ frac {| \ nabla c |} c，u）$收敛到$（\ fracrμ，0,0,0,0）$ in $ l^\ infty（ω）指数为$ t \ rightarrow \ infty $，而在情况下，$ r = 0 $，$（n，c，\ frac {| \ nabla c |} c，u）$ clem clem $（l^\ infty（ω））^4 $ algebrai以$（0,0,0,0,0）$（0,0,0,0,0）$。据我们所知，这些结果提供了有关解决方案在二维中的渐近概况的第一个精确信息。
The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1} \left\{\begin{array}{ll} n_t+u\cdot \nabla n=\triangle n-χ\nabla\cdotp \left(\displaystyle\frac n {c}\nabla c\right)+n(r-μn), c_t+u\cdot \nabla c=\triangle c-nc, u_t+ (u\cdot \nabla) u=Δu+\nabla P+n\nablaϕ, \nabla\cdot u=0, \end{array}\right. \end{equation*} is considered in a bounded smooth domain $Ω\subset \mathbb{R}^2$, where $ϕ\in W^{1,\infty}(Ω)$, $χ>0$, $r\in \mathbb{R}$ and $μ> 0$ are given parameters. It is shown that there exists a value $μ_*(Ω,χ, r)\geq 0$ such that whenever $ μ>μ_*(Ω,χ, r)$, the global-in-time classical solution to the system is uniformly bounded with respect to $x\in Ω$. Moreover, for the case $r>0$, $(n,c,\frac {|\nabla c|}c,u)$ converges to $(\frac r μ,0,0,0)$ in $L^\infty(Ω)\times L^\infty(Ω)\times L^p(Ω)\times L^\infty(Ω)$ for any $p>1$ exponentially as $t\rightarrow \infty$, while in the case $r=0$, $(n,c,\frac {|\nabla c|}c,u)$ converges to $(0,0,0,0)$ in $(L^\infty(Ω))^4$ algebraically. To the best of our knowledge, these results provide the first precise information on the asymptotic profile of solutions in two dimensions.
