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Influence of flux limitation on large time behavior in a three-dimensional chemotaxis-Stokes system modeling coral fertilization

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 Added by Ji Liu
 Publication date 2021
  fields
and research's language is English
 Authors Ji Liu




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In this paper, we consider the following system $$left{begin{array}{ll} n_t+ucdot abla n&=Delta n- ablacdot(nmathcal{S}(| abla c|^2) abla c)-nm, c_t+ucdot abla c&=Delta c-c+m, m_t+ucdot abla m&=Delta m-mn, u_t&=Delta u+ abla P+(n+m) ablaPhi,qquad ablacdot u=0 end{array}right.$$ which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where $mathcal{S}$ is a given function fulfilling $$|mathcal{S}(sigma)|leq K_{mathcal{S}}(1+sigma)^{-frac{theta}{2}},qquad sigmageq 0$$ with some $K_{mathcal{S}}>0.$ Based on conditional estimates of the quantity $c$ and the gradients thereof, a relatively compressed argument as compared to that proceeding in related precedents shows that if $$theta>0,$$ then for any initial data with proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded, and which also enjoys the stabilization features in the sense that $$|n(cdot,t)-n_{infty}|_{L^{infty}(Omega)}+|c(cdot,t)-m_{infty}|_{W^{1,infty}(Omega)} +|m(cdot,t)-m_{infty}|_{W^{1,infty}(Omega)}+|u(cdot,t)|_{L^{infty}(Omega)}rightarrow0 quadtextrm{as}~trightarrow infty$$ with $n_{infty}:=frac{1}{|Omega|}left{int_{Omega}n_0-int_{Omega}m_0right}_{+}$ and $m_{infty}:=frac{1}{|Omega|}left{int_{Omega}m_0-int_{Omega}n_0right}_{+}.$



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252 - Hailong Ye , Chunhua Jin 2021
In this paper, we study the time periodic problem to a three-dimensional chemotaxis-Stokes model with porous medium diffusion $Delta n^m$ and inhomogeneous mixed boundary conditions. By using a double-level approximation method and some iterative techniques, we obtain the existence and time-space uniform boundedness of weak time periodic solutions for any $m>1$. Moreover, we improve the regularity for $mlefrac{4}{3}$ and show that the obtained periodic solutions are in fact strong periodic solutions.
301 - Chunhua Jin 2021
In this paper, we study the consumption-chemotaxis-Stokes model with porous medium slow diffusion in a three dimensional bounded domain with zero-flux boundary conditions and no-slip boundary condition. In recent ten years, many efforts have been made to find the global bounded solutions of chemotaxis-Stokes system in three dimensional space. Although some important progress has been carried out in some papers, as mentioned by some authors, the question of identifying an optimal condition on m ensuring global boundedness in the three-dimensional framework remains an open challenge. In the present paper, we put forward a new estimation technique, completely proved the existence of global bounded solutions for arbitrary slow diffusion case, and partially answered the open problem proposed by Winkler.
The chemotaxis--Navier--Stokes system begin{equation*}label{0.1} left{begin{array}{ll} n_t+ucdot abla n=triangle n-chi ablacdotp left(displaystylefrac n {c} abla cright)+n(r-mu n), c_t+ucdot abla c=triangle c-nc, u_t+ (ucdot abla) u=Delta u+ abla P+n ablaphi, ablacdot u=0, end{array}right. end{equation*} is considered in a bounded smooth domain $Omega subset mathbb{R}^2$, where $phiin W^{1,infty}(Omega)$, $chi>0$, $rin mathbb{R}$ and $mu> 0$ are given parameters. It is shown that there exists a value $mu_*(Omega,chi, r)geq 0$ such that whenever $ mu>mu_*(Omega,chi, r)$, the global-in-time classical solution to the system is uniformly bounded with respect to $xin Omega$. Moreover, for the case $r>0$, $(n,c,frac {| abla c|}c,u)$ converges to $(frac r mu,0,0,0)$ in $L^infty(Omega)times L^infty(Omega)times L^p(Omega)times L^infty(Omega)$ for any $p>1$ exponentially as $trightarrow infty$, while in the case $r=0$, $(n,c,frac {| abla c|}c,u)$ converges to $(0,0,0,0)$ in $(L^infty(Omega))^4$ algebraically. To the best of our knowledge, these results provide the first precise information on the asymptotic profile of solutions in two dimensions.
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