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Register a new userAsymptotic profile of a two-dimensional chemotaxis--Navier--Stokes system with singular sensitivity and logistic source
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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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This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion begin{eqnarray} left{begin{array}{lll} n_t+ucdot abla n= ablacdot(| abla n|^{p-2} abla n)- ablacdot(nchi(c) abla c),& xinOmega, t>0, c_t+ucdot abla c=Delta c-nf(c),& xinOmega, t>0, u_t+(ucdot abla) u=Delta u+ abla P+n ablaPhi,& xinOmega, t>0, ablacdot u=0,& xinOmega, t>0 end{array}right. end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $Omegasubset mathbb{R}^3$ with smooth boundary. Here, $Phiin W^{1,infty}(Omega)$, $0<chiin C^2([0,infty))$ and $0leq fin C^1([0,infty))$ with $f(0)=0$. It is proved that if $p>frac{32}{15}$ and under appropriate structural assumptions on $f$ and $chi$, for all sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least one global weak solution.
In this paper, the existence and uniqueness of strong axisymmetric solutions with large flux for the steady Navier-Stokes system in a pipe are established even when the external force is also suitably large in $L^2$. Furthermore, the exponential convergence rate at far fields for the arbitrary steady solutions with finite $H^2$ distance to the Hagen-Poiseuille flows is established as long as the external forces converge exponentially at far fields. The key point to get the existence of these large solutions is the refined estimate for the derivatives in the axial direction of the stream function and the swirl velocity, which exploits the good effect of the convection term. An important observation for the asymptotic behavior of general solutions is that the solutions are actually small at far fields when they have finite $H^2$ distance to the Hagen-Poiseuille flows. This makes the estimate for the linearized problem play a crucial role in studying the convergence of general solutions at far fields.
This paper deals with a boundary-value problem in three-dimensional smooth bounded convex domains for the coupled chemotaxis-Stokes system with slow $p$-Laplacian diffusion begin{equation} onumber left{ begin{aligned} &n_t+ucdot abla n= ablacdotleft(| abla n|^{p-2} abla nright)- ablacdot(n abla c), &xinOmega, t>0, &c_t+ucdot abla c=Delta c-nc,&xinOmega, t>0, &u_t=Delta u+ abla P+n ablaphi ,&xinOmega, t>0, & ablacdot u=0, &xinOmega, t>0, end{aligned} right. end{equation} where $phiin W^{2,infty}(Omega)$ is the gravitational potential. It is proved that global bounded weak solutions exist whenever $p>frac{23}{11}$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0geq 0$ and $c_0geq 0$.
A well-known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the Navier-Stokes system coupled with a convective Cahn-Hilliard equation. In some recent contributions the standard Cahn-Hilliard equation has been replaced by its nonlocal version. The corresponding system is physically more relevant and mathematically more challenging. Indeed, the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness. In fact, even in the two-dimensional case, uniqueness of weak solutions is still an open problem. Here we take a step forward in the case of regular potentials. First we prove the existence of a (unique) strong solution in two dimensions. Then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data. This result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions. We also demonstrate that each trajectory converges to a single equilibrium, provided that the potential is real analytic and the external forces vanish.
This paper is concerned with traveling wave solutions of the following full parabolic Keller-Segel chemotaxis system with logistic source, begin{equation} begin{cases} u_t=Delta u -chi ablacdot(u abla v)+u(a-bu),quad xinmathbb{R}^N cr tau v_t=Delta v-lambda v +mu u,quad xin mathbb{R}^N, end{cases}(1) end{equation} where $chi, mu,lambda,a,$ and $b$ are positive numbers, and $tauge 0$. Among others, it is proved that if $b>2chimu$ and $tau geq frac{1}{2}(1-frac{lambda}{a})_{+} ,$ then for every $cge 2sqrt{a}$, (1) has a traveling wave solution $(u,v)(t,x)=(U^{tau,c}(xcdotxi-ct),V^{tau,c}(xcdotxi-ct))$ ($forall, xiinmathbb{R}^N$) connecting the two constant steady states $(0,0)$ and $(frac{a}{b},frac{mu}{lambda}frac{a}{b})$, and there is no such solutions with speed $c$ less than $2sqrt{a}$, which improves considerably the results established in cite{SaSh3}, and shows that (1) has a minimal wave speed $c_0^*=2sqrt a$, which is independent of the chemotaxis.
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