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Register a new userGlobal classical small-data solutions for a three-dimensional Keller--Segel--Navier--Stokes system modeling coral fertilization
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We are concerned with the Keller--Segel--Navier--Stokes system begin{equation*} left{ begin{array}{ll} rho_t+ucdot ablarho=Deltarho- ablacdot(rho mathcal{S}(x,rho,c) abla c)-rho m, &!! (x,t)in Omegatimes (0,T), m_t+ucdot abla m=Delta m-rho m, &!! (x,t)in Omegatimes (0,T), c_t+ucdot abla c=Delta c-c+m, & !! (x,t)in Omegatimes (0,T), u_t+ (ucdot abla) u=Delta u- abla P+(rho+m) ablaphi,quad ablacdot u=0, &!! (x,t)in Omegatimes (0,T) end{array}right. end{equation*} subject to the boundary condition $( ablarho-rho mathcal{S}(x,rho,c) abla c)cdot u!!=! abla mcdot u= abla ccdot u=0, u=0$ in a bounded smooth domain $Omegasubsetmathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $mathcal{S}in C^2(overlineOmegatimes [0,infty)^2)^{3times 3}$ satisfies $|mathcal{S}(x,rho,c)|leq C_S $ for some $C_S>0$, and the initial data satisfy certain smallness conditions.
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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 $m_{infty}:=frac{1}{|Omega|}left{int_{Omega}m_0-int_{Omega}n_0right}_{+}.$
In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane $rr^2$, if the total mass of the cells is strictly less than $8pi$, classical solutions exist for any finite time, and their $H^s$-Sobolev norms are almost uniformly bounded in time. For the radially symmetric solutions, this $8pi$-mass threshold is critical. On the torus $mathbb{T}^2$, the solutions are uniformly bounded in time under the same mass constraint.
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.
We study the regularity and large-time behavior of a crowd of species driven by chemo-tactic interactions. What distinguishes the different species is the way they interact with the rest of the crowd: the collective motion is driven by different chemical reactions which end up in a coupled system of parabolic Patlak-Keller-Segel equations. We show that the densities of the different species diffuse to zero provided the chemical interactions between the different species satisfy certain sub-critical condition; the latter is intimately related to a log-Hardy-Littlewood-Sobolev inequality for systems due to Shafrir & Wolansky. Thus for example, when two species interact, one of which has mass less than $4pi$, then the 2-system stays smooth for all time independent of the total mass of the system, in sharp contrast with the well-known breakdown of one specie with initial mass$> 8pi$.
In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.
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