Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGlobal bounded solution in three-dimensional chemotaxis-Stokes model with arbitrary porous medium slow diffusion
302
0
0.0
(
0
)
Authors
Chunhua Jin
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
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 $c_0geq 0$.
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 work we study the degenerate diffusion equation $partial_{t}=x^{alpha}aleft(xright)partial_{x}^{2}+bleft(xright)partial_{x}$ for $left(x,tright)inleft(0,inftyright)^{2}$, equipped with a Cauchy initial data and the Dirichlet boundary condition at $0$. We assume that the order of degeneracy at 0 of the diffusion operator is $alphainleft(0,2right)$, and both $aleft(xright)$ and $bleft(xright)$ are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution $pleft(x,y,tright)$ and prove several properties for $pleft(x,y,tright)$; by conducting a localization procedure, we obtain an approximation for $pleft(x,y,tright)$ for $x,y$ in a neighborhood of 0 and $t$ sufficiently small, where the error estimates only rely on the local bounds of $aleft(xright)$ and $bleft(xright)$ (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of $alpha=1$. Our work extends part of the existing results to cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probability view (e.g., wellposedness of stochastic differential equations).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
