论文标题
一维随机凯勒 - 具有时间均匀空间维纳过程的segel模型
The one-dimensional stochastic Keller--Segel model with time-homogeneous spatial Wiener processes
论文作者
论文摘要
趋化性是细胞和生物体的基本机制，该机制负责吸引微生物进入食物，胚胎细胞进入发育中的组织或免疫细胞到感染部位。数学上的趋化性由Patlak-Keller- segel模型描述。研究限制行为时，这种宏观的方程系统是从微观模型得出的。但是，在将极限并从显微镜方程传递到宏观方程时，忽略了波动。通过高斯随机场扰动系统恢复系统的固有随机性。这使我们有动力研究经典过程受到扰动的经典patlak segel系统。 我们在均质的Neumann边界条件下研究了经典的Patlak-Keller- segel系统的随机版本，以$ \ MATHCAL {O} = [0,1] $。特别是，让$ \ MATHCAL {W} _1 $，$ \ MATHCAL {W} _2 $为两个时间均匀的空间WIENER在一个过滤的概率空间$ \ Mathfrak {a a} $上。令$ u $和$ v $表示化学信号的细胞密度和浓度。我们研究了耦合系统\ begin {align*}＆d {u} - （r_uΔu-χ{\ rm div}（u \ nabla v））\，dt = u \ circ d \ circ d \ circ d \ natercal {w} _1，\ \＆d { v \ circ d \ mathcal {w} _2，\ end {align*}带有初始条件$（u（u（u（0），v（0）），v（0））=（u_0，v_0）$。正项$ r_u $和$ r_v $分别是细胞的扩散率和趋化因素，正值$χ$是趋化敏感性，$α\ ge0 $是所谓的减湿常数。噪音以Stratonovich的意义解释。给定$ t> 0 $，我们将证明存在$ [0，t] $的martingale解决方案。
Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak--Keller--Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak--Keller--Segel system perturbed by random processes. We study a stochastic version of the classical Patlak--Keller--Segel system under homogeneous Neumann boundary conditions on an interval $\mathcal{O}=[0,1]$. In particular, let $\mathcal{W}_1$, $\mathcal{W}_2$ be two time-homogeneous spatial Wiener processes over a filtered probability space $\mathfrak{A}$. Let $u$ and $v$ denote the cell density and concentration of the chemical signal. We investigate the coupled system \begin{align*} & d {u} - ( r_uΔu- χ{\rm div }( u\nabla v) )\, dt =u\circ d\mathcal{W}_1, \\ & d{v} -(r_v Δv -αv)\, dt = βu \, dt+ v\circ d\mathcal{W}_2, \end{align*} with initial conditions $(u(0),v(0))=(u_0,v_0)$. The positive terms $r_u$ and $r_v$ are the diffusivity of the cells and chemoattractant, respectively, the positive value $χ$ is the chemotactic sensitivity, $α\ge0$ is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given $T>0$, we will prove the existence of a martingale solution on $[0,T]$.
