Large-Data Global Generalized Solutions in a Chemotaxis System with Tensor-Valued Sensitivities
Large-Data Global Generalized Solutions in a Chemotaxis System with Tensor-Valued Sensitivities
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DOI：
10.1137/140979708
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发表时间：
2015-06
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通讯作者：
M. Winkler
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文献类型：
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作者：
M. Winkler
趋化系统$u_t=\Delta u - \nabla \cdot（uS（x，u，v）\cdot\nabla v）;\ v_t=\Delta v - uf（v）$（本摘要中简称为（$\星星$）），对于一个细胞群体的密度$u=u（x，t）$和被前者消耗的一种有吸引力的化学物质的浓度$v=v（x，t）$，在有界域$\Omega\subset{\mathbb{R}}^n$，$n\ge 1$中的无通量边界条件下考虑，具有光滑边界，其中$f \in C^1（[0，\infty）;[0，\infty））$和$S \在C^2（\bar\Omega\times [0，\infty）^2;{\mathbb{R}}^{n\times n}）$中是给定的函数，使得f（0）=0。与相关的凯勒-Segel型问题的标量灵敏度，在存在这样的矩阵值$S$的系统（$\星星$）一般显然不具有任何有用的梯度状结构。因此，它的分析需要基于新类型的先验界限。使用时空$L^2$估计$\nabla \ln（u+1）$作为出发点，我们得到了一系列的紧性的解决方案，适当的正则化版本…
The chemotaxis system $u_t=\Delta u - \nabla \cdot (uS(x,u,v)\cdot\nabla v);\ v_t=\Delta v - uf(v)$ (referred to as ($\star$) in this abstract), for the density $u=u(x,t)$ of a cell population and the concentration $v=v(x,t)$ of an attractive chemical consumed by the former, is considered under no-flux boundary conditions in a bounded domain $\Omega\subset{\mathbb{R}}^n$, $n\ge 1$, with smooth boundary, where $f \in C^1([0,\infty);[0,\infty))$ and $S \in C^2(\bar\Omega\times [0,\infty)^2;{\mathbb{R}}^{n\times n})$ are given functions such that f(0)=0. In contrast to related Keller--Segel-type problems with scalar sensitivities, in the presence of such matrix-valued $S$ the system ($\star$) in general apparently does not possess any useful gradient-like structure. Accordingly, its analysis needs to be based on new types of a priori bounds. Using a spatio-temporal $L^2$ estimate for $\nabla \ln (u+1)$ as a starting point, we derive a series of compactness properties of solutions to suitably regularized vers...