arXiv 2006.093452020-06-16math.AP

具有非规则初值的抛物-椭圆型Keller-Segel方程组整体光滑解的存在性

On the existence of global smooth solutions to the parabolic-elliptic Keller-Segel system with irregular initial data

Frederic Heihoff

DOI：: 10.1007/s10884-021-09950-y

摘要：

本文研究了抛物椭圆型Keller-Segel方程组\[ \left\{ \begin{aligned} u_t &= Δu - χ\nabla \cdot (u \nabla v), \\ 0 &= Δv - v + u \end{aligned} \right. \tag{$\star$} \]在光滑有界区域$Ω\subseteq \mathbb{R}^n$，$n\in\mathbb{N}$中的Neumann边界条件.我们在两个和三维的趋化吸引（$χ> 0$）和排斥（$χ< 0$）的情况下。本文的目的感兴趣的关键特征是在何种条件下，该系统仍然承认全球经典解决方案，由于平滑性质的拉普拉斯算子，即使初始数据是非常不规则的。关于这一点，我们表明，对于初始数据$μ\in \mathcal{M}_+(\overlineΩ)$，如果任何\begin{align} \bullet\;& n = 2,\, χ< 0\, \text{ or } \\ \bullet\;& n = 2,\, χ> 0\, \text{ and the initial mass is small or } \\ \bullet\;& n = 3,\, χ< 0\, \text{ and } μ= f \in L^p(Ω), p > 1 \end{align}持有，它仍然是可能的，以（$\star$），这是连续的$t = 0$在$M_+(\overlineΩ)$上的模糊拓扑结构的全球经典解决方案。

英文摘要：

We consider the parabolic-elliptic Keller-Segel system \[ \left\{ \begin{aligned} u_t &= Δu - χ\nabla \cdot (u \nabla v), \\ 0 &= Δv - v + u \end{aligned} \right. \tag{$\star$} \] in a smooth bounded domain $Ω\subseteq \mathbb{R}^n$, $n\in\mathbb{N}$, with Neumann boundary conditions. We look at both chemotactic attraction ($χ> 0$) and repulsion ($χ< 0$) scenarios in two and three dimensions. The key feature of interest for the purposes of this paper is under which conditions said system still admits global classical solutions due to the smoothing properties of the Laplacian even if the initial data is very irregular. Regarding this, we show for initial data $μ\in \mathcal{M}_+(\overlineΩ)$ that, if either \begin{align} \bullet\;& n = 2,\, χ< 0\, \text{ or } \\ \bullet\;& n = 2,\, χ> 0\, \text{ and the initial mass is small or } \\ \bullet\;& n = 3,\, χ< 0\, \text{ and } μ= f \in L^p(Ω), p > 1 \end{align} holds, it is still possible to construct global classical solutions to ($\star$), which are continuous in $t = 0$ in the vague topology on $M_+(\overlineΩ)$.

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