论文标题
关于具有不规则初始数据的抛物线椭圆形凯勒 - 塞格系统的全球平滑解决方案
On the existence of global smooth solutions to the parabolic-elliptic Keller-Segel system with irregular initial data
论文作者
论文摘要
我们考虑抛物线弹药凯勒 - 塞格系统\ [ \左边\{ \ begin {Aligned} u_t＆=Δu -χ\ nabla \ cdot（u \ nabla v），\\ 0＆=ΔV -V + U \ end {Aligned} \正确的。 \ tag {$ \ star $} \]在平滑的界面$ω\ subseteq \ mathbb {r}^n $，$ n \ in \ mathbb {n} $中，带有neumann边界条件。我们在两个维度和三个维度上都考虑了趋化吸引力（$χ> 0 $）和排斥（$χ<0 $）方案。出于本文的目的，感兴趣的关键特征在于条件上所述系统仍在全球经典解决方案中，即使Laplacian的平滑属性也很不规则。关于这一点，我们显示的初始数据$μ\ in \ mathcal {m} _+（\overlineΩ）$，如果这两个 \ bullet \;＆n = 2，\，χ<0 \，\ text {或} \\ \ bullet \;＆n = 2，\，χ> 0 \，\ text {和初始质量很小或} \\ \ bullet \;＆n = 3，\，χ<0 \，\ text {and}μ= f \ in l^p（ω），p> 1 \ end {align}的保留，仍然可以在$ t = 0 $中连续构建全球经典解决方案（$ \ star $），在$ t = 0 $中连续$ t = 0 $（$ m m _）。
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 $M_+(\overlineΩ)$.
