论文标题
古典patlak-keller-segel方程的长期动态
Long-time dynamics of classical Patlak-Keller-Segel equation
论文作者
论文摘要
当空间尺寸$ n = 2 $时，众所周知，当存在且仅当其初始总质量不在超临界状态下时，就存在对经典Patlak-keller-segel方程（简称PKS方程）的全球温和解决方案。但是，要研究全球温和解决方案的长期行为至$ 2 $ d PKS方程通常需要在初始数据上进行有限的能量和有限的瞬间假设。在本文中，我们介绍了一个新颖的论点来推动和伸展时空带。通过这种方式，我们获得了在相似性变量下表达的PKS方程的$ l^1 $ compactness。结果，我们获得了亚临界方程中2D PKS方程的全局动力学，而没有其他假设。至于空间尺寸$ n \ geq 3 $的较高维度的情况，我们还表征了全球轻度解决方案对PKS方程的长期渐近学。通过对细胞密度的有限量质量假设，如果时间较大，对PKS方程的任何全球轻度解决方案都将接近自相似的轮廓，前提是有一系列时间到Infinity的序列，细胞密度的$ l^\ infty $ norm会收敛到零。在较高维度的情况下，自相似的配置文件由函数$ m \ mathcal {g} _n $给出，其中$ m $是单元格的总质量，$ \ mathcal {g} _n $表示标准的$ n $ n $ dimensional-demensional-demensional-demenmentional progussian概率密度。在任何维度上也讨论了与自相似概况的收敛速率。特别是在较高维度的情况下，如果初始数据具有有限的第二刻，则可以提高$ l^1 $内部数据的一般收敛速率。实际上，当时间较大和$ n \ geq 3 $时，如果初始密度的第二瞬间有限，我们以最佳方式提供了全球轻度解决方案的高阶近似值。本文研究的所有收敛率均在[1，\ infty] $中的$ l^p $ norm下。
When the spatial dimension $n =2$, it has been well-known that a global mild solution to classical Patlak-Keller-Segel equation (PKS equation for short) exists if and only if its initial total mass is not in supercritical regime. However, to study long-time behavior of a global mild solution to $2$D PKS equation usually requires finite-free-energy and finite-second-moment assumptions on initial data. In this article, we introduce a novel argument to push and stretch a space-time strip. By this way, we gain $L^1$-compactness of PKS equation expressed under similarity variables. As a consequence, we obtain global dynamics of 2D PKS equation in subcritical regime with no additional assumptions. As for the higher dimensional case in which the spatial dimension $n \geq 3$, we also characterize the long-time asymptotics of global mild solutions to PKS equation. With a finite-total-mass assumption on density of cells, any global mild solution to PKS equation will approach a self-similar profile when time is large, provided that there is a sequence of time going to infinity on which $L^\infty$-norm of the density of cells converges to zero. The self-similar profile in the higher dimensional case is given by the function $M\mathcal{G}_n$, where $M$ is the total mass of cells and $\mathcal{G}_n$ denotes the standard $n$-dimensional Gaussian probability density. Convergence rates to self-similar profiles are also discussed in any dimensions. Particularly in the higher dimensional case, the general convergence rate for $L^1$-initial data can be improved if the initial data has a finite second moment. In fact, when time is large and $n \geq 3$, we provide, in an optimal way, a higher-order approximation of global mild solutions to PKS equation if the initial density has a finite second moment. All convergence rates studied in this article are under the $L^p$-norm with $p \in [1,\infty]$.
