论文标题
凯勒 - 塞格系统的稳定奇异性形成在三个维度
Stable singularity formation for the Keller-Segel system in three dimensions
论文作者
论文摘要
我们考虑dimensions $ d \ geq 3 $的抛物线 - 椭圆形凯勒 - 塞格系统，这是质量超批评的情况。已知该系统表现出丰富的动态行为，包括通过自相似解决方案形成奇异性。 Brenner等人已经发现了一个明确的例子。 \ cite {bcksv99}，并猜想是非线性的径向稳定。我们证明了$ d = 3 $的猜想。我们的方法包括在相似性变量中重新制定问题，并通过半群理论方法研究交叉点Sobolev空间中的Cauchy进化。为了解决潜在的频谱问题，我们至关重要地依赖我们最近在\ cite {glosch20}中开发的技术。据我们所知，这为Keller-Segel系统稳定的自相似爆炸提供了第一个结果。此外，我们的结果向任何更高维度的扩展很简单。我们指出，我们的方法是一般且强大的，因此可以应用于广泛的抛物线模型。
We consider the parabolic-elliptic Keller-Segel system in dimensions $d \geq 3$, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example has been found more than two decades ago by Brenner et al. \cite{BCKSV99}, and is conjectured to be nonlinearly radially stable. We prove this conjecture for $d=3$. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we crucially rely on a technique we recently developed in \cite{GloSch20}. To our knowledge, this provides the first result on stable self-similar blowup for the Keller-Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.
