论文标题
关于Euler-Riesz系统的适应性和奇异性形成
On well-posedness and singularity formation for the Euler-Riesz system
论文作者
论文摘要
在本文中，我们研究了Euler-Riesz系统的初始值问题，其中相互作用强迫由$ \ nabla（-Δ）^{s}ρ$给出，对于某些$ -1 <s <0 $，$ s = -1 $，与经典的Euler-Poisson系统相对应。我们开发了一个功能框架，以建立欧拉 - 里斯系统的经典解决方案的本地存在和独特性。在此框架中，流体密度可能在无穷大时快速衰减，并且可以覆盖Euler-Poisson系统作为特殊情况。此外，当电势令人反感时，我们通过观察系统的双曲性性质，证明了无压力的Euler-riesz系统的本地良好性。最后，我们对具有吸引力或排斥性相互作用的等等速/等温欧拉河河流系统的经典溶液进行有限的时间爆炸介绍了足够的条件。基于对几个物理数量的估计值的证明为大量初始数据建立了有限的时间爆炸；特别是，密度不需要紧凑。
In this paper, we investigate the initial value problem for the Euler-Riesz system, where the interaction forcing is given by $\nabla(-Δ)^{s}ρ$ for some $-1<s<0$, with $s = -1$ corresponding to the classical Euler-Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler-Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler-Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler-Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler-Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support.
