带自由边界的可压缩欧拉与欧拉-泊松方程组径向对称解的爆破

Blowup of Radially Symmetric Solutions to the Compressible Euler and Euler-Poisson Equations with Free Boundaries

本文在R^(N)(N=2,3)中研究描述流向外部真空的可压缩流体的欧拉与欧拉-泊松方程组径向对称解的爆破.在分离流体与真空的连续自由边界条件下考虑其自由边值问题.对于径向对称的欧拉方程组,证明若初始流平均向外流动,则其光滑解将在有限时刻爆破.对于带有斥力与弛豫项的单极与双极径向对称欧拉-泊松方程组,证明若某个与初始动量有关的加权泛函适当大,则其光滑解将在有限时刻爆破。

In this paper,the authors study the blowup of radially symmetric solutions to the compressible Euler and Euler-Poisson equations in R^(N)(N=2,3),which describe compressible fluids moving into outer vacuum.They consider the free boundary value problems under the continuous density condition across the free boundaries separating the fluid from vacuum.For the radially symmetric Euler equations,they show that the smooth solutions will blow up in a finite time if the initial flow moves outward on average.For the unipolar and bipolar radially symmetric Euler-Poisson equations with repulsive force and relaxation term,they prove that the smooth solutions will break down in a finite time if a weighted functional associated with the initial momentum is suitably large.