Blowup for the C¹ Solutions of the Euler-Poisson Equations of Gaseous Stars in R^N · arXivDesk
1012.5364Dec 24, 201012 pages, We merged the result in "M.W. Yuen, Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces II, Pre-print, arXiv:1012.5143" in the updated version. Key Words: Euler-Poisson Equations, Integration Method, Blowup, Repulsive Forces, With Pressure, $C^{1}$ Solutions, No-Slip Boundary Condition, Compact Support, Initial Value Problem, First Boundary Condition
Blowup for the C^1 Solutions of the Euler-Poisson Equations of Gaseous Stars in R^N
The Newtonian Euler-Poisson equations with attractive forces are the classical models for the evolution of gaseous stars and galaxies in astrophysics. In this paper, we use the integration method to study the blowup problem of the N-dimensional system with adiabatic exponent γ>1, in radial symmetry. We could show that the C1 non-trivial classical solutions (ρ,V)
Nearby in the stack
, with compact support in
[0,R]
, where
R>0
is a positive constant with
ρ(t,r)=0
and
V(t,r)=0
for
r≥R
, under the initial condition equation H₀=∫₀^RrⁿV₀dr>√2R²n-N+4Mn(n+1)(n-N+2)% equation with an arbitrary constant
n>max(N−2,0),
blow up before a finite time
T
for pressureless fluids or
γ>1.
Our results could fill some gaps about the blowup phenomena to the classical
C1
solutions of that attractive system with pressure under the first boundary condition. In addition, the corresponding result for the repulsive systems is also provided. Here our result fully covers the previous case for
n=1
in "M.W. Yuen, Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces, Nonlinear Analysis Series A: Theory, Methods & Applications 74 (2011), 1465--1470".