an:02113150
Zbl 1080.35068
Ducomet, Bernard; Feireisl, Eduard; Petzeltov??, Hana; Stra??kraba, Ivan
Global in time weak solutions for compressible barotropic self-gravitating fluids
EN
Discrete Contin. Dyn. Syst. 11, No. 1, 113-130 (2004).
00104038
2004
j
35Q30 85A30 76N10 35D05
compressible self-gravitating fluid; Navier-Stokes-Poisson system; global in time solutions; weak solution
The Navier-Stokes-Poisson system:
\[
\rho_t+ \text{div}(\rho\vec u)= 0,\tag{1}
\]
\[
(\rho u^i)_t+ \text{div}(\rho u^i\vec u)+ p_{x_i}= \mu\Delta u^i+ (\lambda+ \mu)(\text{div\,}\vec u)_{x_i}+ G\rho\partial_x\Phi,\;i= 1,2,3,\tag{2}
\]
\[
-\Delta\Phi= \rho+ g\tag{3}
\]
describing the time evolution of the density \(\rho= \rho(t, x)\) and the velocity \(\vec u=\vec u(t, x)\) of a gaseous star is considered.
Here \(p\) is the pressure, \(\Phi\) is the gravitational potential of the star, \(G\) is a positive constant, \(g= g(x)\) is a given function, and the viscosity coefficients \(\mu\) and \(\lambda\) satisfy conditions: \(\mu> 0\), \(\lambda+{2\over 3}\mu\geq 0\).
The system (1)--(3) is considered with the initial conditions:
\[
\rho(0)= \rho_0,\quad (\rho u^i)(0)= q^i,\quad i= 1,2,3\tag{4}
\]
and the no-slip boundary conditions for the velocity:
\[
u^i|_{\partial\Omega}= 0,\quad i= 1,2,3.\tag{5}
\]
The main result of this paper states that if the data \(\rho_0\), \(q^i\), \(i= 1,2,3\) satisfy the compatibility conditions, the pressure \(p\) is given by the barotropic constitutive law and \(g\) belongs to the class \(L^1\cap L^\infty(\mathbb{R}^3)\), then there exists a global in time weak solution \(\rho\), \(\vec u\) of the problem (1)--(5).
A. Cichocka (Katowice)