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
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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)