Global in time weak solutions for compressible barotropic self-gravitating fluids
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).
scientific article; zbMATH DE number 2113150