×

zbMATH — the first resource for mathematics

Global in time weak solutions for compressible barotropic self-gravitating fluids. (English) Zbl 1080.35068
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).

MSC:
35Q30 Navier-Stokes equations
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35D05 Existence of generalized solutions of PDE (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI