Free Boundary Value Problem to Compressible Viscous Flow

The talk mainly includes two parts. In the first part,  we consider the free boundary value problem to 3D spherically symmetric compressible isentropic Navier–Stokes–Poisson/Navier–Stokes equations. For self-gravitating gaseous stars (NSP) with γ-law pressure density function, the global existence of spherically symmetric weak solutions is shown, and the regularity and long time behavior of global solution are investigated with the total mass smaller than a critical mass when 6/5 < γ ≤ 4/3.  We also gives the global existence and long time behavior of solutions to Navier–Stokes equations. In the second part, we consider compressible Navier–Stokes system with a singular curve. The initial boundary value problem for the compressible barotropic Navier–Stokes equations is investigated in the case that the initial density has a jump discontinuity across an interior closed curve in two-dimensional bounded domain. If the initial data is a small perturbation of the constant state and the interior closed curve is near a circle inside the domain, the global existence and large time behavior of the piecewise strong solution is shown, in particular, the jump of the fluid density across the convecting curve decays exponentially in time.
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