Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity (Q1304926)

A ``hybrid'' system is considered, formed by a Navier-Stokes type equation, the divergence free condition for ``velocity'' and an evolution equation for ``density''. The main point is to study the above system as function of the ``viscosity''. The Cauchy problem is studied with corresponding boundary conditions, in the cases of non-zero and vanishing ``viscosity''. A unicity and existence theorem is given for the two above cases. An interesting theorem is proved, concerning the convergence of the solution when the ``viscosity'' tends to zero. To prove this result, an approximate solution is constructed inductively and ``a priori'' estimates are given.