Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity. (English) Zbl 0943.35075

Itoh, Shigeharu; Tani, Atusi

Tokyo J. Math. 22, No. 1, 17-42 (1999).

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.

Reviewer: G.Paşa (Bucureşti)

MSC:

35Q35	PDEs in connection with fluid mechanics
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids

Keywords:

convergence for vanishing viscosity; evolution equation for density; a priori estimates; Navier-Stokes type equation; unicity; existence; approximate solution