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

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.

MSC:

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