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On non-homogeneous viscous incompressible fluids. Existence of regular solutions. (English) Zbl 0940.35153

The author considers the motion of non-homogeneous viscous fluids in a bounded domain in \( \mathbb{R}^3\) with the viscosity coefficient depending on the density. The existence of weak solutions was obtained by E. Fernández-Cara and F. Guillén [Ann. Fac. Sci. Toulouse, VI. Sér., Math. 2, No. 2, 185-204 (1993; Zbl 0806.35135)] and in a general case by P. L. Lions [Mathematical topics in fluid mechanics. Vol. 1: Incompressible models. Clarendon Press, Oxford (1996; Zbl 0866.76002)]. Concerning the uniqueness of the regular strong solution we refer to the book of P. L. Lions (loc. cit.).
The aim of this work is to prove the existence of a local regular solution in the case of regular data and positive density. First, the author linearizes the problem and shows the existence and uniqueness for the linearized problem. Further, he proves that there exists \(R\) such that, if \(\|f\|_{L^q(Q_t)^3} +\|u_0\|_{W^{2-2/q,q}(\Omega)^3} \leq R\) (where \(f\) – external forces, \(u_0\) – initial density, \(Q_T\) – space-time cylinder, and \(\Omega \) – domain) or \(T\) is small enough, we can apply the Schauder theorem, which gives the existence of a fixed point, and then we have the existence for the original problem.

MSC:

35Q30 Navier-Stokes equations
35B65 Smoothness and regularity of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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