Frolov, N. N. Solvability of a boundary problem of motion of an inhomogeneous fluid. (English. Russian original) Zbl 0815.76024 Math. Notes 53, No. 6, 650-656 (1993); translation from Mat. Zametki 53, No. 6, 130-140 (1993). The author considers the steady-state motion of a viscous, incompressible and inhomogeneous fluid in a bounded domain \(\Omega \subset \mathbb{R}^ 2\), i.e. \(-\nu \Delta V + \rho (V \cdot \nabla)V + \nabla P = \rho f\), \(\text{div }V = 0\), \(\text{div }(\rho V) = 0\) in \(\Omega\), \(V_{| \Gamma} = V_ 0\), \(\rho_{| \gamma_ i} = \rho_ i\), \(i = 1,\dots,n\), where \(\partial \Omega = \Gamma = \Gamma_ 1 \cup \dots \cup \Gamma_ n\), and \(\gamma_ i\) is an arcwise connected set on \(\Gamma_ i\). The unknowns of the problem are the velocity \(V\), the pressure \(P\) and the density \(\rho\). In a first step, the author reformulates the system in such a way that the only unknown becomes the stream function \(\psi\) of \(V\). This new problem is afterwards solved with the help of the Leray-Schauder principle. Finally, the regularity properties of the solution are examined: if \(\Gamma \in C^{k + 3}\), \(V_ 0 \in C^{k + 2}(\Gamma)\), \(f \in W^{k,p}(\Omega)\) \((p \geq 2)\) and \(\rho_ i \in C^{k,\alpha}(\gamma_ i)\), then \(V \in H^ 1(\Omega) \cap W^{k + 2,p} (\Omega)\) and \(\rho \in C^{k,\alpha}(\overline{\Omega})\). Reviewer: K.Deckelnick (Freiburg i.Br.) Cited in 1 ReviewCited in 2 Documents MSC: 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q30 Navier-Stokes equations Keywords:generalized solution; stream function; Leray-Schauder principle; regularity PDFBibTeX XMLCite \textit{N. N. Frolov}, Math. Notes 53, No. 6, 1 (1993; Zbl 0815.76024); translation from Mat. Zametki 53, No. 6, 130--140 (1993) Full Text: DOI References: [1] A. V. Kazhikhov, ?Solvability of the initial-boundary problem for the equations of motion of an inhomogeneous viscous incompressible fluid,? Dokl. Akad. Nauk SSSR,216, No. 5, 1008-1010 (1974). [2] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [Russian translation], Mir, Moscow (1980). [3] O. A. Ladyzhenskaya, Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid [Russian translation], Mir, Moscow (1970). · Zbl 0215.29004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.