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Global solutions to the potential flow equations for a compressible viscous fluid at small Reynolds numbers. (English. Russian original) Zbl 0835.35111
Differ. Equations 30, No. 6, 935-947 (1994); translation from Differ. Uravn. 30, No. 6, 1010-1022 (1994).
The paper is devoted to analyzing the Stokes approximation \[ \rho {\partial \overline u\over \partial t}= \mu \Delta \overline u+ (\mu+ \lambda) \nabla (\text{div } \overline u)- \nabla P, \]
\[ {\partial \rho\over \partial t}+ \text{div}(\rho \overline u)= 0,\qquad P= P(\rho) \] of the Navier-Stokes equations for a compressible viscous fluid under barotropic motion. Here \(\rho\), \(\overline u\), and \(P\) are density, velocity vector, and pressure, respectively, \(\mu\) and \(\lambda\) are coefficients of dynamic and volumetric viscosity. The authors study the correctness of initial-boundary value problems for this system, in particular, for potential flows. The following results are obtained:
1) On the basis of new a priori estimates the existence in the sense of distributions (weak solutions) is proved for the multidimensional case.
2) In the two-dimensional case these solutions are shown to be smooth enough under properly smooth initial and boundary data.
3) Sufficient conditions are found for unique solvability in the sense of distributions.
4) The potentiality assumption is not essential and used for some simplifications of proofs.

35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
35D05 Existence of generalized solutions of PDE (MSC2000)