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The nonhomogeneous boundary value problem for the stationary NavierStokes equations in 2D symmetric multiply connected domain with a cusp point on the boundary is studied. It is assumed that there is a source or sink in the cusp point. A symmetric solenoidal extension of the boundary value satisfying the LerayHopf inequality is constructed. Using this extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak symmetric solution is proved. No restrictions are assumed on the size of fluxes of the boundary value.

The non-stationary equations describing the motion of the second grade fluid are studied in an infinite threedimensional pipe of arbitrary cross-section. For sufficiently small data the existence of the unique Poiseuille type solution having a given time-dependent flow rate (flux) is proved. The velocity field U has all three components. However, we show that components U 1 , U 2 are secondary in comparison with the axial velocity U 3 .

Third order initial boundary value problem is studied in a bounded plane domain σ with C4 smooth boundary ∂σ. The existence and uniqueness of the solution is proved using Galerkin approximations and a priory estimates. The problem under consideration appear as an auxiliary problem by studying a second grade fluid motion in an infinite three-dimensional pipe with noncircular cross-section.