Abstract
In this paper, we study the nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in two-dimensional symmetric domains with finitely many outlets to infinity. The domains may have no self-symmetric outlet (-type domain), one self-symmetric outlet (
-type domain) or two self-symmetric outlets (
-type domain). We construct a symmetric solenoidal extension of the boundary value satisfying the Leray–Hopf inequality. After having such an extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak solution follows. Notice that we do not impose any restrictions on the size of the fluxes over the inner and outer boundaries. Moreover, the Dirichlet integral of the solution can be either finite or infinite depending on the geometry of the domains.
Notes
No potential conflict of interest was reported by the authors.
1 If an outlet contains the -axis, then this outlet itself is symmetric with respect to the
-axis and it is called self-symmetric outlet.
2 The concept of virtual drain function was introduced by Fujita [Citation12].
3 Notice that the integral in (Equation4.2(4.2) ) over
is equal to zero since
in
4 Here we use the fact that due to the symmetry assumptions, the second component of
vanishes on the
-axis (in trace sense).
5 Let us remind that is a virtual drain function which removes the fluxes from the inner boundaries
to the outer boundary
.