Nonhomogeneous boundary value problem for Navier–Stokes equations in 2D symmetric unbounded domains

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.

Keywords:

• Stationary Navier–Stokes equations
• nonhomogeneous boundary value problem
• nonzero flux
• 2-dimensional domains with outlets to infinity
• symmetry

AMS Subject Classifications:

• 35Q30
• 35J65
• 76D03
• 76D05

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 [12].

3 Notice that the integral in (4.2(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 .

Additional information

Funding

The research leading to these results has received funding from Lithuanian-Swiss cooperation programme to reduce economic and social disparities within the enlarged European Union [grant number CH-3-SMM-01/01].