We consider the stationary Navier-Stokes equations with nonhomogeneous boundary condition in a domain with several boundary components. If the boundary value satisfies only the necessary flux condition (GOC), Leray's inequality does not holds true in general and we cannot prove the existence of a solution. But for a 2-D domain which is symmetric with respect to a line and where the data is also symmetric, C. Amick showed the existence of solutions by reduction to absurdity; later H. Fujita proved Leray-Fujita's inequality and hence the existence of symmetric solutions. In this paper we give a new short proof of Leray-Fujita's inequality and hence a proof of the existence of weak solutions.
"Leray's Inequality for Fluid Flow in Symmetric Multi-connected Two-dimensional Domains." Tokyo J. Math. 35 (1) 63 - 70, June 2012. https://doi.org/10.3836/tjm/1342701344