Prandtl-Batchelor flows on a disk (Q2684870)

The paper studies the steady Navier-Stokes equation \(u^\epsilon\cdot\nabla u^\epsilon+\nabla p^\epsilon-\epsilon^2 \Delta u^\epsilon=0\), \(\nabla\cdot u^\epsilon=0\) in the two-dimensional unit disk \(B_1\) with the rotating boundary condition \(u^\epsilon =[\alpha+\eta f(\theta)]t\) on \(\partial B_1\). Here \(\epsilon>0\), \(\alpha>0\), \(\eta\) is a small number, \(t\) is the unit tangential vector to \(\partial B_1\) and \(f(\theta)\) is a \(2\pi\)-periodic smooth function. It is shown the existence of a solution \((u^\epsilon,p^\epsilon)\) which converges to a solution of the steady Euler equation with constant vorticity as \(\epsilon\to 0\).