Some considerations on the nonlinear stability of stationary planar Euler flows (Q580098)

The authors investigate the stability of stationary solutions of two- dimensional Euler equations \(\partial_ t\omega +(u,\nabla)\omega =0\), \(-\Delta \psi =\omega\), where \(\omega =curl u=\partial_ xu^{(2)}- \partial_ yu^{(1)}\), \(\psi\) is a stream function, i.e. \(u=\nabla^{\perp}\psi =(\partial_ y\psi,-\partial_ x\psi)\). They prove that the nonincreasing solution \({\bar \omega}\)(x,y)\(={\bar \xi}(y)\in L_{\infty}\), in the domain \(\Lambda =(0,L)\times (0,A)\), is stable in the \(L_ 1\) sense with respect to \(L_{\infty}\) perturbations. Further, the authors give sufficient conditions ensuring the stability of a stationary Euler flow with respect to perturbations including small deformations of the domain occupied by the fluid.