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A recent prominent result asserts that steady incompressible Euler flowsstrictly away from stagnation in a two-dimensional infinitely long strip mustbe shear flows. On the other hand, flows with stagnation points, verychallenging in analysis, are interesting and important phenomenon in fluids. Inthis paper, we not only prove the uniqueness and existence of steady flows withstagnation points, but also obtain the regularity of the boundary of stagnationset, which is a class of obstacle type free boundary.  First, we prove a global uniqueness theorem for steady Euler system withPoiseuille flows as upstream far field state in an infinitely long strip. Dueto the appearance of stagnation points, the nonlinearity of the semilinearequation for the stream function becomes non-Lipschitz. This creates achallenging analysis problem since many classical analysis methods do not applydirectly. Second, the existence of steady incompressible Euler flows, tendingto Poiseuille flows in the upstream, are established in an infinitely longnozzle via variational approach. A very interesting phenomenon is theregularity of the boundary of non-stagnant region, which can be regarded as anobstacle type free boundary and is proved to be globally $C^1$. Finally, theexistence of stagnation region is proved as long as the nozzle is wider thanthe width of the nozzle at upstream where the flows tend to Poiseuille flows.

Analysis on the steady Euler flows with stagnation points in an infinitely long nozzle