Abstract:

This paper considers the rigidity of steady inhomogeneous incompressible Euler flows in two-dimensional annular domains. Under the assumption of no stagnation points and the slip boundary condition, with additional asymptotic conditions at infinity for unbounded domains and near the origin for punctured domains, the smooth fluids are proved to be circular shear flows, which extends the rigidity theorem for the homogeneous case to the inhomogeneous case. First, by establishing geometric properties of streamlines and the gradient of the stream function, the original system is transferred to a semilinear elliptic equation depending on the gradient terms. Then, by the moving plane method, the comparison principles are established in the corresponding domains, from which the radial symmetry properties of the stream function and streamlines are derived. Finally, for free boundary problems, a Serrin-type theorem for the inhomogeneous case is proved, based on which the rigidity theorem for contact discontinuity solutions is established.

Key words: Euler equations, annular flow, semilinear elliptic equations, free boundary problem