Rigidity of shear flows of the Euler equations in the plane

In this paper we show that steady states $u$ of the pressureless Euler equation which belong to $L^3_{loc}(\mathbb{R}^2,\mathbb{R}^2)$ are shear flows. This is achieved by combining results of degenerate Monge-Amp\`ere-type equations with the theory of two dimensional transport equations. We also show that the problem of rigidity and flexibility for the associated differential inclusion is rigid for sequences equibounded in $L^{4+}$ and flexible for sequences equibounded in $L^{4-}$, thus displaying a gap in the rigidity exponent between the exact and the approximate problem.