The Euler equations in planar nonsmooth convex domains

As a model problem for the barotropic mode of the primitive equations of the oceans and atmosphere, we consider the Euler system on a bounded convex planar domain Ω , endowed with non-penetrating boundary conditions. For 4 3 ≤ p ≤ 2 , and initial and forcing data with L p ( Ω ) vorticity we show the existence of a weak solution, enriching and extending the results of Taylor (2000) [32]. physical case of a rectangular domain Ω = [ 0 , L 1 ] × [ 0 , L 2 ] , a similar result holds for all 2 < p < ∞ as well. Moreover, by means of a new BMO -type regularity estimate for the Dirichlet problem on a planar domain with corners, we prove uniqueness of solutions with bounded initial vorticity.