Traveling Waves Near Couette Flow for the 2D Euler Equation

Abstract

In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in $$H^s$$ , with $$s<3/2$$ , at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces [see Bedrossian and Masmoudi (Publ Math Inst Hautes Etudes Sci 122:195–300, 2015)], where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the $$H^{\frac{3}{2}+}$$ neighborhoods of Couette flow [see Lin and Zeng (Arch Ration Mech Anal 200:1075–1097, 2011)].