Traveling waves near Couette flow for the 2D Euler equation. (English) Zbl 1516.35314
Summary: 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 [J. Bedrossian and N. Masmoudi, Publ. Math., Inst. Hautes Étud. Sci. 122, 195–300 (2015; Zbl 1375.35340)], 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 [Z. Lin and C. Zeng, Arch. Ration. Mech. Anal. 200, No. 3, 1075–1097 (2011; Zbl 1229.35197)].
MSC:
| 35Q31 | Euler equations |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 35C07 | Traveling wave solutions |
| 35B20 | Perturbations in context of PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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