The article ‘Traveling Waves Near Couette Flow for the 2D Euler Equation’, signed by Ángel Castro (ICMAT-CSIC) and Daniel Lear (University of Cantabria) has been published in the journal Communications in Mathematical Physics.

The study of the mathematical stability of the Couette flow for the 2D Euler equation started in the 19th century with the works of Kelvin, Orr, Reynolds, Rayleigh, Stokes, Sommerfeld and others. In their pioneering investigations they found that the linear problem is stable. However, experiments showed instabilities and transition to turbulence for any size of disturbance when the Reynolds number is large. This contradiction between theory and experiment is nowadays referred as the “Sommerfeld paradox”.

The stability of the Couette flow for 2D Euler has been intensely studied in the last decade and substantial progress has been made.  J. Bedrossian and N. Masmoudi proved that solutions starting close to the Couette flow (at the vorticity level) in a Gevrey space,   tend to a shear flow close to the Couette flow. The assumption on the Gevrey regularity is essential to get this result.  Y. Deng and M. Masmoudi showed that the previous result does not hold, in general.

If the distance to the Couette flow is measured in Sobolev spaces, there are several results in the literature studying the existence of nontrivial stationary or traveling solutions. Here trivial means that the velocity takes the form (u(y), 0).

Li and Z. Lin proved the existence of smooth nontrivial traveling waves close to the Couette flow in L2 (at the level of the velocity). Z. Lin and C. Zeng proved the existence of nontrivial smooth stationary solutions of 2D Euler close to the Couette flow in Hs, for s<3/2 (at the level of the vorticity). In addition, they proved that, for s>3/2, if there is a traveling wave close to the Couette flow then this traveling wave is trivial. In our paper, we prove the existence of nontrivial smooth traveling waves arbitrarily close to the Couette in Hs (at the level of the vorticity), with s<3/2, and with a speed of traveling of order 1.

Reference: Castro, Á., Lear, D. Traveling Waves Near Couette Flow for the 2D Euler Equation. Commun. Math. Phys. 400, 2005–2079 (2023). https://doi.org/10.1007/s00220-023-04636-6