E.E. Lokharu | Fine properties of steady water waves
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2022-08-01

O.A. Ladyzhenskaya centennial conference on PDE’s, St. Petersburg, July 16 – July 22, 2022
Evgeniy E. Lokharu (Lund University) 
Fine properties of steady water waves

In this talk we will discuss some recent results on two dimensional steady
water waves; see [1]. We will explain how the Benjamin and Lighthill conjecture can be significantly refined and will prove a new bound for the amplitude of an arbitrary Stokes wave in terms of the non-dimensional Bernoulli constant. Our result, in particular, implies the inequality 𝑎 ≤ 𝐶𝑐2/𝑔, where 𝑎 is the amplitude, 𝑐 is the speed of the wave, and 𝑔 is the gravitational constant. This fact is valid for arbitrary Stokes waves irrespectively of the amplitude with an absolute constant 𝐶. Another observation is that any extreme Stokes wave over a sufficiently deep stream has necessarily a small amplitude, provided the non-dimensional mass flux is much smaller than the depth.
References
[1] E. Lokharu, On the amplitude and the flow force constant of steady water
waves, J. Fluid Mech. 921 (2021).