Mathematics

The water waves system was introduced by Euler in the 18th century to describe the motion of a fluid with a free surface under the action of gravity and surface tension, and it is still a very active research area in fluid dynamics. The unknowns of the problem are two time-dependent functions describing the velocity field of the fluid and the free-surface profile (giving the shape of the waves). A limitation of the classical formulations of the problem is that they are restricted to irrotational flows, whereas some important physical phenomena, like rip currents (and in general wave-currents interactions), can be described only by taking into account vorticity effects. The course is concerned with the mathematical analysis of three-dimensional water waves with vorticity. 

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• Part 1: Water waves system in the irrotational case, Zakharov-Craig-Sulem formulation, the Dirichlet-Neumann operator [1]
• Part 2: Non-existence of 3D travelling water waves with constant non-zero vorticity [2]
• Part 3: Existence of doubly-periodic gravity-capillary steady water waves on Beltrami flows [3]
• Part 4: Local well-posedness for three-dimensional water waves with general vorticity [4]

Exam:

The exam will consist in the presentation of an advanced topic agreed with the lecturer in a talk approximately 50 minutes long.

• [1] D. Lannes, The Water Waves problem: mathematical analysis and asymptotics, 2013
• [2] E. Wahlén, Non-existence of three-dimensional travelling water waves with constant non-zero vorticity, 2014
• [3] M.D. Groves, D. Nilsson, S. Pasquali and E. Wahlén, Analytical study of a generalised Dirichlet-Neumann operator and application to three-dimensional water waves on Beltrami flows, 2024
• [4] A. Castro and D. Lannes, Well-posedness and shallow-water stability for a new Hamiltonian formulation of the water waves equation with vorticity, 2015

 

Lectures:

Wed November  5 h10:00-12:00 room 133
Thu  November   6 h10:00-12:00 room 134
Thu  November 13 h10:00-12:00 room 134
Mon November 17 h14:00-16:00 room 134
Mon November 24 h14:00-16:00 room 134
Thu November  27 h11:00-13:00 room 134
Thu  December   4  h11:00-13:00 room 134
Thu  December 11 h14:00-16:00 room 134
Mon December 15 h14:00-16:00 room 134
Thu  December 18 h11:00-13:00 room 134