Two dimensional water waves

16.06.2021 09:50 - 10:35

Mihaela Ifrim (University of Wisconsin)


Abstract: The classical water-wave problem consists of solving the Euler equations in the presence of a free fluid surface (e.g the water-air interface). This talk will provide an overview of recent developments concerning the motion of a two-dimensional incompressible fluid with a free surface. There is a wide range of problems that fall under the heading of water waves, depending on a number of assumptions that can be applied: surface tension, gravity, finite bottom, infinite bottom, rough bottom, etc., and combinations thereof. We will present the physical motivation for studying such problems, followed by the discussion of several interesting mathematical questions related to them.

We will focus on the 2D gravity water waves in infinite depth, with zero vorticity. The first step in the analysis is the choice of coordinates, where multiple choices are available. Once the equations of motion are derived we will discuss the main issues arising in the study of local well-posedness, as well as the long time behaviour of solutions with small, or small and localized data. Explicitly, we are concerned with low regularity solutions for the water wave equations in two space dimensions both for the local and on global theory.  Such solutions have been proved to exist earlier in much higher regularity.

In the last part of the talk we will introduce a new, very robust method which allows one to obtain enhanced lifespan bounds for the solutions at very low regularity. One key ingredient here is represented by the newly developed balanced cubic energy estimates. Another is the nonlinear vector field Sobolev inequalities, an idea we first introduced in the context of the Benjamin-Ono equation. If time permits, we will also introduce an alternative approach to the scattering theory, which in some cases yields a straightforward route to proving global existence results and obtaining an asymptotic description of solutions. This is joint work with Daniel Tataru, and in part with Albert Ai.

Fakultät für Mathematik, Dekan R. I. Bot
Zoom Meeting