Water Waves with Angled Crests: Preprint Available

At some point, I hope I’ll have time to write a longer exposition of my thesis research for the general public. For now, though, I’d like to announce that a preprint of the paper containing my research, A Priori Estimates for Two-Dimensional Water Waves with Angled Crests, jointly written with my advisor, Sijue Wu, is now available on the arxiv.1

Let me give two quick summaries, one for mathematicians and one for everyone else.

For mathematicians, here’s the abstract2:

We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-right angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with the inward-facing normal derivative of the pressure non-negative, instead of the strong Taylor stability criterion, which requires that the inward-facing normal derivative of the pressure be bounded below by a strictly positive constant.

For everyone else:

Draw the cartoon version of water in the ocean. What does it look like? You probably have drawn something with waves containing sharp, pointed crests. (Something like this.) This is our intuitive notion of what water waves look like. And yet, despite the tremendous advances in water waves over the past two decades, most current research3 cannot handle such waves, because those angled crests make the problem harder. (For those of you who remember calculus, this is because the sharp angle means that things aren’t differentiable. Given that the problem we’re solving uses [partial] differential equations, this is bad.) Our paper is a step in the direction of understanding that problem; it does this by proving a technical result suggesting that, in some sense, that picture can be handled by our basic mathematical model of water waves in the ocean. In addition, this research helps move towards an understanding of the role of bottoms and coasts in how water moves in oceans, by considering a simplified model of the effect of a vertical wall on water waves.

Thanks to everyone who supported me during grad school!

Update: Also, you can see the video and slides of a talk my co-author Sijue Wu gave about this and related papers.

Update: The paper has now been published (sorry, paywall) in the Cambridge Journal of Mathematics. The newer version of the arxiv preprint is very close to the journal version, except for some typos.

  1. This preprint contains a few minor refinements of the dissertation, so it’s probably more useful. But if you want to see a copy of the dissertation, email me and I’d be happy to send you a copy. 
  2. Slightly modified only to remove the dependence on mathematical symbols. 
  3. I should note that I’m referring specifically to the mathematical research in water wave equations from a theoretical perspective; I’m not familiar with numerical and experimental research.