I have always been mesmerized by waves. Here is an example of the kind of wave that I like. Of course, there are all sorts of waves cruising around the universe, but if you're speaking with a surfer about waves, then you will undoubtedly be discussing ocean waves, and not any type of ocean wave but those large, fast-moving, cylindrical waves, that pitch-out into a perfect barrel and break down the line.
Remarkably, the motion of that translucent blue-green salt-water is rather nicely described by the oldest set of partial differential equations known to man, the Euler equations of fluid dynamics. These are a coupled set of nonlinear conservation laws for the three components of the velocity as well as the pressure. They are set in a domain that is changing in time, and is itself one of the basic unknowns of the problem; hence, this is a free-boundary problem.
It is of great importance to ascertain if solutions to the Euler equations exist. A solution must solve the Euler equations inside the moving domain, while simultaneously transporting the moving interface between the air and water. A variety of interesting questions can be asked: If solutions exist, are they unique? Is the free-boundary smooth? Do solutions exist for all time, or is there a finite-time singularity? These basic problems are central to my research program.
To answer them we must move away from the classical picture of breaking waves introduced a half century ago, in which a wave "breaks" when its tangent plane becomes vertical. Instead we use a more realistic definition, which allows the wave's slope to become vertical, then turn over and become cylindrical (the ideal), and then finally break, in the sense of the crest of wave falling down onto its trough. In this view the free-boundary is not a graph over the horizontal plane. We must employ a description of the fluid that follows the trajectories of the fluid particles, and views the moving fluid domain as a diffeomorphic copy of its initial state. We have found the Lagrangian description of fluid dynamics to be ideally suited for this purpose. This allows us to control the geometry of the moving free-boundary, and hence also provides a priori control of the velocity and pressure. This “control” involves a new structure obtained from the nonlinear interaction between the cofactor matrix of the diffeomorphism taking the initial boundary to its current location and specially constructed derivatives of the velocity field. Having this geometric control in hand, we introduced a new class of solutions to the Euler equations that are well-defined all the way to the breaking-singularity. (This is when the crest touches the trough so the fluid domain has a self-intersecting boundary with a singular cusp.) This method has shown that unique solutions exist to the Euler equations which propagate the free-boundary and, in finite-time, turn over and break.
There are many other physical scenarios in which similar surfaces of discontinuity are propagated. One intriguing example from astrophysics is the expansion of the cloud produced by a supernova explosion. Here a compressible gas, modeled by the compressible Euler equations, expands into a vacuum, the moving free boundary of the gas is the surface along which the density vanishes. Mathematically, this vanishing of the density function introduces tremendous degeneracy into this nonlinear system, making traditional methods for hyperbolic equations useless for proving existence or regularity. To address this problem we have developed new, high order inequalities new energy structures associated with nonlinear wave interactions. This establishes a framework for analyzing degenerate, hyperbolic moving free-boundary problems that can be used in a variety of contexts.
There are vast number of physical systems that give rise to free-boundary problems. For example, fluid-structure problems treat interactions between an elastically deformable solid and a viscous fluid. Here the deformations of the solid produce a moving, time-dependent material interface between the structure and the fluid that is itself an unknown in the problem. The swimming of a dolphin is a very interesting example that led me into this class of free-boundary problems. In 1936 the British zoologist Sir James Gray showed that the muscles of dolphins simply were not strong enough to propel them at the speeds they in fact achieve. This came to be known as “Gray's Paradox.” Early attempts to resolve this paradox failed to correctly include the nonlinear interaction responsible for accelerating the fluid and allowing the dolphin to swim. In simplest mathematical terms, the viscous fluid can be viewed as governed by a heat (or diffusion) equation and the dolphin by a wave (or conservative) equation. Intuition suggests that when coupling these two regimes diffusion will win out and dominate the wave equation. But in fact, to our great surprise, we found that the wave regime takes over, and makes the viscous fluid behave like a conservative wave equation. This showed that it is essential, when treating nonlinear fluid-structure interaction problems, to analyze the structure simultaneously with the fluid, keeping all nonlinearities intact. Our analysis showed that wave-like vibrations of the dolphin's rough skin creates tiny vortices in the fluid, which in turn, accelerate the dolphin.