Research / Free Boundaries
Quadrature domains: a class of free boundary problems arising in fluid dynamics
A basic property of harmonic functions (solutions of Laplace's equation) is that their average over a sphere or a ball equals point evaluation at the center. A quadrature domain is a domain that admits a more general formula for integration of any harmonic function u: \begin{equation} \int_{\Omega} {u(x) dV} = \sum_{m=1}^{N} {a_m u (x_m)}. \end{equation} Instead of a single point evaluation, there is a sum of them (sometimes the formula also includes evaluations of derivatives of u). The coefficients do not depend on u, so this is a finite formula prescribed for an infinite-dimensional space of functions.
Quadrature domains and viscous fingering: A variety of problems in fluid dynamics admit solutions that turn out to be some type of quadrature domain (perhaps modifying the quadrature formula above depending on the context, see the survey by D. Crowdy). One important such problem concerns viscous fingering, where an inviscid fluid displaces a viscous fluid within a Hele-Shaw cell or porous medium. In this context the growth law is known as Darcy's law, and can be derived from averaging the Navier-Stokes equations across the gap in a Hele-Shaw cell or across each pore in a porous medium. After further simplification (such as neglecting surface tension at the interface), the boundary of the domain moves with velocity V equal to the normal derivative of the fluid pressure which is is a harmonic Green's function G (with a fixed singularity at s): \begin{equation}\label{LG} \left\{ \begin{array}{l} \Delta G = \delta_{s}, \text{ in } \Omega_t \\ G(t,x,s)|_{\partial \Omega_t} = 0 \\ V = -\partial_n G(t,x,s) \end{array}\right. \end{equation} S. Richardson (1972) discovered infinitely many conserved quantities of this evolution, and showed that the problem of determining the domain at some final time can be reduced in principle to solving the inverse potential problem (a classical problem to recover the shape of a uniform massive object given the gravitational field it creates outside itself).
The problem of constructing explicit solutions As a consequence of Richardson's theorem, if the initial domain is a quadrature domain, then it will remain a quadrature domain throughout the evolution, and the viscous fingering problem is reduced to the problem of constructing a quadrature domain with prescribed quadrature formula. This statement holds for any number of spatial dimensions. As for the problem of constructing explicit quadrature domains, there are two successful methods in the planar case. The first is based on conformal mapping, and the second method is based on the fact that the boundary of a quadrature domain is an algebraic curve with coefficients related to the quadrature formula. The first method (conformal mapping) cannot be extended to higher dimensions due to Liouville's rigidity theorem, and the second method (algebraic geometry) also fails in light of the fact that the boundary of a higher-dimensional quadrature domain need not be algebraic, a result due to Eremenko and I that answered a 1992 question of H.S. Shapiro. We constructed explicit non-algebraic examples in four spatial dimensions. It is an open problem to construct a non-trivial example in three dimensions.
Thus, the viscous fingering problem is an instance of an infinite-dimensional nonlinear dynamical system with a complete set of conservation laws, yet the integrable structure has not been implemented effectively in the three-dimensional case. As a model problem, consider the evolution of an initially round (three-dimensional) drop under extraction of fluid from an off-center sink. The problem is well-known in the two-dimensional case, and appears in the title of a book "Why the boundary of a round drop becomes a curve of order four" by Varchenko and Etingof. In the three-dimensional case the explicit shape of the evolving drop is unknown. Another simple example is to construct a three-dimensional solid that generates an external gravitational field equivalent a point source superimposed on a dipole source. When the strength of the dipole is sufficiently small, the existence and uniquencess of such a domain is classical, but the explicit parameterization in three dimensions is an open problem. In the plane, the solution is another curve of order four known as Pascal's limacon, and in four dimensions, the solution is a transcendental surface parameterized using elliptic integrals.
Arclength quadrature domains and vortex dynamics: By considering integration over the boundary of the domain, one arrives at the related notion of an arclength null-quadrature domain (ALNQD). The classification problem in this case remains open and is much more intricate. Partial results and constructions have mainly focused on so-called exceptional domains and more generally quasi-exceptional domains. The methods used include transcendental and non-elementary conformal mapping, sophisticated function theory (univalent functions whose derivatives are purely singular inner functions), Riemann surfaces, a non-trivial correspondence to minimal surfaces, and interesting connections with fluid dynamics (hollow vortex equilibria).
The classification problem for Arclength null-quadrature domains: A. Eremenko and I used Abelian differentials on the Schottky double Riemann surface in order to classify so-called quasi-exceptional domains. This led to new moduli spaces of periodic ALNQDs. These results hold promise for completing the general classification problem provided the following question can be answered affirmatively.
Question: Is every ALNQD a quasi-exceptional domain?
That is, does every ALNQD admit a so-called ``roof function'', a positive, harmonic function with zero Dirichlet data and constant Neumann data (possibly a different constant on different boundary components)? The null-quadrature domains constructed in provide new examples of vortex equilibria that generalize the periodic vortex array constructed by Baker, Saffman, and Sheffield in 1976. From the point of view of applications it would be interesting to investigate the stability of such equilibrium solutions.