I'm an analyst, specializing in nonlinear partial differential equations. Although I am broadly interested in the subject, my work has concentrated on equations that model some sort of physical phenomenon, and in particular equations coming from fluid mechanics, interfacial dynamics, and the Ginzburg-Landau model of superconductivity. The lifeblood of modern PDE is estimates, which ultimately flow from various functional inequalities valid for functions belonging to certain spaces. As such, I am also interested in basic analysis tools and techniques such as function spaces, inequalities and embedding theorems, and harmonic analysis. What follows is a rough summary of some of the things I've worked on. See the Publications section, my arXiv postings, or my mathscinet listing for the source material.
Viscous surface waves
The viscous surface wave problem concerns the dynamics of a layer of viscous incompressible fluid with a fixed lower boundary and a free upper boundary. This situation occurs, for instance, in the ocean, when we think of the ocean bottom as being fixed and think of the ocean surface as the free interface. Continuing with the ocean analogy, we usually assume that the fluid is subject to a uniform gravitational force pointing in the direction of the fixed lower bottom. Depending on what type of fluid we want to model, at the free boundary we sometimes include interesting interfacial physics terms, such as surface tension.
One of my first projects related to the VSW problem concerned the problem with gravity but no surface tension. In particular, with Yan Guo, we proved that solutions exist globally in time provided that the initial data are sufficiently small. We also derived precise decay information for the solutions, and showed that the decay rate is strongly tied to whether or not the flow is horizontally periodic. An interesting feature of the analysis was that the decay information played an essential role in proving that the solutions existed globally, and the global-in-time bounds on the solution played an essential role in proving the decay. This is contrast to many other problems where one can prove decay as a secondary argument after existence has been established.
In later work I extended this sort of analysis to handle the problem of a viscous fluid flowing down an inclined plane. Together with my undergraduate researchers David Altizio, Xinyu Wu, Taisuke Yasuda, we also proved similar results for the Faraday problem, which concerns a layer of fluid that is vibrated from below.
I have also studied many interesting variants of surface tension and their effect on the flow. With Chanwoo Kim and Lei Wu we studied surfactant-driven flows. Surfactants are chemicals that collect at an interface, dynamically change the strength of surface tension, and induce tangential surface stresses called Marangoni forces. With my MS student Samuel Zbarsky we studied fractional generalizations of surface tension and characterized the decay properties of solutions in terms of the strength of the fractional operator. With my Ph.D. student Antoine Remond-Tiedrez we studied flexural fluids, which are fluids for which the interface is subject to bending forces. In work with Juhi Jang we studied how passive scalars are affected by being advected in a viscous fluid with a free boundary.
In work with Giovanni Leoni we constructed the first traveling wave solutions to the free boundary incompressible Navier-Stokes equations. These solutions require either an external bulk force or applied surface stress that is time-independent when viewed in the traveling-wave frame. For instance, one can think of the latter as modeling a tube moved uniformly across the fluid surface while blowing air onto it. In work with Noah Stevenson we extended this result to the setting of multiple layers of incompressible fluid.
The contact line problem
The contact line problem concerns the dynamics of a triple-phase interface such as the interface between air, fluid, and solid in a cup. This is a notoriously tricky problem because standard modeling assumptions (namely the no-slip condition) give rise to contradictory behavior. My work on the contact line problem employs a fairly new model of contact line behavior that allows for fully dynamic contact angles as well as dynamic lines. The main goal is to probe the new model by proving well-posedness and stability results.
In work with Yan Guo we studied a two-dimensional version of the problem in which the inertia term in the Navier-Stokes equations is set to zero. This can be thought of as Stokes flow or the zero Reynolds number limit. The geometric configuration was that of an open-top vessel, essentially a two-dimensional cup. The analysis of this problem is quite tricky due to the presence of corners in the equilibrium domain, which wreak havoc in standard elliptic theory. Yan and I were able to prove that the unique equilibrium solution is asymptotically stable for this new model, which adds evidence to the soundness of model. I developed the corresponding local existence theory for this problem in joint work with Yunrui Zheng.
In work with my former postdoc, Lei Wu, we pushed the Stokes flow analysis to another, trickier geometric configuration: the droplet configuration. Here the technical headache comes from the extra degree of freedom of a droplet: it can slide along the rigid interface while its endpoints simultaneously move.
The micropolar fluid model takes into account the structure of the microscopic objects that comprise a fluid. Here one should think of examples such as red blood cells in blood, fat globules in milk, the rod-like molecules in a liquid crystal, or the tiny magnetic particles in a ferrofluid. The key novelty in this model is that it treats each point in the fluid as having a copy of the microstructure, which is treated as a rigid body. Movement of the center of mass of the rigid bodies corresponds to the motion of the bulk fluid, but each copy of the microstructure is free to rotate independently. The microstructure is capable of carrying microangular momentum, and this is dynamically coupled with the bulk angular momentum.
In work with my graduate student Antoine Remond-Tiedrez, we examined the role of anisotropy in the micropolar model. This means that the microstructure's inertia tensor is not a scalar multiple of the identity, and so there are preferred directions of rotation. For simplicity we considered the case in which the microstructure has exactly two distinct eigenvalues, which is like having an axis of symmetry. One can think of these are rod-like or pancake-like depending on the relative size of the eigenvalues. This setup admits equilibrium solutions with no bulk motion and with uniformly spinning microstructure when a uniform microtoque is applied to the system. We studied the stability of these configuration and proved that rod-like configurations are unstable and pancake-like configurations are stable.
In a summer undergraduate research project with Noah Stevenson we studied special solutions in the isotropic case, in which the inertia tensor is a multiple of the identity and there is no preferred direction of microspin. We constructed special solutions in this case, which we called potential microflows. These consist of solutions in which the bulk velocity vanishes and the microangular velocity is conservative and satisfies a special linear parabolic PDE. We proved that near these special solutions the micropolar equations are globally well-posed, and we derived various stability and asymptotic stability results for such solutions.
In modern PDE we often prove that solutions to a nonlinear equation exist through indirect means such as fixed point arguments. These rely crucially on having the right space, or functional setting, for the problem at hand. As such, a key tool in the analyst's toolbox is a good understanding of function spaces.
In joint work with Giovanni Leoni, we studied a new type of fractional Sobolev space that arises in considering the traces of functions belonging to homogeneous Sobolev spaces on infinite strip-like domains. Such domains and homogeneous spaces appear naturally, for instance, in studying strips of fluids. As with Besov spaces or fractional Sobolev spaces, the new spaces can be characterized in terms of difference quotients, but for the new spaces there is a curious screening effect (inherited from the fact that the strip-like domain is bounded in one direction) that screens the size of the perturbation used in the difference quotient. After defining these spaces and studying some of their basic properties, we proved that they characterize the trace space of the aforementioned homogeneous Sobolev spaces. These trace results then allowed us to prove the existence of solutions to a class of quasilinear elliptic equations on strip-like domains. In work with my MS students David Altizio and Annie Xu, we derived further properties of the screened spaces and extended them to cylindrical geometries.
In work with Noah Stevenson we studied the fractional spaces within the context of interpolation theory. This required developing a new abstract interpolation technique for pairs of seminormed spaces, which we dubbed the truncated method. Using this technique, we characterized the screened spaces as truncated interpolation spaces. Along the way we introduced a more general Besov scale of screened spaces as well.
An extremely simple model of a star can be obtained by thinking of the star as blob of self-gravitating compressible fluid. To further simplify things, one can consider only the non-relativistic setting, which leads to the Navier-Stokes-Poisson system of equations. In this setting we use the classical Poisson equation to solve for the gravitational potential generated by the star and then use the associated gravitational field as a force acting on the fluid itself. For polytropic pressure laws with polytropic constants in a certain range, the equations admit compactly supported spherical equilibrium solutions known as Lane-Emden solutions. A long-standing conjecture from astrophysics identified a range of polytropic constants that should lead to instability due to gravitational collapse. In joint work with Juhi Jang we proved this conjecture and showed that the fully nonlinear problem is unstable.
The Rayleigh-Taylor instability occurs when a heavy fluid is accelerated into a lighter fluid. The canonical example is observed when one rapidly flips over a cup with liquid in it. In that instant the liquid is heavier than air, and the instability leads to the subsequent crying over spilled liquid. Although this is the example of R-T instability most familiar to us, it actually occurs over a huge range of scales. For instance, the instability wreaks havoc in certain designs of fusion reactors, and it plays a role in determining the structure of supernova remnants such as the Crab Nebula.
The R-T instability was well-understood mathematically for the linearized equations of inviscid fluid mechanics since the original work of Rayleigh. However, a rigorous theory for the viscous problem at the nonlinear level was missing. In work with Yanjin Wang and Chanwoo Kim we closed this gap and fully characterized the instability for the fully nonlinear and viscous problem for incompressible fluids. Among other things, we showed that if surface tension is present between the fluids, then there exists a critical threshold below which the instability sets in and above which it is prevented. This is a nonlinear viscous version of the classic linear, inviscid long-wave instability.
In work with Juhi Jang and Yanjin Wang, building on some earlier work with Yan Guo, we proved that the same viscous instabilities persist for the compressible problem. The critical threshold is the same. In work with Yan Guo we showed that the compressible inviscid problem is beyond unstable: it is ill-posed in Sobolev spaces.
The Ginzburg-Landau model
Superconductors are special materials in which electrical resistivity disappears below a critical temperature, allowing for the flow of permanent supercurrent. When subjected to electromagnetic fields and currents, superconductors begin to fail through the appearance of localized defects known as vortices, whose motion disrupts the flow of supercurrent. To combat the motion, material impurities are added to devices in order to attract or pin the vortices. It is then important to have mathematical models capable of determining vortex dynamics in the presence of pinning and electromagnetic fields, and of describing static equilibrium configurations. Ginzburg and Landau (both awarded Nobel prizes) developed such a model, which although phenomenological in nature, has been shown to be valid as a sort of limit of the underlying quantum theory.
My work on the Ginzburg-Landau model has focused on questions of vortex dynamics and of understanding the static G-L free energy in terms of vortices in two-dimensional superconductors. I proved the first rigorous results about the behavior of vortices subjected to electric currents and fields. This work characterized the time scale of vortex dynamics in terms of the strength of the applied fields as well as identified the dynamical law for the evolution of the vortices. In subsequent work with Sylvia Serfaty we studied the case of vortex dynamics with pinning and strong fields. We showed that without strong fields, the pinning forces dominate and the dynamics are trivial. Strong fields, however, can de-pin the vortices and induce motion. This work identified the critical de-pinning current at which vortices first move away from the pinning sites, and in this case we identified the resulting vortex dynamics. In solo work and joint work with Sylvia Serfaty we also worked on refinements of the ''vortex balls construction,'' which provides lower bounds of the G-L energy in terms of the number and location of the vortices.