Curriculum Vitae (updated!)
The main focus of my research is nonlinear hyperbolic waves and their interactions. Such waves typically arise in solutions of conservation laws, which express the fundamental balance laws of continuum physics in the absence of dissipation and other higher order effects, and represent the extension of Newton's law of motion to continuous media. An essential feature of nonlinear hyperbolic PDE is the formation of shock waves, in the presence of which classical analysis breaks down. Another is finite speed of propagation, which means that disturbances remain localized and we can identify a detailed local wave structure.
I am most interested in the interactions of these nonlinear waves and the beautiful interaction patterns that occur when there are many such waves. In particular, I try to identify physical phenomena associated with certain wave patterns and from these derive interesting properties of solutions.
My earliest work concerned the nonlinear effects of weak wave interactions [2][3]. These can be expressed by asymptotic expansions, and one can obtain explicit formulae for the second (and third) order effects of interactions. I introduced the Method of Reorderings, which provides a bookkeeping mechanism for reflected waves by identifying a partial ordering on these reflected waves.
Using the Method of Reorderings, I obtained the L^{∞}-stability of solutions given by Glimm's method and identified a critical value V_{c} of the total variation of the initial data, below which global solutions are guaranteed to exist [2]. Then, in joint work with Blake Temple, we derived a large time large variation existence result for solutions of the Euler equations of gas dynamics [4][5][6]. This existence result has not been improved upon, for reasons which are becoming clear with our current work on periodic solutions.
Global existence results for large initial variation do not hold in general because of the possible onset of resonance. In an effort to understand this phenomenon from a purely local point of view, I constructed and analyzed several explicit 3×3 examples of resonant solutions [9][10][11]. These (nonphysical) solutions exhibit a wide range of behavior not seen in 2×2 systems, including time-periodicity and exponential growth and decay of solutions. From these explicit examples, even for weak waves, if many waves are present, multiple interaction effects can balance or dominate genuine nonlinearity.
One of the degenerate resonant systems I constructed was modified to produce the first example of solutions which blow up in finite time [J]. Shortly afterwards, I found another example of blowup of solutions, in a pair of Burgers equations coupled through either linear boundary conditions or through an "entropy" field [20][21]. This is a particularly simple example which can be completely analyzed, giving blowup for arbitrarily small data, and leads to a clarification of the assumptions of Glimm's celebrated existence theorem [G], namely that the system be uniformly hyperbolic in a connected neighborhood of the data [23]. Also, in joint work with Kris Jenssen, we obtained blowup of smooth (C^{k} for any k) solutions in finite time, and identified the blowup mechanism in terms of a "nonlocal loss of strong hyperbolicity" [22].
Having concluded that "anything is possible" for general systems, I made the conscious decision to focus my research on specific systems of physical interest. The first such system I studied in detail was the motion of an idealized nonlinear elastic string [14][15][16]. This had previously been studied in two dimensions, giving a 4×4 system which appeared to support anomalous shocks which satisfy entropy conditions. However, in three dimensions, this is a 6×6 nonstrictly hyperbolic system with degenerate waves, and correctly counting characteristics rules these anomalous waves out [17]. This is an example where the correct physics informs and simplifies the mathematics, and I like to think this is a common thread to much of my analytical work.
I gave a complete description of the Riemann problem and pairwise wave interactions, proved a Glimm theorem, and identified several different resonances in the system. Also, in a joint paper with Wlodek Domanski, we defined interaction coefficients for nonlinear elasticity in three dimensions, and calculated these for plane waves in an isotropic medium [24][25].
The p-system of 2×2 isentropic gas dynamics can be regarded as the prototypical system of conservation laws, being of intrinsic interest, as well as a subsystem of many larger physical systems. The major open problem for this system is (BV) existence and decay of solutions having large amplitude. I have been working on this problem for the last several years. Under certain regularity assumptions, I showed that a vacuum cannot form in finite time: that is, if a vacuum is present at time t=T>0, then it is embedded in the data and is present at time t=0+ [12][13].
I have recently given a complete description of nonlinear global wave interactions, obtaining estimates on wave strength uniform up to the vacuum state [26]. This can be used to obtain global existence of solutions provided the data has monotone pressure, via an interaction potential. It is expected that this potential can be combined with the uniform shock interaction estimate to prove global existence for general data.
When measured correctly, simple waves interact "linearly" in the p-system. In ongoing joint work with Blake Temple, we observed that in the full 3×3 Euler equations, a simple wave can be made to change its compressive character when crossing a contact discontinuity. We exploited this fact to generate a (formal) shock-free time-periodic wave pattern [27]. These beautiful wave patterns have many new and surprising features. Moreover, these patterns are realized in a linearization for which we can explicitly calculate the complete spectrum of the linearized operator, and recast the existence of periodic solutions as a bifurcation problem, which in turn reduces to an Implicit Function Theorem with small divisors [28][29]. We have thus found a deep connection between conservation laws and KAM theory. This in turn opens up a host of new questions which have not previously been considered in conservation laws.
I am currently focussing on my studies of the p-system and the Euler equations. For the p-system, I have constructed global exact weak solutions which shed new light on the behavior of solutions near vacuum, and in particular demonstrate the various possible scenarios when a vacuum collapses [30]. I have also constructed a modified Front Tracking Scheme which can handle vacuums and compressions [31]. I expect to be able to combine these to obtain global existence of solutions with arbitrary BV data. The decay of solutions is an equally important problem; here there is an interplay between genuine nonlinearity and distance to the vacuum, which is given by time-dependent bounds.
Blake Temple and I are actively continuing our study of periodic solutions of Euler. There is much interesting structure in the linearizations which is yet to be analyzed, and we are searching for a way in which to apply a Nash-Moser technique to our small divisor problem. Because these phenomena are so new, we expect them to open up the field to a host of fascinating new problems. Foremost among these are questions of stability: although our solutions are not stable under perturbations of the data, we believe the mechanism is stable and we expect that our solutions belong to some sort of attractor. These new solutions will provide a basis of fascinating research for years to come.
My student, Geng Chen, is currently analyzing pairwise interactions of nonlinear waves in the 3×3 Euler equations. Beginning with as few as three waves, complicated and beautiful wave patterns are generated, and it is a real challenge to extract order out of these patterns. In particular, our work on periodic solutions shows that entropy fluctuations can delay shock formation, and it appears that this may occur on a local level for generic wave interactions.
Support of the National Science Foundation under grants DMS-0104485 and DMS-0507884 is gratefully acknowledged.
October, 2008