Introduction to my world of research

My research interest is in the theoretical aspects of partial differential equations. While the motion of a finite set of particles, such as the motion of our solar system can be modeled by a set of ordinary differential equations, partial differential equations are for the motion of a set of uncountable number of particles. Many physical processes can be modeled with a set of partial differential equations. For example, air and water can be modeled by Euler equations. Plasma motion in a confined Tokamak can be modeled by Vlasov-Fokker-Planck system of partial differential equations. Once the initial state of the physical process is known, the solution of the partial differential equations with the given initial datum can be used to predict the physical process. The prediction of orbital motions of planets in our solar system is an excellent example for modeling by ordinary differential equations. Weather forecasting in the most part is due to modeling by partial differential equations. While there are many excellent successes via models of partial differential equations, the prediction of weather and the stability of a Tokamak are still not dependable. The reason is believed to be the turbulent nature of the materials, which is not well understood physically and its mathematical modeling is a challenge. Overall, there are many unsolved issues on century-old models such as the Euler equations and many more models are being produced nowadays because of their useful applications in industry and numerous branches of scientific research. Questions related to the modeling process and its resultant set of partial differential equations are as follows.

a. Does the set of partial differential equations have a solution for a given physically reasonable initial or boundary data?

While it is true that the physical process will always move forward, the mathematical model may not possess a solution because the model is always an approximation for the physical mechanism. Ignored effects may render the model pathological.

b. Is the solution unique?

Pathological models may possess multiple solutions for a single datum, which gives ambiguity in prediction. In addition, some physical processes do have multiple solutions.

c. Is the solution stable?

If a solution changes greatly when its initial data changes only slightly, more care should be taken in selecting the initial data and in the process of obtaining the solution from the equations.

d. Does the solution tend to some asymptotic state as time goes by?

e. Are there solutions with new phenomena?

More and more models are now set up to probe into scientifically new territories in which scientists depend on the models to discover new patterns.

f. How does one (efficiently) calculate the solution?

Exact solution formulas are rare and hard to find. It is sufficient in most situations to find an approximate solution. While a number of standard numerical methods are available for lots of partial differential equations that we are familiar with, it is most likely that for a situation in the research front of nonlinear partial differential equations, none of the known methods can be used to approximate the solution. The 2-d Euler equations with vortex sheet initial data is such an example. When standard methods are available, efficient algorithms are needed for relatively large problems since powerful and user-friendly computers are now still limited.

g. Does the solution predict the physical process?

A model may be perfect for its own sake. It is not useful if its solutions are far from the physical motions of the process that it is supposed to model. More mechanisms should be incorporated into the next level of models if a model fails to behave as it is expected. Of course the more sophisticated models are harder to analyze and simple models may be sufficient.

I am mainly concerned with the first six questions (a-f) on equations coming from fluids, plasmas, liquid crystals, combustion/energy, quantum mechanics, probability theory/ life sciences, financials, etc. I mention selectively some of my results as follows for the more interested readers.

(1) Two-dimensional incompressible Euler equations with vortex sheet initial data.

Vortex sheet initial data are very singular data. The wings of an airplane are prototypical places to produce vortex sheets. The vorticity of vortex sheet data is zero everywhere in a region except for a surface on which the vorticity concentrates. The evolution of a vortex sheet in the three dimensional setting is too complicated to have any mathematical theory so far. You can be convinced of its complexity by simply closely observing the motions of white wakes left in the sky behind an airplane. The white materials are steam vaporized by the fast moving wings of the airplane. The steam forms vortex sheets when it is produced and changes to turbulent rings a few minutes later and eventually diffuses into the ambient background. Sometimes vortex sheets left by big jets are fatal places for small planes. In the two-dimensional case, vortex sheets degenerate into curves which are still called vortex sheets by convention. These curves can stretch indefinitely and can self-twine to form complicated patterns or no patterns at all. The evolution of a vortex sheet, including its singularity formation, has been studied since the beginning of the last century. Although more and more insights are obtained, most basic questions remain open. Only recently in 1986 DiPerna and Majda started to address the question of existence of a global (in time) solution for the 2-D Euler equations with this vortex sheet initial data. They introduced a method known as ``Concentration-cancellation'' for this problem. They showed that a typical sequence of approximate solutions can concentrate its energy on a set of (cylindrical)Hausdorf dimension at most one in space time. Furthermore they showed that the limit of a sequence of approximate solutions will be a global solution for the vortex sheet initial value problem if the sequence concentrates its energy on a set of Hausdorf dimension less than one. The gap is that there might be a sequence of approximate solutions which concentrates their energy on a set of Hausdorf dimension one. My work [5] is for filling the gap. I showed that the limit of a sequence of approximate solutions will be a global solution for the vortex sheet initial value problem if the sequence concentrates its energy on a set which has finite one dimensional Hausdorf measure. In the proof of the result of work [5], new techniques such as the geometric measure theory play important roles.

(2) One-dimensional Vlasov-Poisson System with measures as initial data.

Connected closely in methodology to the problem mentioned in the vortex sheet initial value problem for the 2-D incompressible Euler equations, we consider the one-dimensional Vlasov-Poisson system with measures as initial data. This system is extremely similar in structure to the Euler equations, and is only slightly easier as it turns out. We showed here that a sequence of approximate solutions can in fact concentrate on a set large enough so that its limit is not a solution in either classical or Schwarz distributional senses. So the so-called measure-valued solutions are indeed needed here for studying approximate solutions, see work [12]. The success here comes from our key observation that there are exact and explicit solutions consisting of electron particles. Because of this, we are able to show rigorously that ``electron sheets'', the analogue of vortex sheets, develop singularities in finite time, and multiple solutions can co-exist with the same initial data. Our highly accurate numerical simulations show that there is no selection principle to pick up a unique solution: i.e., two physically reasonable but different sequences of approximate solutions can converge to different solutions, see work [11]. All these are answers to our questions posed earlier in (a-f), and these answers clearly raise more questions.

(3) Singularity formation in a nonlinear variational wave equation from liquid crystals.

We are the first to study rigorously the wave motion of liquid crystals. There are quite a few people studying the steady states and a few who study the heat flow of liquid crystals. When inertial motion dominates viscous forces, it is more appropriate to use the wave model than the heat model. It is however much more difficult to study the wave motion than the heat flow of liquid crystals. For example, the wave motion may result in singularities in finite time while the heat flow does not. The main result in this area is our proof that solutions of the wave flow of liquid crystals develop singularities in finite time, see work [15]. The proof comes a long way. First there was numerical evidence for the singularity formation performed by Hunter and Saxton. Then a geometric optics asymptotic equation was derived around the possible singularity. A complete study of the asymptotic equation is done in work [9, 10, 13]. We finally find a way to make the asymptotics rigorous (work [15]). While it is hard to say that any of the steps above is super ingenious, the whole process of approach is typical of the new research method in nonlinear applied sciences at the present: it combines numerical experiments, asymptotic probes, and rigorous mathematics into one giant powerful tool.

(4) Structures of solutions to two-dimensional Riemann problems of gas dynamics.

Riemann studied the one-dimensional compressible Euler equations with a special kind of initial data which consists of two constant states of pressure, density, and velocity separated at the position x=0 at time t=0. He found all the solutions which are shock waves, rarefaction waves, and contact discontinuities. In the middle of the twentieth century, the study of similar equations, which are called conservation laws, with similar initial values, which are called Riemann problems, was quite extensive in one dimension. The solutions of the Riemann problems contain almost all the structures of solutions that the conservation law can have. Besides, any solution of the conservation law may be approximated by a finite set of these simple solutions. It was not until the middle of 1980s that people tried to solve similar problems for two dimensional conservation laws and made some remarkable progress. My collaborators and I were among the first to have significant results on two dimensional Riemann problems. Our first result is on the scalar equation. We constructed all the solutions to the suitably generalized Riemann problem which consists of four constant states separated by the origin. The complete set consists of over thirty different configurations of solutions, see work [1].

We then proceeded to systems of conservation laws. The up-front model is the gas dynamics system of Euler equations. We propose a set of conjectures on the structure of solutions, see work [3]. Subsequent numerical calculations by C. W. Schulz-Rinne, J.P. Collins, and H.M.Glaz, `Numerical solutions of the Riemann problem for two dimensional gas dynamics', preprint, 1992, and by Tong Zhang, G. Q. Chen, and Shuli Yang, `On the 2-D Riemann problem for the compressible Euler equations I. interaction of shocks and rarefaction waves', preprint 1993, both confirm our conjectures with only slight modifications.

Just recently, we are able to construct some rigorous solutions to the Riemann problem of 2-D gas dynamics. We have never seen any rigorous solutions of this kind before. These solutions have swirling motions in them. Some of them have close similarities with the hurricanes and tornadoes that we are familiar with, see work [17, 18, 20-21, 24].

Since the last update, there is tremendous progress in this area. Please take a look at the publication list.

(5) Strongly singular solutions of conservation laws.

In constructing solutions to the Riemann problem of 2-D gas dynamics, a new wave called delta-shock wave was introduced by my collaborator Tong Zhang and his team of students. Delta-shock waves are characterized by its strong singularity which cannot be described by any ordinary functions. The use of this wave is highly questionable. In a model problem, we manage to prove that the delta-shock waves in the model is the limit of vanishing viscosity of smooth solutions to the viscous perturbations of the model, see [7, 23]. Therefore we provide some form of justification to the validity of delta-shock waves.

(6) Transient analysis of linear birth-death processes with immigration and emigration.

Partial differential equations arise in these birth-death stochastic processes with immigration and emigration. We find explicit solutions for some textbook models. See [36, 37].

September 13, 1996. Modified June 6, 2002. Modified again December 18, 2009. My home page