**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.

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.

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