What is the purpose of PDE research/study

[mordechai9]

I have flipped through the first few pages of Evan's PDE book lately, and I am considering taking a graduate PDE course in the fall. However I don't really understand the purpose of PDE research. Not that I really understand the purpose of ODE research or even analysis research for that matter. Is analysis research basically the same thing as PDE research?

What is the purpose/goal of studying PDE's? Why are they interested in existence/uniqueness? What else are the objectives, and why? What do they hope to obtain from doing PDE research? Do you study one particular PDE and find classes of solutions to it? Or do you study general PDE theory, and what do you hope to find?

 

[g_edgar]

Do you know the purpose (in your sense) of research in any branch of mathematics?

 

[jambaugh]

"Purpose" is an individual motivation. Maybe you can make money or maybe you have an overbearing mother that insists you study PDE's or maybe (usually) you enjoy the subject.

Mathematical research per se is just for the knowledge. How it will be useful elsewhere is subject to the results of the research and can't always be known before-hand. But PDE's are fundamental to a lot of different subjects. (Which is why the public funds it.) Obviously it applies to physics but also chemistry, engineering, environmental sciences, biology, medicine, ...)

More toward your question: (And here I will just blather on until I run out of steam)
There are mutliple directions one may research w.r.t. PDE's.
Uniqueness and existence are important for obvious reasons but consider even if you are trying to crank out a numerical method, the convergence of the method and whether its results are meaningful will depend both on existence and uniqueness. For example if a solution is not unique the numerical algorithm may not converge but rather drift around the space of solutions.

One example of a an area of PDE research is my masters thesis work. I applied symmetry group analysis to non-linear heat equations (reaction-diffusion equations). If you can find an abstract symmetry then this means you can eliminate a variable and/or reduce the degree. (I found a couple of scaling symmetries) This is very important given how numerical methods' or simulations' computing requirements grow with the inverse of the scale raised to the power of the numer of variables. A PhD student had a wonderful thesis on using symmetries to substanitively increase the order of precision in numerical calculations.

Anyway probably the biggest area of research in PDE's are non-linear PDE's either in terms of general theory (such as the aformentioned symmetry analysis, perturbation analysis, stability of solutions...) or in terms of specific domains of application (reaction diffusion models, solitons, fluid dynamics, ...) or in terms of numerical methods.

These of course overlap and interact. Note much insight into the math is acquired from the application and of course the results in the mathematical theory apply. For example Emma Noether's work with action principle based dynamics has been generalize mightily spawning the symmetry methods of which I spoke.

Another random thought: One method of representing Lie algebras is with differential operators (i.e. you embed a given Lie Group in the diffeomorphism group on some manifold). You then find that the invariants of the Group/Algebra are differential operators and their action of the representation gives you a differential equation.

Effectively the Leibniz relations for differentials can be identified with the Jacobi identities of a Lie product so the two topics have a lot to say about each other. A Lie algebre = an algebra of derivations.

A final point, when dealing with linear PDE's one can apply standard linear algebraic techniques e.g. diagonalizing the operator to resolve solutions in terms of a spectrum. Once the PDE is non-linear this doesn't work but one can look for parallel transformation methods in specific cases.

 

[mordechai9]

Alright thanks that was very interesting and informative blathering :) . A lot of what you said sort of goes over my head but OK that gives me a more concrete perspective.

- I hadn't thought about the importance of uniqueness for numerical methods, I was thinking it was purely mathematically relevant, so that's very cool

- So I take it that a lot of the research goes into studying the solutions, e.g., putting estimates on the solution, perturbing the initial value and seeing what happens to the solution, etc.,

- Also I guess a lot of research goes into generalizing/reformulating the PDE using algebraic methods, like the Lie group stuff that you said -- but that is really starting to go over my head there

Thanks!

 

[HallsofIvy]

Here's an interesting application of "uniqueness":

Suppose you have a wire fastened to a wall at one point, pulled taut, and then "flipped" so as to send a "hump", deforming the wire upward, along the wire. When it hits the place where the wire is attached to the wall, it will "reflect" and come back. Will it come back above or below the wire?

Imagine extending the wire symmetrically on the other side of the wall. Your hump and a mirror-image hump on the other side of the wall will both satisfy the PDE wave equation. Further, if that mirror-image hump is below the wire rather than above it, when the both reach the wall, they cancel leaving that point fixed. Since that is a solution for the entire on both sides of the wall, and leaves the wall point fixed, by "uniqueness it is the only solution that has that property. In particular, that means that the reflected wave will come back below the wire.

(If you have the wire attached to a pole such that the wire can slide up and down, but the tangent remains the same, the same kind of argument will show that the hump will return above the wire.)

 

[mordechai9]

AHHHH that is awesome ... more!

 

[ephedyn]

^Yes, I agree... that's way awesome.