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I am a 2nd year mathematics undergraduate in the UK and recently took an introductory elective in partial differential equations. The focus was on solving some classical examples, all arising from an application - this seems to be typical of most such introductory courses. From what I can see of more advanced PDE-focused electives, my impression is they seem to follow a similar trend, 'inspecting' or solving the PDE more rigorously with more advanced analysis, but mostly still application focused ie 'this type of PDE arises in this modelling problem/application, we study these properties and use these techniques to solve it'.

  1. My main question is, despite the general motivation of application, are there any well-known PDEs that were first discovered and studied from a pure perspective and then later found applications? In other words, are there any notable examples of PDEs that could belong here?
  2. A followup question from this would then be, are there any examples currently in the first stage, but not the second (ie derived and studied entirely in pure mathematics, but don't really have any obvious application yet)?
TCWS
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  • My PDE lecturer said after the course dont go home and try to come up with your own PDE/ODE. Perhaps he meant that PDEs drops via applications? – voroshilov Dec 28 '23 at 17:14

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There are a collection of ideas stemming from John Nash’s work that initially had a purely geometric character, but later became useful in a variety of PDE settings, including applications to PDEs which directly describe physical phenomena.

For example, Nash-Moser iteration arose as a technique to prove the Nash embedding theorem, but now it’s a very powerful tool in nonlinear PDE. One physically relevant example of its application was in Cedric Villani’s work on nonlinear Landau damping, which concerns the decay of the electric field in a plasma.

Another example (which I’m admittedly less familiar with the details of): the technique of convex integration (which was originally used in isometric embedding problems) has seen use in fluid dynamics to resolve questions related to turbulence, like Onsager’s conjecture.

I would however say that modern work on PDEs isn’t “application focused.” PDE analysis is very much its own field with its own set of interesting questions and techniques, although many questions are highly connected to physics, geometry, other sciences, etc.

kieransquared
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