TITLE: Simulating Turing machines with {O,P}DEs.
QUESTION [28 upvotes]: Qiaochu Yuan in his answer to this question recalls a blog post (specifically, comment 16 therein) by Terry Tao:

For instance, one cannot hope to find an algorithm to determine the existence of smooth solutions to arbitrary nonlinear partial differential equations, because it is possible to simulate a Turing machine using the laws of classical physics, which in turn can be modeled using (a moderately complicated system of) nonlinear PDE


Is this "it is possible" an application of a Newton thesis, much as the usual Church-Turing thesis, going along the lines of «if something is imaginably doable in real life, one can simulate it with the usual equations of physics», or has the simulation actually been actually carried out? 
Does one need PDEs to simulate Turing machines, or are ODEs good enough?

REPLY [8 votes]: I believe that the reference is to seminal work of Pour-El and Richards where they measure the computational contents of various types of classical systems of partial differential equations. They have a series of papers where they carry this out; I believe much of it can be found in their book Computability in analysis and physics (Perspectives in Mathematical Logic, Springer-Verlag, 1989). There are other approaches to this, but they all reach essentially the same conclusions. Pour-El and Richards have the advantage that they concentrate on specific systems that actually arise in physics, such as the wave equation.