Is there a theory for infinite dimensional PDEs?

[jostpuur]

Is there any established theory concerning infinite dimensional PDE?

 

[ObsessiveMathsFreak]

Do you mean that the function has infinitely many variables, or that it is an infinite dimensional function of a finite number of variables?

 

[jostpuur]

Infinitely many variables.

For example a quantum mechanical real Klein-Gordon field, if I have understood correctly, can be pretty much described by the infinite dimensional non-homogenous heat equation (the Shrodinger's equation, with certain constants and with the harmonic potential). Something like this

$$i\partial_t \Psi(t,\phi) = \sum_{k\in\mathbb{R}^3} \Big(-\alpha \partial^2_{k} + \beta |k|^2\Big)\Psi(t, \phi)$$

where

$$\Psi:\mathbb{R}\times\mathbb{R}^{\mathbb{R}^3}\to\mathbb{C}.$$

It can be solved by a separation attempt

$$\Psi(t,\phi) = \prod_{k\in\mathbb{R}^3} \Phi_k(t) \Psi_k (\phi(k)),$$

where

$$\Phi_k,\;\Psi_k:\mathbb{R}\to\mathbb{C}$$

This is total honest pseudo mathematics, motivated by physics, don't complain about it!

In fact his is a very vague example with uncountable set of variables. There could be more rigor examples with only countably many variables.

 

[jostpuur]

It could be these are supposed to be called functional differential equations, but I'm not sure. Some quick google hits were slightly confusing.