Differential equations coming from physics have a very peculiar aspect: given initial conditions that physically make sense, there must exist a (unique) solution.

This is just because we know that in nature the phenomenon actually happens, and in deterministic theories there is only one way in which the phenomenon can evolve from initial conditions.

Are there examples of (partial) differential equations from physics for which is hard to show existence and uniqueness? As an example, take n planets that interacts following the gravitational law. The associated differential equation will contain distances between points in the denominator, so it is not clear a priori that solutions will not blow up. However, discarding cases in which planets collide, the solution actually exist, but it is not manifest from regularity of the equations. Another example could be taken from the heat flow equations.

In case such an example is found, I would like to know which actual methods are used to prove uniqueness and existence, and if the physical intuition actually plays a role.

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    $\begingroup$ "given initial conditions that physically make sense, there must exist a (unique) solution." That's not true in quantum mechanics, e.g. the double-slit experiment. But this fact doesn't make your question meaningless: we can just restrict your question to deterministic scenarios. $\endgroup$ Feb 2 at 16:27
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    $\begingroup$ You have to prove existence and uniqueness mathematically. Just because something happens in physics, how do you know that your mathematical model behaves just like the physical situation that it's supposed to model? $\endgroup$ Feb 2 at 16:44
  • $\begingroup$ @HansLundmark: of course, mathematics cannot be proven with experiments: it is a self contained logical system. But it can give a strong hint that solutions exist globally. I would be very surprised however if a system in classical mechanics turns out to be not well modeled by our theory, which is well established. I am just intrigued by the fact that a purely mathematical statement can actually be " verified" by looking at what happens in the real world. Aren't you? :D $\endgroup$ Feb 2 at 18:04
  • $\begingroup$ @AdamRobinson: that's why I said "in deterministic theories". $\endgroup$ Feb 2 at 18:04
  • $\begingroup$ Surely physics can be very useful for suggesting what you should try to prove, but that's different from providing the actual proof. Anyway, here are some links that might be relevant: ncatlab.org/nlab/show/…, mathoverflow.net/questions/116531. $\endgroup$ Feb 2 at 18:29

Your question, as I understand it, is:

In deterministic models, does there exist a solution (to the equations of motion or the differential equations governing the system e.g. the heat flow equations), and is it unique?

And I would say that this is the very definition of deterministic.

Wikipedia says:

In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.

$$$$ If your question instead is, "under what conditions are the solutions of a mathematical model of a physical system unique?", then this is more interesting mathematically, but probably physcically irrelevant. For example, take the heat flow equations. As long as the initial starting conditions of temperature - as a function of space and time - $T(x,y,z,t),$ are smooth in a differentiable sense, then the time evolution of the system will surely be unique.

However, if the initial starting conditions of temperature, $T(x,y,z,t),$ are discontinuous, then strange stuff might happen. You could even take it to the extreme and make it everywhere discontinuous. Then the time evolution of the system is probably not deterministic because the rules of the game, i.e. what should happen are probably not well-defined (although you could come up with your own rules for what should happen). This corresponds to the situation where every atom has a temperature which is not close to any of it's surrounding atoms - and for this it is probably very difficult, if not computationally impossible to even approximate the time evolution of the system due to the chaotic nature of "everywhere discontinuous".

So basically the differentiable equations in physics require a nice smooth (or at least, continuous or differentiable?) function in order to determine the time evolution of the system. Once this continuity/differentiability breaks down, you can no longer use the differential equations to predict the time evolution: you need to look to other laws (like laws of particle physics) to see what's going on.

Conclusion: Some systems will not be deterministic when the starting conditions are discontinuous, but will be deterministic when the starting system is continuous; other systems will not be deterministic when the starting conditions are continuous, but will be deterministic when the starting system is differentiable; other systems will not be deterministic when the starting conditions are differentiable, but will be deterministic when the starting system is twice differentiable, and so on. Which one of these is the case for a given system depends on what kind of equations govern the physical system and you would have to look at each system on a case-by-case basis...

  • $\begingroup$ Thank you! You righteously addressed the issue of uniqueness as a matter of "physical conditions that make sense"; this probably applies also to the existence issue. For me, also assuming smooth initial conditions it's not clear that a solution should exist, unique or not. Also, I am not quite sure what do you mean by smoothness constraints on the initial data, since I think them as a bunch of numbers (like in the case of temperature, pressione and volume, these are just three numbers). But probably you are thinking about systems that starts from a "distribution" like in the case of atoms $\endgroup$ Feb 9 at 14:15
  • $\begingroup$ "For me, also assuming smooth initial conditions it's not clear that a solution should exist, unique or not." Actually you may be right. It depends on how well-defined the rules of your game are. You're saying that if we start with a smooth initial condition (e.g. a perfect unit sphere room $x^2 + y^2 + z^2 \leq 1$, which doesn't externally leak heat; Define the initial temperature as $T(x,y,z,t=0) = x+y+z$), then it might be possible to get a discontinuous function of temperature with respect to space at some point in time, and then we cannot physically determine what happens after this point $\endgroup$ Feb 9 at 15:19
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    $\begingroup$ in time, without more rules (laws). This seems plausible to me, especially if we start out with rules that only allow us to determine the "next" state of temperature if if the current spatial distribution of temperature with respect to time is continuous. So long as the rules of our game allow us to determine what happens next for each possible distribution of temperature, then the system is deterministic. I'm not sure you can really say anything else, like whether or not a continuous distribution of temperature will remain continuous.You would have to look to vector calculus/fluid dynamics. $\endgroup$ Feb 9 at 15:20

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