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I encountered the following statement in Boyce's Elementary Differential Equations and Boundary Value Problems :

Not all differential equations have solutions; nor is the question of existence purely mathematical. If a meaningful physical problem is correctly formulated mathematically as a differential equation, then the mathematical problem should have a solution.

Is this true?

StefanGill
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    Navier Stokes equation pde is well formulated fluid flow problem but.. – Narasimham Aug 27 '17 at 19:21
  • What do you think? – sammy gerbil Aug 27 '17 at 19:39
  • Title question (v2) is different from quoted sentence. – Qmechanic Aug 27 '17 at 20:06
  • You can't formulate "a physical problem" as a differential equation, or any other math. You can only formulate "a mathematical model of a physical problem." But "a model of a thing" is not the same as "the thing itself". It makes assumptions, which are not always true. For example a simple model of projectile motion is the SUVAT equations, but that model ignores air resistance, the effects of relativity at high velocities, the change in gravity at high altitude, etc. SUVAT is often a useful model, but it's not the same as "the correctly formulated and meaningful physical problem". – alephzero Aug 28 '17 at 01:03
  • @Narasimham Are the N-S equations even a "correctly formulated" description of fluid flow? I don't know of any fluids that are continuous. There are these pesky finite-sized things we call "atoms" and "molecules" ... the N-S millennium prize might be asking the wrong question, for all I know!! – alephzero Aug 28 '17 at 01:14
  • @alephzero if you're considering individual atoms for Navier-Stokes, you're probably doing something wrong. – Kyle Kanos Aug 28 '17 at 01:30
  • @Narasimham: Could you please explain your "but.." a little? To my knowledge analytically solving the Navier Stokes equations typically is not possible. But this does not mean that it has no solution or that the solution is not "mathematical" (What does that mean anyway?). Or are there other problems that I am not aware of? I assume that simulation software is used to calculate numerical solutions to the Navier Stokes equations on a daily basis. – Gregor Michalicek Aug 28 '17 at 06:02
  • The "but..." is a reference to the fact that proving the existence, uniqueness, and regularity of the general N-S equations is one of the most important open mathematics problems. – Chris Rackauckas Sep 27 '17 at 02:05

2 Answers2

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Maybe there is more context that qualifies this statement, but taken as is, it's completely false. In general, when we talk about existence of solutions to a differential equation, we're talking about existence given a certain set of boundary conditions. It's perfectly possible, in practical real-world problems, that we can have constraints on the boundary conditions, and if our boundary conditions don't satisfy those constraints, there is no solution.

For example, I could write down a differential equation representing the free motion of a body in a viscous medium. I could then specify the following boundary conditoins: at $t=0$ its velocity is zero, and at $t=t_f>0$ its velocity is nonzero. There is no such solution.

In physical problems where we specify the initial conditions, we want not just existence but uniqueness of solutions.

Less trivially, we can have examples of inconsistency or indeterminism (solution exists, but is not unique) in physical problems that come up in interesting, "meaninful" contexts. Examples include Norton's dome, naked singularities, and the Novikov consistency principle.

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    I wonder if the kinds of situations you allude to are implicitly excluded via the "If a meaningful physical problem is correctly formulated mathematically" part of the original statement. But otherwise I agree the context of the statement should be made much clearer. – ZeroTheHero Aug 27 '17 at 19:32
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    I agree with ZeroTheHero. Your answer is wrong while the book quote is correct. Real physical conditions clearly result in real physical outcomes. Therefore, if such systems are correctly described by differential equations (where correctly includes the proper boundary conditions), then these equations must have solutions that produce these outcomes. This doesn't mean the solutions are always known or can be expressed in elementary functions. Finally, a "naked singularity" is a mathematical abstraction, but not a physical system, as none have been observed. A critical difference to understand. – safesphere Aug 27 '17 at 19:51
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    I'm not sure what exactly you mean by "we want uniqueness of solutions". Bi-stable systems are known to have non-unique solutions (pretty much by definition). and they're physically not at all problematic. – MSalters Aug 27 '17 at 23:30
  • No. The solution of a bistable system is unique after the initial condition is specified. This is why boundary conditions are important for fully specifying the model. – Chris Rackauckas Sep 27 '17 at 02:07
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In the best of all possible worlds we know the laws of physics perfectly and can write them as differential equations (or something similar). But we do not live in that world. Instead we create models of the physical world that may not correspond to the actual laws (due to ignorance or just approximation). Good models give informative predictions: there is a mapping between what happens in reality and the model that is close to a bijection, so we can use the model to predict physical responses. How close it has to be depends on the application.

Now, physics as far as we know never fails to produce a "solution" of what the future state of the world will be. But models clearly can fail at this, even models that are accurate in large domains. Depending on the application this might disqualify them - only use models that use differential equations that have solutions! - or be OK if model failures occur far away from the problems of interest. When the Schwarzschild metric in GR predicts geodesics that end up in a singularity it may not be a problem if one is working on orbits or stuff going on far away from the central mass.

The issue isn't whether physics "runs" on differential equations that always have solutions, but all about what properties a useful model should have in the case at hand. In short, it is not about differential equations but modelling problems well.

Anders Sandberg
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  • You don't seem to be answering the OP's question. He asked if the statement in the quote was true and the answer is yes. The question was not about physics in general or mathematical modeling of anything. The quote specifically selects correctly formulated real problems and as such they must have real solutions. – safesphere Aug 27 '17 at 22:42