Does an ODE (or Initial value problem) modelling a real physical problem always have a solution, whatever the initial conditions? If so, can this be extended to Partial differential equations as well?
2026-03-26 06:28:46.1774506526
Existence of solutions for an Ordinary differential equation modelling a real physical problem
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Famously, the Navier‒Stokes equations are modelling a real physical problem, yet whether or not they may be solved (and to which extent) is as yet unknown. This is of course a set of PDEs.
Regarding ODEs, if your model is sufficiently dumb, you'll obtain a solution that isn't logical. I think I remember a model for 1-d car traffic where the cars sped up to infinity. I also imagine that phenomena like feedbacks are instances where the solution explodes to infinity, yet I doubt that they do so in finite time. I can't think of an example with finite blowup though.