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If you have a PDE that models physical phenomena, it is fundamental that:
  • there must exist a solution for every physically valid initial condition, otherwise it means that the equation does not describe certain cases of reality
  • the solution must be unique, otherwise how are we to choose between the multiple solutions?
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Unlike for ordinary differential equations which have the Picard–Lindelöf theorem, the existence and uniqueness of solution is not well solved for PDEs.
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Ancestors

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