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This question already has an answer here:

I would tou like to investigate two things:

I) What is the motivation for the study of viscosity solutions of elliptic linear (fully non-linear) Pde's

II) The relationship between viscosity solutions and distributional solutions.

For example, distributional solutions are defined a.e and contains the space of classical solutions. The viscosity solutions also enjoys the same property and with good coefficients are also classical solutions. There exists relation of inclusion between the set of viscosity and weak solutions ? $Lu=A_{ij}D_{ij}u$?

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marked as duplicate by Harry49, Alexander Gruber Apr 29 at 2:00

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ George Stokes wrote a paper introducing the navier stokes eq with a brief intro that might answer ur question $\endgroup$ – Saketh Malyala Apr 28 at 22:10