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I'm currently trying to get into PDE's and as part of a course i'm focusing on proofs on existence of solutions to the Navier-Stokes Equations. Although existence of solutions has been proved for 1D and 2D my major problem is that i can't find one or more references/books/papers which proof the existence for the same Navier-Stokes Equations system.

There are papers which use periodic solutions, others have different settings on viscosity and others examine various conditions which help on existence proof's like weak Leray-Hopf solutions or Serrin conditions. In the end, since i'm very new at PDE's and Navier-Stokes itself, i dont see the wood for the trees.

Also, if i understood a few remarks in my reading's right, it's true that weak solutions are also strong solutions - but i couldnt find a proof or even a reference to a proof of this statement. So it would also be helpful if someone could give me a hint where to find this connection too so the weak-solution proofs make more sense?

So, which references would be helpful to read to create a paper for my course which sum's up existence proofs for 1D,2D and current interesting proof of a 3D setting of the Navier-Stokes equations?

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Some years ago I had the same question; I found J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’éspace. Acta Mathematica, 63, 7 1934, surprisingly easy to read and absorb (English translation here: https://arxiv.org/pdf/1604.02484.pdf). If you've been trained in functional analysis, you could start from P. Constantin and C. Foias. Navier–Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.

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