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Anyone can explain the main idea, or recommend some paper or book on that?

For the whole space case, or the periodic case, the proofs are everywhere. But those do not seem to apply to the bounded case. I wasn't able to find a readable proof for the bounded domain case in English reference. Please help.

Thanks.

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When you say you cannot find a proof in English reference, I assume you know about the work of Yudovitch, that of Wolibner, and of Bardos.

Are you, however, aware of the work of Kato, which is in English?

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  • $\begingroup$ Yes, I know those which are not in English. Kato's work is on Holder continuous case(for the bounded domain). I should have said I expected one proof in Sobolev framework. Thanks a lot. $\endgroup$
    – pde_bk
    Commented Aug 7, 2013 at 13:52

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