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First question, if solutions are found to be non-unique, would the NS equations have to be corrected?

Second question, since uniqueness is unknown, if someone finds a blow-up solution would he or she have to prove that it's unique to answer the problem? (e.g. turbulent and smooth solution possible for exactly the same IC, BC, etc..? kinda strange.)

I'm curious what uniqueness means in relation to smoothness and turbulence. I'm not an expert in math so sorry if the answers are obvious. haha. Thanks in advance.