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Cosmic Censorship

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7. Critical well-posedness for

Wave Maps and Yang-Mills

Well posed for

Hs(loc)- data for any s > sc.

Weakly globally well posed for

s = sc and small initial data

WELL POSED

Hs(loc)-initial data _ local in time, unique Hs(loc)-solutions. Continuous dependence on the data:
strong analytically
weak non-analytically

CRITICAL EXPONENT s = sc

Hs is invariant under the non-linear scaling of the equations.

Wave maps in Rn+1 sc =n/2

Yang-Mills in Rn+1 sc =(n-2)/2

There has been a lot of progress in treating the case s > sc

CONJECTURE