PPT Slide

? ?0T ||? ?(t)|| L? ? CT E s (T)

if ?0T ||? ?(t)|| L? ? C

E s [?](t)= E[?... ? ?](t)

E[?](t) = ? (|? t ?(t,x)|2 + ? i |? i ? (t,x)| 2 )dx

? = 0

METHODS: Energy Estimates, Sobolev Ineq.

E(t)=E(0)

[?? , ]=0

E s (t)=E s (0)

? = ?(? ) • ? ? • ? ?

E s (t) ? CE s (0) , t ? [0,T]

Translation symmetry of Minkowsky

Conservation of energy

||f || L? ? C|| f|| H s , s > n/2

Sobolev Inequality