MAIN OPEN PROBLEMS
2 Break-down for 3D Euler Equations
There may in fact exist plenty of global smooth solutions which are, however, unstable. More precisely the set of all smooth initial data which lead to global in time smooth solutions may have measure zero, yet, it may be dense in the set of all regular initial conditions, relative to a reasonable topology.
_t u+u __ u= -_p
div u= 0
CONJECTURE
Weak Form
There exists:
||_(t)||L_ __ as t _ T* .
Strong Form
Most regular data lead to such behavior. More precisely the set of initial data which lead to finite time break-down is dense in the set of all regular data with respect to a reasonable topology.
Most unstable equation.
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