Zakharov system on R^2

Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (-1/2,-1).

LWP for (s0,s1) = (1/2,0) GiTsVl1997

For (s0,s1) = (1,0) this was proven in BoCo1996, Co1997.

GWP for small (1,0) data BoCo1996; the smallness is needed to control the nonquadratic portion of the energy.

As long as the H1 norm remains bounded (which is automatic for small data), the higher H^s norms of u grow by at most |t|(s-1)+ CoSt-p

Explicit blowup solutions have been constructed with a blowup rate of t-1 in H1 norm GgMe1994, GgMe1994b. This is optimal in the sense that no slower blowup rate is possible Me1996b