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
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