Zakharov system on R

Zakharov on R and T

Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (-1,-3/2). LWP on R for (0,-1/2), or more generally whenever -1/2 < s0 - s1 £ 1 and 2s0³ s1 + 1/2 ³ 0 [GiTsVl1997] On T (with irrational period) the same result holds but one must place the additional restriction 0 £ s0 - s1 [Tk-p3]. For general period, a slightly weaker version of this (assuming weighted pointwise bounds on Fourier co-efficients, and establishing an invariant Gibbs measure), is in [Bo1994] GWP on R for (9/10+,0) [Pe-p]. In the energy class (1,0) this was shown in [GiTsVl1997].