Zakharov system on R

• 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].