Zakharov system on R

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  • 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.