Zakharov system on R: Difference between revisions

Latest revision as of 00:00, 3 February 2007

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.

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-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]].
-Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (-1,-3/2).
+[[Category:Equations]]
-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].