Cubic NLW/NLKG on R3: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
No edit summary
 
No edit summary
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
* Scaling is <math>s_c = 1/2</math>.
* Scaling is <math>s_c = 1/2</math>.
* LWP for <math>s \geq 1/2</math> by Strichartz estimates (see e.g. [[Bibliography#LbSo1995|LbSo1995]]; earlier references exist)
* LWP for <math>s \geq 1/2</math> by Strichartz estimates (see e.g. [[LbSo1995]]; earlier references exist)
** When <math>s=1/2</math> the time of existence depends on the profile of the data and not just on the norm.
** When <math>s=1/2</math> the time of existence depends on the profile of the data and not just on the norm.
** One can improve the critical space <math>H^{1/2}</math> to a slightly weaker Besov space [Pl-p2].
** One can improve the critical space <math>H^{1/2}</math> to a slightly weaker Besov space ([[Pl-p2]]).
** For <math>s<1/2</math> one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
** For <math>s<1/2</math> one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case ([[CtCoTa-p2]]).
* GWP for <math>s>3/4</math> [[references:KnPoVe-p2 KnPoVe-p2]] for defocussing NLKG.(An alternate proof is in [[Bibliography#GalPl2003|GalPl2003]]).
* GWP for <math>s>3/4</math> ([[KnPoVe-p2]]) for defocussing NLKG.(An alternate proof is in [[GalPl2003]]).
** For <math>s\geq1</math> this is clear from energy conservation (for both NLKG and NLW).
** For <math>s\geq1</math> this is clear from energy conservation (for both NLKG and NLW).
** One also has GWP and scattering for data with small <math>H^{1/2}</math> norm for general cubic non-linearities (and for either NLKG or NLW).
** One also has GWP and scattering for data with small <math>H^{1/2}</math> norm for general cubic non-linearities (and for either NLKG or NLW).
** In the defocussing case one has scattering for large <math>H^1</math> data [[Bibliography#BaeSgZz1990|BaeSgZz1990]], see also [Hi-p3].
** In the defocussing case one has scattering for large <math>H^1</math> data ([[BaeSgZz1990]]), see also ([[Hi-p3]]).
** Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
** Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
** In the focussing case there is blowup from large data by the ODE method.
** In the focussing case there is blowup from large data by the ODE method.
* For periodic defocussing NLKG there is a weak turbulence effect in <math>H^s</math> for <math>s > 5</math> (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} [[Bibliography#Kuk1995b|Kuk1995b]].In particular <math>H^s</math> cannot be a symplectic phase space for <math>s > 5</math>.
* For periodic defocussing NLKG there is a weak turbulence effect in <math>H^s</math> for <math>s > 5</math> (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} [[Kuk1995b]].In particular <math>H^s</math> cannot be a symplectic phase space for <math>s > 5</math>.


[[Category:Wave]]
----  [[Category:Equations]]
[[Category:Equations]]

Latest revision as of 04:55, 2 August 2006

  • Scaling is .
  • LWP for by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
    • When the time of existence depends on the profile of the data and not just on the norm.
    • One can improve the critical space to a slightly weaker Besov space (Pl-p2).
    • For one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case (CtCoTa-p2).
  • GWP for (KnPoVe-p2) for defocussing NLKG.(An alternate proof is in GalPl2003).
    • For this is clear from energy conservation (for both NLKG and NLW).
    • One also has GWP and scattering for data with small norm for general cubic non-linearities (and for either NLKG or NLW).
    • In the defocussing case one has scattering for large data (BaeSgZz1990), see also (Hi-p3).
    • Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
    • In the focussing case there is blowup from large data by the ODE method.
  • For periodic defocussing NLKG there is a weak turbulence effect in for (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} Kuk1995b.In particular cannot be a symplectic phase space for .