Difference between revisions of "Quadratic NLS"
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* For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[CaWe1990]], [[Ts1987]]. | * For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[CaWe1990]], [[Ts1987]]. | ||
** In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [[KnPoVe-p]] | ** In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [[KnPoVe-p]] | ||
− | * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[St1997]], [[ | + | * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[St1997]], [[CoDeKnSt2001]]. |
** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [[MurTao2004]]. | ** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [[MurTao2004]]. | ||
* If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Ta2001]]. | * If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Ta2001]]. | ||
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* Scaling is <math>s_c = -1/2.\,</math> | * Scaling is <math>s_c = -1/2.\,</math> | ||
* For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[CaWe1990]], [[Ts1987]]. | * For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[CaWe1990]], [[Ts1987]]. | ||
− | * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[St1997]], [[ | + | * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[St1997]], [[CoDeKnSt2001]]. |
* If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Ta2001]]. | * If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Ta2001]]. | ||
* In the Hamiltonian case (a non-linearity of type <math>|u| u\,</math>) we have GWP for <math>s \ge 0\,</math> by <math>L^2\,</math> conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data. | * In the Hamiltonian case (a non-linearity of type <math>|u| u\,</math>) we have GWP for <math>s \ge 0\,</math> by <math>L^2\,</math> conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data. |
Latest revision as of 01:30, 17 March 2007
Description | |
---|---|
Equation | |
Fields | |
Data class | |
Basic characteristics | |
Structure | non-Hamiltonian |
Nonlinearity | semilinear |
Linear component | Schrodinger |
Critical regularity | |
Criticality | N/A |
Covariance | N/A |
Theoretical results | |
LWP | varies |
GWP | - |
Related equations | |
Parent class | NLS |
Special cases | Quadratic NLS on R, T, R^2, T^2, R^3, T^3 |
Other related | - |
Contents
Quadratic NLS
Equations of the form
which a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.
Quadratic NLS on R
- Scaling is
- For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
- If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
- This can be improved to the Besov space MurTao2004. The bilinear estimates fail for NaTkTs2001.
- If the quadratic non-linearity is of type then one can push LWP to KnPoVe1996b.
- Since these equations do not have conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
- If the non-linearity is then there is GWP in thanks to conservation, and ill-posedness below by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.
Quadratic NLS on
- For any quadratic non-linearity one can obtain LWP for Bo1993. In the Hamiltonian case () this is sharp by Gallilean invariance considerations KnPoVe-p
- If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
- In the Hamiltonian case (a non-linearity of type ) we have GWP for by conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
Quadratic NLS on
- Scaling
- For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
- In the Hamiltonian case () this is sharp by Gallilean invariance considerations KnPoVe-p
- If the quadratic non-linearity is of or type then one can push LWP to St1997, CoDeKnSt2001.
- This can be improved to the Besov space MurTao2004.
- If the quadratic non-linearity is of type then one can push LWP to Ta2001.
- In the Hamiltonian case (a non-linearity of type ) we have GWP for by conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
Quadratic NLS on T^2
- If the quadratic non-linearity is of type then one can obtain LWP for Gr-p2
Quadratic NLS on
- Scaling is
- For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
- If the quadratic non-linearity is of or type then one can push LWP to St1997, CoDeKnSt2001.
- If the quadratic non-linearity is of type then one can push LWP to Ta2001.
- In the Hamiltonian case (a non-linearity of type ) we have GWP for by conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
Quadratic NLS on
- If the quadratic non-linearity is of type then one can obtain LWP for Gr-p2.