# 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]]. | ||

* If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[KnPoVe1996b]]. | * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[KnPoVe1996b]]. | ||

− | ** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [[ | + | ** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [[MurTao2004]]. The <math>X^{s,b}\,</math> bilinear estimates fail for <math>H^{-3/4}\,</math> [[NaTkTs2001]]. |

* If the quadratic non-linearity is of <math>\underline{u}u\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[KnPoVe1996b]]. | * If the quadratic non-linearity is of <math>\underline{u}u\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[KnPoVe1996b]]. | ||

* Since these equations do not have <math>L^2\,</math> conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data. | * Since these equations do not have <math>L^2\,</math> conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data. | ||

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** 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]], [[CoDeKnSt-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]], [[CoDeKnSt-p]]. | ||

− | ** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [[ | + | ** 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> [[Ta-p2]]. | * If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Ta-p2]]. | ||

* 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. |

## Revision as of 06:45, 15 February 2007

## 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, CoDeKnSt-p.
- This can be improved to the Besov space MurTao2004.

- If the quadratic non-linearity is of type then one can push LWP to Ta-p2.
- 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, CoDeKnSt-p.
- If the quadratic non-linearity is of type then one can push LWP to Ta-p2.
- 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