Cubic NLW/NLKG on R: Difference between revisions
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* Scaling is <math>s_c = -1/2</math>. | * Scaling is <math>s_c = -1/2</math>. | ||
* LWP for <math>s \geq 1/6</math> by energy estimates and Sobolev (solution is in <math>L^3_x</math>). | * LWP for <math>s \geq 1/6</math> by energy estimates and Sobolev (solution is in <math>L^3_x</math>). | ||
** For <math>s<1/6</math> one has ill-posedness [CtCoTa-p2], indeed it is not even possible to make sense of solutions in the distributional sense. | ** For <math>s<1/6</math> one has ill-posedness ([[CtCoTa-p2]]), indeed it is not even possible to make sense of solutions in the distributional sense. | ||
* GWP for <math>s>1/3</math> for defocussing NLKG [[ | * GWP for <math>s>1/3</math> for defocussing NLKG ([[Bo1999]]) | ||
** For <math>s \geq 1</math> this is clear from energy conservation (for both NLKG and NLW). | ** For <math>s \geq 1</math> this is clear from energy conservation (for both NLKG and NLW). | ||
** 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. | ||
* ''Remark'': NLKG can be viewed as a symplectic flow with the symplectic form of <math>H^{1/2}</math>. NLW is similar but with the homogeneous <math>H^{1/2}</math>. | * ''Remark'': NLKG can be viewed as a symplectic flow with the symplectic form of <math>H^{1/2}</math>. NLW is similar but with the homogeneous <math>H^{1/2}</math>. | ||
* Small global solutions to NLKG (either focusing or defocusing) have logarithmic phase corrections due to the critical nature of the nonlinearity (neither short-range nor long-range).However there is still an asymptotic development and an asymptotic completeness theory, see [[ | * Small global solutions to NLKG (either focusing or defocusing) have logarithmic phase corrections due to the critical nature of the nonlinearity (neither short-range nor long-range).However there is still an asymptotic development and an asymptotic completeness theory, see [[De2001]], [[LbSf-p]]. | ||
[[Category:Wave]] | |||
[[Category:Equations]] | |||
Latest revision as of 04:49, 2 August 2006
- Scaling is .
- LWP for by energy estimates and Sobolev (solution is in ).
- For one has ill-posedness (CtCoTa-p2), indeed it is not even possible to make sense of solutions in the distributional sense.
- GWP for for defocussing NLKG (Bo1999)
- For this is clear from energy conservation (for both NLKG and 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.
- Remark: NLKG can be viewed as a symplectic flow with the symplectic form of . NLW is similar but with the homogeneous .
- Small global solutions to NLKG (either focusing or defocusing) have logarithmic phase corrections due to the critical nature of the nonlinearity (neither short-range nor long-range).However there is still an asymptotic development and an asymptotic completeness theory, see De2001, LbSf-p.
