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 [[Bibliography#Bo1999|Bo1999]]
* 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 [[Bibliography#De2001|De2001]], [LbSf-p]
* 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]].


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[[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.