Septic NLS: Difference between revisions

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====Septic NLS on <math>R</math>====
====Septic NLS on <math>R</math>====


* Scaling is s<sub>c</sub> = 1/6.
* Scaling is <math>s_c = 1/6\,</math>.
* LWP is known for s <font face="Symbol">³</font> s<sub>c</sub> [[Bibliography#CaWe1990|CaWe1990]].
* LWP is known for <math>s \ge s_c\,</math> [[CaWe1990]].
** For s=s<sub>c</sub> the time of existence depends on the profile of the data as well as the norm.
** For <math>s=s_c\,</math> the time of existence depends on the profile of the data as well as the norm.
** For s<s_c we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly [CtCoTa-p2]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
** For <math>s<s_c\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the [[virial identity]] and scaling.
* GWP for s <font face="Symbol">³</font> 1 by Hamiltonian conservation.
* GWP for <math>s \ge 1\,</math> by Hamiltonian conservation.
** This has been improved to s > 1-<font face="Symbol">e</font> in [[Bibliography#CoKeStTkTa2003b|CoKeStTkTa2003b]] in the defocusing case. This result can of course be improved further.
** This has been improved to <math>s > 1-e\,</math> in [[CoKeStTkTa2003b]] in the defocusing case. This result can of course be improved further.
** Scattering in the energy space [[Bibliography#Na1999c|Na1999c]]
** Scattering in the energy space [[Na1999c]]
** One also has GWP and scattering for small H^{s<sub>c</sub>} data for any septic non-linearity.
** One also has GWP and scattering for small <math>H^{s_c}\,</math> data for any septic non-linearity.


====Septic NLS on <math>R^2</math>====
====Septic NLS on <math>R^2</math>====


* Scaling is s<sub>c</sub> = 2/3.
* Scaling is <math>s_c = 2/3\,</math>.
* LWP is known for s <font face="Symbol">³</font> s<sub>c</sub> [[Bibliography#CaWe1990|CaWe1990]].
* LWP is known for <math>s \ge s_c\,</math> [[CaWe1990]].
** For s=s<sub>c</sub> the time of existence depends on the profile of the data as well as the norm.
** For <math>s=s_c\,</math> the time of existence depends on the profile of the data as well as the norm.
** For s<s_c we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly [CtCoTa-p2]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
** For <math>s<s_c\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
* GWP for s <font face="Symbol">³</font> 1 by Hamiltonian conservation.
* GWP for <math>s \ge 1\,</math> by Hamiltonian conservation.
** This has been improved to s > 1-<font face="Symbol">e</font> in [[Bibliography#CoKeStTkTa2003b|CoKeStTkTa2003b]] in the defocusing case. This result can of course be improved further.
** This has been improved to <math>s > 1-e\,</math> in [[CoKeStTkTa2003b]] in the defocusing case. This result can of course be improved further.
** Scattering in the energy space [[Bibliography#Na1999c|Na1999c]]
** Scattering in the energy space [[Na1999c]]
** One also has GWP and scattering for small H^{s<sub>c</sub>} data for any septic non-linearity.
** One also has GWP and scattering for small H^{s<sub>c</sub>} data for any septic non-linearity.


====Septic NLS on <math>R^3</math>====
====Septic NLS on <math>R^3</math>====


* Scaling is s<sub>c</sub> = 7/6.
* Scaling is <math>s_c = 7/6\,.</math>
* LWP is known for s <font face="Symbol">³</font> s<sub>c</sub> [[Bibliography#CaWe1990|CaWe1990]].
* LWP is known for <math>s \ge s_c\,</math> [[CaWe1990]].
** For s=s<sub>c</sub> the time of existence depends on the profile of the data as well as the norm.
** For <math>s=s_c\,</math> the time of existence depends on the profile of the data as well as the norm.
** For s<s_c we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly [CtCoTa-p2]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
** For <math>s<s_c\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
* GWP and scattering for small data by Strichartz estimates [[Bibliography#CaWe1990|CaWe1990]].
* GWP and scattering for small data by Strichartz estimates [[CaWe1990]].
** For large data one has blowup in the focusing case by the virial identity; in particular one has ill-posedness in the energy space.
** For large data one has blowup in the focusing case by the virial identity; in particular one has ill-posedness in the energy space.
** It is not known (and would be extremely interesting to find out!) what is going on in the defocusing case; for instance, is there blowup from smooth data? Even for radial data nothing seems to be known. This may be viewed as an extremely simplified model problem for the global regularity issue for Navier-Stokes.
** It is not known (and would be extremely interesting to find out!) what is going on in the defocusing case; for instance, is there blowup from smooth data? Even for radial data nothing seems to be known. This may be viewed as an extremely simplified model problem for the global regularity issue for Navier-Stokes.

Latest revision as of 06:25, 21 July 2007

Septic NLS on

  • Scaling is .
  • LWP is known for CaWe1990.
    • For the time of existence depends on the profile of the data as well as the norm.
    • For we have ill-posedness, indeed the norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
  • GWP for by Hamiltonian conservation.
    • This has been improved to in CoKeStTkTa2003b in the defocusing case. This result can of course be improved further.
    • Scattering in the energy space Na1999c
    • One also has GWP and scattering for small data for any septic non-linearity.

Septic NLS on

  • Scaling is .
  • LWP is known for CaWe1990.
    • For the time of existence depends on the profile of the data as well as the norm.
    • For we have ill-posedness, indeed the norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
  • GWP for by Hamiltonian conservation.
    • This has been improved to in CoKeStTkTa2003b in the defocusing case. This result can of course be improved further.
    • Scattering in the energy space Na1999c
    • One also has GWP and scattering for small H^{sc} data for any septic non-linearity.

Septic NLS on

  • Scaling is
  • LWP is known for CaWe1990.
    • For the time of existence depends on the profile of the data as well as the norm.
    • For we have ill-posedness, indeed the norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
  • GWP and scattering for small data by Strichartz estimates CaWe1990.
    • For large data one has blowup in the focusing case by the virial identity; in particular one has ill-posedness in the energy space.
    • It is not known (and would be extremely interesting to find out!) what is going on in the defocusing case; for instance, is there blowup from smooth data? Even for radial data nothing seems to be known. This may be viewed as an extremely simplified model problem for the global regularity issue for Navier-Stokes.
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