Septic NLS

Septic NLS on $R$

• Scaling is $s_{c}=1/6\,$.
• LWP is known for $s\geq s_{c}\,$ CaWe1990.
  • For $s=s_{c}\,$ 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.
• GWP for $s\geq 1\,$ by Hamiltonian conservation.
  • This has been improved to $s>1-e\,$ 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 $H^{s_{c}}\,$ data for any septic non-linearity.

Septic NLS on $R^{2}$

• Scaling is $s_{c}=2/3\,$.
• LWP is known for $s\geq s_{c}\,$ CaWe1990.
  • For $s=s_{c}\,$ 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.
• GWP for $s\geq 1\,$ by Hamiltonian conservation.
  • This has been improved to $s>1-e\,$ 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^{s_c} data for any septic non-linearity.

Septic NLS on $R^{3}$

• Scaling is $s_{c}=7/6\,.$
• LWP is known for $s\geq s_{c}\,$ CaWe1990.
  • For $s=s_{c}\,$ 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.
• 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.