Septic NLW/NLKG on R3

• Scaling is $s=7/6$.
• LWP for $s\geq 7/6$ by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
  • When $s=7/6$ the time of existence depends on the profile of the data and not just on the norm.
  • For $s<s_{c}$ one has instantaneous blowup in the focusing case, and unbounded growth of $H^{s}$ norms in the defocusing case (CtCoTa-p2).
• Global existence of large smooth solutions is unknown in the defocussing case; in the focussing case one certainly has blowup by ODE methods.
  • In the energy class $s=1$, one has ill-posedness in the sense that the solution map is not uniformly continuous Leb2000; for higher dimensions see BrKum2000. This is despite an a priori bound on the $H^{1}xL^{2}$ norm in the defocussing case from energy conservation. A variant of this result appears in CtCoTa-p2.
  • For small data one of course has GWP and scattering (LbSo1995).

Global regularity problem

It is not known what happens to large smooth solutions in the defocusing case, even in the radial case. One may tentatively conjecture that global smooth solutions exist for generic large data, though perhaps not for exceptional large data.

This problem can be viewed as an extremely simplified (but still incredibly difficult) model problem for the global regularity issue for Navier-Stokes. By far the main difficulty is that all the known conserved and monotone quantities are supercritical with respect to scaling, and so we have no effective mechanism for long-term control of the solution.