Cubic NLS on R3

Cubic NLS on $\mathbb {R} ^{3}$

Description
Equation	$iu_{t}+\Delta u=\pm |u|^{2}u$
Fields	$u:\mathbb {R} \times \mathbb {R} ^{3}\to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} ^{3})$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{1/2}(\mathbb {R} ^{3})$
Criticality	mass-supercritical; energy-subcritical; scattering-subcritical
Covariance	Galilean
Theoretical results
LWP	$H^{s}(\mathbb {R} )$ for $s\geq 1/2$
GWP	$H^{s}(\mathbb {R} )$ for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 4/5} (+) or for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 0} , small norm
Related equations
Parent class	cubic NLS
Special cases	-
Other related	-

The theory of the cubic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \R^3} is as follows.

• LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 1/2\,} CaWe1990.
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=1/2\,} the time of existence depends on the profile of the data as well as the norm.
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s<1/2\,} we have ill-posedness, indeed the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 1/2\,} there is unconditional well-posedness FurPlTer2001
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s >= 2/3\,} this is in Ka1995.
• GWP and scattering for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 4/5\,} in the defocussing case CoKeStTkTa-p8
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 5/6\,} GWP is in CoKeStTkTa2002
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>11/13\,} GWP is in Bo1999
  • For radial data and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 5/7\,} GWP and scattering is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>5/7\,} Bo1998b, Bo1999.
  • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s\ge 1\,} this follows from Hamiltonian conservation. One also has scattering in this case GiVl1985.
  • For small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{1/2}\,} data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^5\,} in spacetime.
  • In the focusing case one has blowup whenever the energy is negative Gs1977, OgTs1991, and in particular one has blowup arbitrarily close to the ground state BerCa1981, SaSr1985.If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s_{loc}\,} remaining bounded for all time Bo1996c, Bo1998b. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component Ta2004b