NLS wellposedness

In order to establish the well-posedness of the NLS 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 H^s} one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume

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 p} is an odd integer, or 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 p > [s]+1.}

With this assumption, one has 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 0, s_c\,} , CaWe1990; see also Ts1987; for the case 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\,,} see GiVl1979. In 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 L^2\,} -subcritical cases one has GWP for all 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 0\,} by 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^2\,} conservation; in all other cases one has GWP and scattering for small data 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 H^s\,} , 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 s_c.\,} These results apply in both the focussing and defocussing cases. At the critical exponent one can prove Besov space refinements Pl2000, Pl-p4. This can then be used to obtain self-similar solutions, see CaWe1998, CaWe1998b, RiYou1998, MiaZg-p1, MiaZgZgx-p, MiaZgZgx-p2, Fur2001.

Now suppose we remove the regularity assumption that 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 p\,} is either an odd integer or larger than 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\,.} Then some of the above results are still known to hold:

• In 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^1\,} subcritical case one has GWP 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 H^1\,,} assuming the nonlinearity is smooth near the origin Ka1986
  • 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 R^6\,} one also has Lipschitz well-posedness BuGdTz2003

In the periodic setting these results are much more difficult to obtain. On the one-dimensional torus T one has LWP for $s>0,s_{c}\,$ if $p>1\,$, with the endpoint $s=0\,$ being attained when $1\leq p\leq 4\,$ Bo1993. In particular one has GWP in $L^{2}\,$ when $p<4\,,$ or when $p=4\,$ and the data is small norm.For $6>p\geq 4\,$ one also has GWP for random data whose Fourier coefficients decay like $1/|k|\,$ (times a Gaussian random variable) Bo1995c. (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 p=6\,} one needs to impose a smallness condition on 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 L^2\,} norm or assume defocusing; 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 p>6\,} one needs to assume defocusing).

• For the defocussing case, one has GWP in 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^1\,} -subcritical case if the data 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 H^1\,.}

Many of the global results 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 H^s\,} also hold true 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 L^{2}(|x|^{2s} dx)\,} . Heuristically this follows from the pseudo-conformal transformation, although making this rigorous is sometimes difficult. Sample results are in CaWe1992, GiOzVl1994, Ka1995, NkrOz1997, NkrOz-p. See NaOz2002 for further discussion.