NLS wellposedness: Difference between revisions
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* In the <math>H^1\,</math> subcritical case one has GWP in <math>H^1\,,</math> assuming the nonlinearity is smooth near the origin [[Ka1986]] | * In the <math>H^1\,</math> subcritical case one has GWP in <math>H^1\,,</math> assuming the nonlinearity is smooth near the origin [[Ka1986]] | ||
** In <math>R^6\,</math> one also has Lipschitz well-posedness [[ | ** In <math>R^6\,</math> one also has Lipschitz well-posedness [[BuGdTz2003]] | ||
<br /> In the periodic setting these results are much more difficult to obtain. On the one-dimensional torus T one has LWP for <math>s > 0, s_c\,</math> if <math>p > 1\,</math>, with the endpoint <math>s=0\,</math> being attained when <math>1 \le p \le 4\,</math> [[Bo1993]]. In particular one has GWP in <math>L^2\,</math> when <math>p < 4\,,</math> or when <math>p=4\,</math> and the data is small norm.For <math>6 > p \ge 4\,</math> one also has GWP for random data whose Fourier coefficients decay like <math>1/|k|\,</math> (times a Gaussian random variable) [[Bo1995c]]. (For <math>p=6\,</math> one needs to impose a smallness condition on the <math>L^2\,</math> norm or assume defocusing; for <math>p>6\,</math> one needs to assume defocusing). <br /> | <br /> In the periodic setting these results are much more difficult to obtain. On the one-dimensional torus T one has LWP for <math>s > 0, s_c\,</math> if <math>p > 1\,</math>, with the endpoint <math>s=0\,</math> being attained when <math>1 \le p \le 4\,</math> [[Bo1993]]. In particular one has GWP in <math>L^2\,</math> when <math>p < 4\,,</math> or when <math>p=4\,</math> and the data is small norm.For <math>6 > p \ge 4\,</math> one also has GWP for random data whose Fourier coefficients decay like <math>1/|k|\,</math> (times a Gaussian random variable) [[Bo1995c]]. (For <math>p=6\,</math> one needs to impose a smallness condition on the <math>L^2\,</math> norm or assume defocusing; for <math>p>6\,</math> one needs to assume defocusing). <br /> | ||
Latest revision as of 20:20, 9 August 2006
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
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 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 > 0, s_c\,}
if 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 > 1\,}
, with the endpoint 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=0\,}
being attained when 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 1 \le p \le 4\,}
Bo1993. In particular 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 L^2\,}
when 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 < 4\,,}
or when 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=4\,}
and the data is small 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 6 > p \ge 4\,}
one also has GWP for random data whose Fourier coefficients decay like 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 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.
