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 if , with the endpoint being attained when Bo1993. In particular one has GWP in when or when and the data is small norm.For one also has GWP for random data whose Fourier coefficients decay like (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.
