| 000 | 02774cam a22003497 4500 | ||
|---|---|---|---|
| 001 | on1432712088 | ||
| 003 | OCoLC | ||
| 005 | 20240624111422.0 | ||
| 008 | 240507s2024 riu 0|| 0 eng d | ||
| 020 | _a1470469790 | ||
| 020 | _a9781470469795 | ||
| 035 | _a(OCoLC)1432712088 | ||
| 040 |
_aYDX _beng _cYDX _dOCLCO _dYSM _dOCLCO _dUNBCA _dOCLCO |
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| 066 | _c(S | ||
| 100 | 1 |
_aBurq, Nicolas, _eauthor |
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| 245 | 1 | 0 |
_aAlmost sure scattering for the one dimensional nonlinear Schrödinger equation / _cNicolas Burq, Laurent Thomann |
| 264 | 1 |
_aProvidence, RI : _bAmerican Mathematical Society, _c2024. |
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| 264 | 4 | _c©2024 | |
| 300 |
_avii, 87 pages ; _c26 cm |
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| 336 |
_atext _btxt _2rdacontent |
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| 337 |
_aunmediated _bn _2rdamedia |
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| 338 |
_avolume _bnc _2rdacarrier |
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| 490 | 1 |
_aMemoirs of the American Mathematical Society ; _vno. 1480 |
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| 504 | _aIncludes bibliographic references. | ||
| 650 | 0 | _aScattering (Mathematics) | |
| 650 | 6 | _aDispersion (Mathématiques) | |
| 700 | 1 |
_aThomann, Laurent, _eauthor |
|
| 830 | 0 |
_aMemoirs of the American Mathematical Society ; _vv. 1480. |
|
| 880 |
_6520-00/(S _aWe consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p>1. On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On Rd, the large time dispersion shows that the only invariant measure is the δ measure on the trivial solution u=0, and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their Lp regularity. We deduce from this precise description the global well-posedness of the equation for p>1 and scattering for p>3 (actually even for 1<p≤3, we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their Lp+1 norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution. |
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_2ddc _cBK _n0 |
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| 948 | _hNO HOLDINGS IN P5A - 7 OTHER HOLDINGS | ||
| 999 |
_c41354 _d41354 |
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