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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.12582 |
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| _version_ | 1866915541990506496 |
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| author | Knezevitch, Alexis |
| author_facet | Knezevitch, Alexis |
| contents | We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity $p \geq 5$, which is known to be globally well-posed in the Sobolev space $H^σ(\mathbb{T})$, for every $σ\geq 1$, thanks to the conservation and finiteness of the energy. For regularities $σ< 1$, where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures $μ_s$, with covariance operator $(1-Δ)^s$, for $s$ in a range $(s_p,\frac{3}{2}]$. We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures $μ_s$, with additional $L^q$-bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These $L^q$-bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain's invariant argument and to a recent work by Forlano-Tolomeo. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12582 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative quasi-invariance of Gaussian measures below the energy level for the 1D generalized nonlinear Schrödinger equation and application to global well-posedness Knezevitch, Alexis Analysis of PDEs We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity $p \geq 5$, which is known to be globally well-posed in the Sobolev space $H^σ(\mathbb{T})$, for every $σ\geq 1$, thanks to the conservation and finiteness of the energy. For regularities $σ< 1$, where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures $μ_s$, with covariance operator $(1-Δ)^s$, for $s$ in a range $(s_p,\frac{3}{2}]$. We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures $μ_s$, with additional $L^q$-bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These $L^q$-bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain's invariant argument and to a recent work by Forlano-Tolomeo. |
| title | Quantitative quasi-invariance of Gaussian measures below the energy level for the 1D generalized nonlinear Schrödinger equation and application to global well-posedness |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.12582 |