On Growth of Sobolev norms for cubic Schrödinger equation with harmonic potential in dimensions $d=2,3$
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911295798771712 |
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| author | Song, Yilin Zhang, Ruixiao Zheng, Jiqiang |
| author_facet | Song, Yilin Zhang, Ruixiao Zheng, Jiqiang |
| contents | In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schrödinger equation with harmonic potential in dimensions $d=2,3$, \begin{align}\label{PNLS} \begin{cases}\tag{PNLS} i\partial_tu-Hu=|u|^{2}u,&(t,x)\in\mathbb{R}\times\mathbb{R}^d,\\ u(0,x)=u_0(x), \end{cases} \end{align} where $H=-Δ+|x|^2$. Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear Strichartz estimates, which removes the $\varepsilon$-loss of Burq-Poiret-Thomann [Preprint, arXiv: 2304.10979]. To show the polynomial growth of Sobolev norm, our proof relies on the upside-down $I$-method associated to the harmonic oscillator. Due to the lack of Fourier transform or expansion, we need to carefully control the freqeuncy interaction of the type "high-high-low-low". To overcome this difficulty, we establish the explicit interaction for products of eigenfunctions. Our bound covers the result of Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436] in dimension two and is new in dimension three. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Growth of Sobolev norms for cubic Schrödinger equation with harmonic potential in dimensions $d=2,3$ Song, Yilin Zhang, Ruixiao Zheng, Jiqiang Analysis of PDEs In this article, we study the growth of higher-order Sobolev norms for solutions to the defocusing cubic nonlinear Schrödinger equation with harmonic potential in dimensions $d=2,3$, \begin{align}\label{PNLS} \begin{cases}\tag{PNLS} i\partial_tu-Hu=|u|^{2}u,&(t,x)\in\mathbb{R}\times\mathbb{R}^d,\\ u(0,x)=u_0(x), \end{cases} \end{align} where $H=-Δ+|x|^2$. Motivated by Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436], we first establish the bilinear Strichartz estimates, which removes the $\varepsilon$-loss of Burq-Poiret-Thomann [Preprint, arXiv: 2304.10979]. To show the polynomial growth of Sobolev norm, our proof relies on the upside-down $I$-method associated to the harmonic oscillator. Due to the lack of Fourier transform or expansion, we need to carefully control the freqeuncy interaction of the type "high-high-low-low". To overcome this difficulty, we establish the explicit interaction for products of eigenfunctions. Our bound covers the result of Planchon-Tzvetkov-Visciglia [Rev. Mat. Iberoam., 39 (2023), 1405-1436] in dimension two and is new in dimension three. |
| title | On Growth of Sobolev norms for cubic Schrödinger equation with harmonic potential in dimensions $d=2,3$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.17731 |