Decay estimates for Nonlinear Schrödinger equation with the inverse-square potential
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912158563958784 |
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| author | Wang, Jialu Xu, Chengbin Zhang, Fang |
| author_facet | Wang, Jialu Xu, Chengbin Zhang, Fang |
| contents | In this paper, we study the dispersive decay estimates for solution to the $3\mathrm{D}$ energy-critical nonlinear Schrödinger equation with an inverse-square operator $\mathcal{L}_a$ where the operator is denoted by $\mathcal{L}_{a}:=-Δ+\frac{a}{|x|^2}$ with the constant $a\geq0$. Inspired by the work of \cite{KMVZZ1,K}, we first establish that the solutions exhibit $\dot{H}^1(\R^3)$ uniform regularity, derive the Lorentz-Strichartz estimates, and then obtain the desired decay estimates using the bootstrap argument. The key ingredients of our approach include the equivalence of Sobolev norms and the fractional product rule. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11424 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decay estimates for Nonlinear Schrödinger equation with the inverse-square potential Wang, Jialu Xu, Chengbin Zhang, Fang Analysis of PDEs In this paper, we study the dispersive decay estimates for solution to the $3\mathrm{D}$ energy-critical nonlinear Schrödinger equation with an inverse-square operator $\mathcal{L}_a$ where the operator is denoted by $\mathcal{L}_{a}:=-Δ+\frac{a}{|x|^2}$ with the constant $a\geq0$. Inspired by the work of \cite{KMVZZ1,K}, we first establish that the solutions exhibit $\dot{H}^1(\R^3)$ uniform regularity, derive the Lorentz-Strichartz estimates, and then obtain the desired decay estimates using the bootstrap argument. The key ingredients of our approach include the equivalence of Sobolev norms and the fractional product rule. |
| title | Decay estimates for Nonlinear Schrödinger equation with the inverse-square potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.11424 |