Low-regularity global solution of the inhomogeneous nonlinear Schrödinger equations in modulation spaces
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| Format: | Preprint |
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2024
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| author | Bhimani, Divyang G. Dhingra, Diksha Sohani, Vijay Kumar |
| author_facet | Bhimani, Divyang G. Dhingra, Diksha Sohani, Vijay Kumar |
| contents | The study of low regularity Cauchy data for nonlinear dispersive PDEs has successfully been achieved using modulation spaces $M^{p,q}$ in recent years. In this paper, we study the inhomogeneous nonlinear Schrödinger equation (INLS) $$iu_t + Δu\pm |x|^{-b}|u|^αu=0,$$
where $α, b>0,$ on whole space $\mathbb R^n$ in modulation spaces. In the subcritical regime $(0<α< \frac{4-2b}{n}),$ we establish local well-posedness in $L^{2}+M^{α+2,\frac{α+2}{α+1}}( \supset L^2 + H^s \ \text{for} \ s>\frac{nα}{2(α+2)}).$ By adapting Bourgain's high-low decomposition method, we establish global well-posedness in $M^{p,\frac{p}{p-1}}$ with $2<p$ and $p$ sufficiently close to 2. This is the first global well-posedness result for INLS on modulation spaces, which contains certain Sobolev $H^s$ $(0<s<1)$ and $L^p_s-$Sobolev spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00869 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Low-regularity global solution of the inhomogeneous nonlinear Schrödinger equations in modulation spaces Bhimani, Divyang G. Dhingra, Diksha Sohani, Vijay Kumar Analysis of PDEs 35Q55 The study of low regularity Cauchy data for nonlinear dispersive PDEs has successfully been achieved using modulation spaces $M^{p,q}$ in recent years. In this paper, we study the inhomogeneous nonlinear Schrödinger equation (INLS) $$iu_t + Δu\pm |x|^{-b}|u|^αu=0,$$ where $α, b>0,$ on whole space $\mathbb R^n$ in modulation spaces. In the subcritical regime $(0<α< \frac{4-2b}{n}),$ we establish local well-posedness in $L^{2}+M^{α+2,\frac{α+2}{α+1}}( \supset L^2 + H^s \ \text{for} \ s>\frac{nα}{2(α+2)}).$ By adapting Bourgain's high-low decomposition method, we establish global well-posedness in $M^{p,\frac{p}{p-1}}$ with $2<p$ and $p$ sufficiently close to 2. This is the first global well-posedness result for INLS on modulation spaces, which contains certain Sobolev $H^s$ $(0<s<1)$ and $L^p_s-$Sobolev spaces. |
| title | Low-regularity global solution of the inhomogeneous nonlinear Schrödinger equations in modulation spaces |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/2410.00869 |