Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.13561 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908272968073216 |
|---|---|
| author | Yan, Xiangqian Li, Yongsheng Yan, Wei |
| author_facet | Yan, Xiangqian Li, Yongsheng Yan, Wei |
| contents | In this article, we investigate the orthonormal Strichartz estimates and the convergence
problem of the density function associated with $\partial_{x}^{3}+\partial_{x}^{-1}$. Firstly,
when $γ_{0}\in\mathfrak{S}^β(\dot{H}^{s})$ with $\frac{1}{4}\leq s<\frac{1}{2},\,
0<α\leq 1$, and $1\leqβ<\fracα{1-2s}$, we prove that
$\lim\limits_{t\longrightarrow0}\sum\limits_{j=1}^{+\infty}λ_{j} \left|e^{-t(\partial_{x}^{3}+\partial_{x}^{-1})}f_{j}\right|^{2}=\sum\limits_{j=1}^{+\infty}λ_{j} \left|f_{j}\right|^{2}.$
This extends the Theorem 1.1 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.).
Moreover, we present the Hausdorff
dimension of the divergence set of the density function related to $\partial_{x}^{3}+\partial_{x}^{-1}$,
namely ${\rm dim_{H}}D(γ_{0})\leq (1-2s)β$, which extends the Theorem 1.1 of Zhao et al.
(Acta Math. Sci. Ser. B (Engl. Ed.) 42(2022), 1607-1620.).
%
Secondly, we present the orthonormal Strichartz estimates and the Schatten bounds with space-time norms on $\mathbf{R}$. %
Finally, by using full randomization,
we establish the probabilistic convergence of the density function related to $\partial_{x}^{3}+\partial_{x}^{-1}$
on $\R$, which extends the Theorem 1.3 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13561 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The orthonormal Strichartz estimates and convergence problem of density functions related to $\partial_{x}^{3}+\partial_{x}^{-1}$ Yan, Xiangqian Li, Yongsheng Yan, Wei Analysis of PDEs Primary-35Q53 In this article, we investigate the orthonormal Strichartz estimates and the convergence problem of the density function associated with $\partial_{x}^{3}+\partial_{x}^{-1}$. Firstly, when $γ_{0}\in\mathfrak{S}^β(\dot{H}^{s})$ with $\frac{1}{4}\leq s<\frac{1}{2},\, 0<α\leq 1$, and $1\leqβ<\fracα{1-2s}$, we prove that $\lim\limits_{t\longrightarrow0}\sum\limits_{j=1}^{+\infty}λ_{j} \left|e^{-t(\partial_{x}^{3}+\partial_{x}^{-1})}f_{j}\right|^{2}=\sum\limits_{j=1}^{+\infty}λ_{j} \left|f_{j}\right|^{2}.$ This extends the Theorem 1.1 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.). Moreover, we present the Hausdorff dimension of the divergence set of the density function related to $\partial_{x}^{3}+\partial_{x}^{-1}$, namely ${\rm dim_{H}}D(γ_{0})\leq (1-2s)β$, which extends the Theorem 1.1 of Zhao et al. (Acta Math. Sci. Ser. B (Engl. Ed.) 42(2022), 1607-1620.). % Secondly, we present the orthonormal Strichartz estimates and the Schatten bounds with space-time norms on $\mathbf{R}$. % Finally, by using full randomization, we establish the probabilistic convergence of the density function related to $\partial_{x}^{3}+\partial_{x}^{-1}$ on $\R$, which extends the Theorem 1.3 of Yan et al. (Indiana Univ. Math. J. 71(2022), 1897-1921.). |
| title | The orthonormal Strichartz estimates and convergence problem of density functions related to $\partial_{x}^{3}+\partial_{x}^{-1}$ |
| topic | Analysis of PDEs Primary-35Q53 |
| url | https://arxiv.org/abs/2503.13561 |