Strichartz estimates for orthonormal functions and convergence problem of density functions of Boussinesq operator on manifolds

TL;DR

This paper establishes new maximal-in-time and Strichartz estimates for orthonormal functions related to the Boussinesq operator on manifolds, addressing convergence and divergence set properties of density functions with probabilistic results.

Contribution

It introduces novel maximal-in-time and Strichartz estimates for the Boussinesq operator on manifolds, including optimal bounds and probabilistic convergence results.

Findings

Proved pointwise convergence of density functions on \\mathbb{R} and unit ball.

Established Hausdorff dimension bounds for divergence sets.

Derived optimal Strichartz estimates for orthonormal functions.

Abstract

This paper is devoted to studying the maximal-in-time estimates and Strichartz estimates for orthonormal functions and convergence problem of density functions related to Boussinesq operator on manifolds. Firstly, we present the pointwise convergence of density function related to Boussinesq operator with $\gamma_{0}\in\mathfrak{S}^{\beta}(\dot{H}^{\frac{1}{4}}(\mathbf{R}))(\beta<2)$ with the aid of the maximal-in-time estimate related to Boussinesq operator with orthonormal function on $\R$. Secondly, we present the pointwise convergence of density function related to Boussinesq operator with $\gamma_{0}\in\mathfrak{S}^{\beta}(\dot{H}^{s})(\frac{d}{4}\leq s<\frac{d}{2},\, 0<\alpha\leq d, 1\leq\beta<\frac{\alpha}{d-2s})$ with the aid of the maximal-in-time estimates related to Boussinesq operator with orthonormal function on the unit ball $\mathbf{B}^{d}(d\geq1)$ established in this paper; we also present the Hausdorff dimension of the divergence set of density function related to Boussinesq operator $dim_{H}D(\gamma_{0})\leq (d-2s)\beta$. Thirdly, we show the Strichartz estimates for orthonormal functions and Schatten bound with space-time norms related to Boussinesq operator on $\mathbf{T}$ with the aid of the noncommutative-commutative interpolation theorems established in this paper, which are just Lemmas 3.1-3.4 in this paper; we also prove that Theorems 1.5, 1.6 are optimal. Finally, by using full randomization, we present the probabilistic convergence of density function related to Boussinesq operator on $\R$, $\mathbf{T}$ and $\Theta=\{x\in\R^{3}:|x|<1\}$ with $\gamma_{0}\in\mathfrak{S}^{2}$.