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Liyun Zhao

Publications and source records attributed to Liyun Zhao.

4 recordsLinked to original sources
arXiv2022

Generalize Hilbert operator acting on Dirichlet spaces

Let $μ$ be a positive Borel measure on the interval $[0,1)$. For $γ>0$, the Hankel matrix $\mathcal{H}_{μ,γ}=(μ_{n,k})_{n,k\geq0}$ with entries $μ_{n,k}=μ_{n+k}$, where $μ_{n+k}=\int_{0}^{\infty}t^{n+k}dμ(t)$. formally induces the operator $$\mathcal{H}_{μ,γ}=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}μ_{n,k}a_k\right)\frac{Γ(n+γ)}{n!Γ(γ)}z^n,$$ on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}{a_k}{z^k}$ in the unit disc $\mathbb{D}$. Following ideas from \cite{author3} and \cite{author4}, in this paper, for $0\leqα<2$, $2\leqβ<4$, $γ\geq1$. we characterize the measure $μ$ for which $\mathcal{H}_{μ,γ}$ is bounded(resp.,compact)from $\mathcal{D}_α$ into $\mathcal{D}_β$.

math.CV↗
arXiv2022

The range of Hilbert operator and Derivative-Hilbert operator acting on $H^1$

Let $μ$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_μ=(μ_{n,k})_{n,k\geq0}$ with entries $μ_{n,k}=μ_{n+k}$, where $μ_n=\int_{[0,1)}t^{n}dμ(t)$. For $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function in $\mathbb{D}$, the Hilbert operator is defined by $$\mathcal{H}_μ(f)(z)=\sum_{n=0}^{\infty}\Bigg(\sum_{k=0}^{\infty}μ_{n,k}a_k\Bigg)z^n, \quad z\in \mathbb{D}.$$ The Derivative-Hilbert operator is defined as $$\mathcal{DH}_μ(f)(z)=\sum_{n=0}^{\infty}\Bigg(\sum_{k=0}^{\infty}μ_{n,k}a_k\Bigg)(n+1)z^n, \quad z\in \mathbb{D}.$$ In this paper, we determine the range of the Hilbert operator and Derivative-Hilbert operator acting on $H^{\infty}$.

math.CV↗
arXiv2011

Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is cν^{1/2}.

math.AP↗