arXiv subjects

Xiaonan Hao

Publications and source records attributed to Xiaonan Hao.

3 recordsLinked to original sources

arXiv2024

Asymptotic behavior of 3-D evolutionary model of Magnetoelasticity for small data

In this article, we consider the evolutionary model for magnetoelasticity with vanishing viscosity/damping, which is a nonlinear dispersive system. The global regularity and scattering of the evolutionary model for magnetoelasticity under small size of initial data is proved. Our proof relies on the idea of vector-field method due to the quasilinearity and the presence of convective term. A key observation is that we construct a suitable energy functional including the mass quantity, which enable us to provide a good decay estimates for Schrödinger flow. In particular, we establish the asymptotic behavior in both mass and energy spaces for Schrödinger map, not only for gauged equation.

math.AP↗

arXiv2023

On the hydrodynamics of hyperbolic liquid crystal elastomers

Liquid crystal elastomers are special cross-linked polymer materials combining the large elastic deformability of elastomers with the orientational orders of liquid crystals. This model exhibits markedly different phenomena than the liquid crystal model due to the strong coupling between mechanical elastic deformation and orientation vector. Our results are threefold. (i) First we derive the hydrodynamics of liquid crystal elastomers with inertial effect by energetic variational approach inspired by Chun Liu [33]. (ii) Then we study the hyperbolic liquid crystal elastomers from mathematical point, which is a fully quasilinear hyperbolic system. The local well-posedness for large data is proved by zero-viscosity limit method. (iii) Finally, we show the global regularity for small and smooth initial data near the constant equilibrium in three dimensions, which is achieved by carefully investigating the structure of second-order material derivatives and the cancellations in the system. This is the first global result on hyperbolic liquid crystal elastomers.

math.AP↗

arXiv2018

Global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism

We prove the global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism on the stress tensor in $\mathbb{R}^d$ for the small initial data. Our proof is based on the observation that the behaviors of Green's matrix to the system of $\big(u,(-Δ)^{-\frac12}\mathbb{P}\nabla\cdotτ\big)$ as well as the effects of $τ$ change from the low frequencies to the high frequencies and the construction of the appropriate energies in different frequencies.

math.AP↗