arXiv subjects

Shunhang Zhang

Publications and source records attributed to Shunhang Zhang.

3 recordsLinked to original sources

arXiv2025

Global strong solutions to the $3$D rotating compressible Navier--Stokes--Korteweg system for large data in the critical $\widehat{L^p}$ framework

Let us consider the $3$D compressible Navier--Stokes--Korteweg system in the rotational framework. Although there is a wealth of literature on the weak solutions to this system, there seem to be no results on the strong solutions. In this paper, we show the unique existence of global solutions for {\it large} initial data in the critical Besov-type spaces based on the Fourier--Lebesgue spaces $\widehat{L^p}(\mathbb{R}^3)$ with $2 \leq p < 3$, provided that the rotation speed and the Mach number are sufficiently large and small, respectively. The key ingredient of the proof is to establish the Strichartz-type estimates due to the dispersion caused by the mixture of the rotation and acoustic waves in the Fourier--Lebesgue spaces, and focus on the better structure of dissipation from the Korteweg term and the nonlinear terms of the divergence form in the momentum formulation.

math.AP↗

arXiv2024

Global well-posedness and large-time behavior for a special $2\frac{1}{2}$D full compressible viscous non-resistive MHD system

In this paper, we consider the full compressible, viscous, non-resistive MHD system under the assumption that the fluids move on a plane while the magnetic field is oriented vertically. Within the framework of Besov spaces, by introducing several new unknown quantities and exploiting the intrinsic structure of the system, we prove the global well-posedness of strong solutions for initial data close to a constant equilibrium state. Furthermore, under some suitable additional conditions involving only the low-frequency part of the initial perturbation, we develop a Lyapunov-type energy argument, which yields the optimal time-decay rates of the global solution. To the best of our knowledge, our result is the first one on global solvability to the full compressible, viscous, non-resistive MHD system in multi-dimensional whole space.

math.AP↗

arXiv2023

Well-posedness for the incompressible Hall-MHD system with initial magnetic field belonging to $H^{\frac{3}{2}}(\mathbb{R}^3)$

In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system for initial data $(u_0,B_0)\in H^{\frac{1}{2}+σ}(\mathbb{R}^3)\times H^{\frac{3}{2}}(\mathbb{R}^3)$ with $σ\in (0,2)$. In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $σ$ to $0$ with the aid of the new formulation of the Hall-MHD system observed by Danchin and Tan (Commun Partial Differ Equ 46(1):31-65, 2021). Compared with the previous works, our local well-posedness results improve the regularity condition on the initial data. Moreover, we establish the global well-posedness for small initial data in $H^{\frac{1}{2}+σ}(\mathbb{R}^3)\times H^{\frac{3}{2}}(\mathbb{R}^3)$ with $σ\in (0,2)$, and get the optimal time-decay rates of solutions.

math.AP↗