arXiv subjects

Binqiang Xie

Publications and source records attributed to Binqiang Xie.

8 recordsLinked to original sources

arXiv2025

Ill-posedness of incompressible Kelvin-Helmholtz problem with transverse magnetic field

In this paper, we prove the linear and nonlinear ill-posedness of the well-known Kelvin-Helmholtz problem of the incompressible ideal magnetohydrodynamics (MHD) equations with transverse magnetic field. Our proof rigorously verifies that "the development of the Kelvin-Helmholtz instability, in the direction of the streaming, is uninfluenced by the presence of the magnetic field in the transverse direction" which was proposed by S. Chandrasekhar' book named by Hydrodynamic and Hydromagnetic stability.

math.AP↗

arXiv2025

The stability of current vortex sheets with transverse magnetic field

Compared to the results in \cite{Shivamoggi}, using the normal mode method, we have rigorously confirmed that a transverse magnetic field reduces the stability of the system. Specifically, a larger velocity is required for stability in the presence of a magnetic field than in its absence. More precisely, when the magnitude of the magneto-acoustic Mach number $M_{B}:=\frac{\dot{v}_1^{+}}{\bar{C}_{B}}>\sqrt{2}$, we proved the well-posedness of the current vortex sheet problem for compressible MHD flows with a transverse magnetic field.

math.AP↗

arXiv2024

Effect of weak elasticity on Kelvin-Helmholtz instability

In this paper, we present an analysis of the Kelvin-Helmholtz instability in two-dimensional ideal compressible elastic flows, providing a rigorous confirmation that weak elasticity has a destabilizing effect on the Kelvin-Helmholtz instability. There are two critical velocities, $U_{\text{low}}$ and $U_{\text{upp}}$, where $U_{\text{low}}$ and $U_{\text{upp}}$ represent the lower and upper critical velocities, respectively. We demonstrate that when the magnitude of the rectilinear solutions satisfies $U_{\text{low}}+cε_{0}\le |\dot{v}^{+}_{1}| \le U_{\text{upp}}-cε_{0}$, the linear and nonlinear ill-posedness of the piecewise smooth solutions of the Kelvin-Helmholtz problem for two-dimensional ideal compressible elastic fluids is established uniformly, where $c$ is the sound speed and $ε_{0}$ is some small enough positive constant.

math.AP↗

arXiv2024

Ill-posedness of the Kelvin-Helmholtz problem for compressible Euler fluids

In this paper, when the magnitude of the Mach number is strictly between some fixed small enough constant and $\sqrt{2}$, we can prove the linear and nonlinear ill-posedness of the Kelvin-Helmholtz problem for compressible ideal fluids. To our best knowledge, this is the first reslult that proves the nonlinear ill-posedness to the Kelvin-Helmholtz problem for the compressible Euler fluids.

math.AP↗

arXiv2017

On the dynamics of Navier-Stokes-Fourier equations

In this paper we are concerned with a non-isothermal compressible Navier-Stokes-Fourier model with density dependent viscosity that vanish on the vacuum. We prove the global existence of weak solutions with large data in the three-dimensional torus $Ω=T^{3}$. The main point is that the pressure is given by $P=Rρθ$ without additional cold pressure assumption.

math.AP↗

arXiv2016

On a nonisothermal ideal gas Navier-Stokes-Fourier equations

In this paper we are concerned with a non-isothermal compressible Navier-Stokes-Fourier model with density dependent viscosity that vanish on the vacuum. We prove sequential stability of variational weak solutions in periodic domain Ω= T3. The main point is that the pressure is given by P = Rρθ.

math.AP↗

arXiv2016

Global existence of weak solutions for generalized quantum MHD equation

We prove the existence of a weak solution to a generalized quantum MHD equation in a 2-dimensional periodic box for large initial data. The existence of a global weak solution is established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent γ>1.

math.AP↗

arXiv2016

On a compressible non-isothermal model for nematic liquid crystals

We prove the existence of a weak solution to a non-isothermal compressible model for nematic liquid crystals. An initial-boundary value problem is studied in a bounded domain with large data. The existence of a global weak solution is established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent γ>3/2.

math.AP↗