arXiv subjects

Liening Qiao

Publications and source records attributed to Liening Qiao.

4 recordsLinked to original sources

arXiv2026

Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping

We study the Cauchy problem for the three-dimensional isentropic compressible ideal (inviscid and non-resistive) magnetohydrodynamic equations with velocity damping on the periodic torus $\mathbb{T}^3$. The system admits a steady equilibrium consisting of a constant density $\barρ$ and a uniform background magnetic field $ω\in\mathbb{R}^3$. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of $(\barρ,\mathbf{0},ω)$ in the Sobolev space $H^N(\mathbb{T}^3)$ with $N\geq 6r+4$, and if $ω$ satisfies a Diophantine condition, then the system admits a unique global smooth solution. Moreover, the perturbations decay algebraically in time. To the best of our knowledge, this is the first global well-posedness result for the multi-dimensional isentropic compressible ideal MHD system. The proof reveals a hidden dissipation mechanism: although neither the density equation nor the magnetic field equation contains explicit diffusion or damping, the coupling between the velocity and the magnetic field through the background field $ω$, combined with a Diophantine--Poincaré inequality, generates effective dissipation for both the density perturbation and the magnetic field perturbation, which together with the velocity damping yields global regularity and time decay.

math.AP↗

arXiv2026

Stabilization by a background magnetic field: global well-posedness of the full compressible viscous non-resistive MHD system without heat-conductivity

We consider the three-dimensional full compressible magnetohydrodynamic(MHD) system on the periodic torus $\mathbb T^3$ in the regime where the only dissipative mechanism acting on the system is the viscosity of the fluid: the magnetic field is non-resistive and the flow is non-heat-conducting. We prove that this system admits a unique global smooth solution, together with explicit algebraic decay rates, provided that the perturbation $(\mathbf u_0,\,P_0-\bar P,\,\mathbf H_0-\mathbf n)$ of the equilibrium state $(\mathbf 0,\bar P,\mathbf n)$ is sufficiently small in a high-order Sobolev space and the background magnetic field $\mathbf n\in\mathbb R^3$ satisfies a Diophantine condition. No smallness whatsoever is imposed on the initial density: it is only required to be bounded away from vacuum and from infinity, and may exhibit arbitrarily large variations. The proof uncovers a hidden dissipation mechanism. Although neither the density, nor the pressure, nor the magnetic field is endowed with any diffusion or damping of its own, the coupling of these quantities with the velocity through the background field $\mathbf n$, combined with a Poincaré-type inequality of Diophantine origin, generates effective dissipation for both the pressure and the magnetic field perturbations. The large variations of the density are handled by a two-tier energy argument, in which weighted time-decay estimates for the intermediate-order energy compensate exactly for the linear-in-time growth of the highest-order norm of the density.

math.AP↗

arXiv2025

Global well-posedness of the inviscid resistive isentropic compressible MHD system

Due to the absence of dissipation mechanism to the inviscid compressible systems, it is a challenging problem to prove their global solvability. In this paper, we are concerned with the initial-boundary value problem to the inviscid and resistive isentropic compressible magnetohydrodynamic (MHD) system on three dimensional torus $\mathbb T^3$. Global well-posedness and large time behavior of solutions are established in the first time for the isentropic setting, under the condition that the initial data $(ρ_0, u_0, H_0)$ is a small perturbation around the constant state $(1, 0, w)$, with $w$ satisfying the Diophantine condition. The main observation of this paper is that the spatial derivatives of the density along directions perpendicular to $w$ are dissipated. Such dissipation mechanism is generated from the interaction between the velocity field and the background magnetic field. This verifies the weak stabilizing effects of the magnetic filed on the dynamics in the scenario of inviscid isentropic flows. Due to different dissipation mechanisms for the density, velocity, and magnetic field, three ties of dissipative energies are designed, that is, high order Sobolev norms of the perturbed magnetic field, intermediate order Sobolev norms of the perturbed density, and low order Sobolev norms of the velocity field.

math.AP↗

arXiv2020

The unique global solvability of the nonhomogeneous incompressible asymmetric fluids with vacuum

The present paper deals with the nonhomogeneous incompressible asymmetric fluids equations in dimension $d= 2,3$. The aim is to prove the unique global solvability of the system with only bounded nonnegative initial density and $H^{1}$ initial velocities. We first construct the global existence of the solution with large data in 2-D. Next, we establish the existence of local in time solution for arbitrary large data and global in time for some smallness conditions in 3-D. Finally, the uniqueness of the solution is proved under quite soft assumptions about its regularity through a Lagrangian approach. In particular, the initial vacuum is allowed.

math.AP↗