arXiv subjects

Fuyi Xu

Publications and source records attributed to Fuyi Xu.

9 recordsLinked to original sources

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 Poincar\'e-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.

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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 $\bar{\rho}$ and a uniform background magnetic field $\omega\in\mathbb{R}^3$. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of $(\bar{\rho},\mathbf{0},\omega)$ in the Sobolev space $H^N(\mathbb{T}^3)$ with $N\geq 6r+4$, and if $\omega$ 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 $\omega$, combined with a Diophantine--Poincar\'{e} 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.

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arXiv2025

On the uniqueness of strong solution to the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system

This paper is mainly concerned with an initial-boundary value problem of the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system with the Landau potential in a two and three dimensions. The existence of strong solutions with bounded and strictly positive density for this system was constructed by Giorgini and Temam \cite{GT}. However, whether uniqueness holds has remained an open question. The present work solves this question and we prove the uniqueness of strong solution. Our method mainly relies on some extra time weighted estimates and the Lagrangian approach.

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arXiv2025

Magnetic Stabilization of Compressible Flows: Global Existence in 3D Inviscid Non-Isentropic MHD Equations

Solutions to the compressible Euler equations in all dimensions have been shown to develop finite-time singularities from smooth initial data such as shocks and cusps. There is an extraordinary list of results on this subject. When the inviscid compressible flow is coupled with the magnetic field in the 3D inviscid non-isentropic compressible magnetohydrodynamic (MHD) equations in $\mathbb{T}^3$, this paper rules out finite-time blowup and establishes the global existence of smooth and stable solutions near a suitable background magnetic field. This result rigorously confirms the stabilizing phenomenon observed in physical experiments involving electrically conducting fluids.

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arXiv2022

The maximal regularity and its application to a multi-dimensional non-conservative viscous compressible two-fluid model with capillarity effects in $L^{ p}$-type framework

The present paper is the continuation of work \cite{XC}, devoted to extending it to a critical functional framework which is not related to the energy space. Employing the special dissipative structure of the non-conservative viscous compressible two-fluid model with capillarity effects, we first exploit the maximal regularity estimates for the corresponding linearized system in all frequencies which behaves like the heat equation. Then we construct the global well-posedness for the multi-dimensional model when the initial data are close to a stable equilibrium state in the sense of suitable $L^{ p}$-type Besov norms. As a consequence, this allows us to work in the framework of Besov space with negative regularity indices and this fact is particularly important when the initial data are large highly oscillating in physical dimensions $N= 2, 3$. Furthermore, based on a refined time weighted inequalities in the Fourier spaces, we also establish optimal time decay rates for the constructed global solutions under a mild additional decay assumption involving only the low frequencies of the initial data.

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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.

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arXiv2020

On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.

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arXiv2019

The unique global solvability and optimal time decay rates for a multi-dimensional compressible generic two-fluid model with capillarity effects

The present paper deals with the Cauchy problem of a compressible generic two-fluid model with capillarity effects in any dimension $N\geq2$. We first study the unique global solvability of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. Due to the presence of the capillary terms, we exploit the parabolic properties of the linearized system for all frequencies which enables us to apply contraction mapping principle to show the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the optimal time decay rates for the constructed global solutions.

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arXiv2018

Existence, uniqueness and optimal decay rates for the 3D compressible Hall-magnetohydrodynamic system

We are concerned with the study of the Cauchy problem to the 3D compressible Hall-magnetohydrodynamic system. We first establish the unique global solvability of strong solutions to the system when the initial data are close to a stable equilibrium state in critical Besov spaces. Furthermore, under a suitable additional condition involving only the low frequencies of the data and in $L^{2}$-critical regularity framework, we exhibit the optimal time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a mixed parabolic-hyperbolic system, and on a refined time-weighted energy functional.

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