arXiv subjects

Youyi Zhao

Publications and source records attributed to Youyi Zhao.

9 recordsLinked to original sources

arXiv2026

Global well-posedness for 3D incompressible magneto-micropolar fluids without resistivity and spin viscosity in strip domains

The global existence of classical solutions to the 3D compressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain was recently established by Feng, Hong, and Zhu [Sci. China Math., 2024]. While Lin and Xiang [Sci. China Math., 2020] established global well-posedness for the 2D incompressible counterpart, the global well-posedness for the 3D incompressible case remains open. The analysis is rendered difficult by three major obstacles which are further compounded in the 3D case: the degeneracy induced by the lack of magnetic diffusion and spin viscosity; the coupling between micro-rotation and velocity fields characterized by a non-dissipative anti-symmetric structure; and the interaction between the velocity, magnetic field, and pressure, where the pressure acts as a non-state variable. In this paper, by adapting the two-layer energy method of Guo and Tice [Arch. Ration. Mech. Anal., 2013] and the techniques employed in Feng et al., together with refined trace estimates, we overcome these difficulties and establish the global well-posedness of classical solutions to the 3D incompressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain. Moreover, we demonstrate the algebraic time-decay of solutions toward the equilibrium.

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arXiv2026

Global well-posedness for 3D compressible and incompressible micropolar fluids without angular viscosity in strip domains

This paper investigates an initial-boundary value problem for three-dimensional (3D) micropolar fluids in a strip domain, including both the compressible and the (homogeneous and inhomogeneous) incompressible cases in the absence of angular viscosity. The analysis is rendered difficult by two major obstacles: the degeneracy induced by vanishing angular viscosity, and the strong coupling between micro-rotation and velocity fields characterized by a non-dissipative anti-symmetric structure. Moreover, the presence of physical boundaries in the strip domain further compounds these obstacles. While the global well-posedness of the 2D incompressible Cauchy problem has been established in the literature, no results are available for the 3D system and the initial-boundary value problem in both two and three dimensions, particularly in the compressible case. By exploiting the intrinsic structure of the system and establishing delicate energy estimates, we overcome these difficulties and prove the global well-posedness of strong solutions near equilibrium in a strip domain.

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arXiv2025

Nonlinear Stability of the Rayleigh-Taylor Problem in Quantum Navier-Stokes Equations

It is well-known that the Rayleigh--Taylor (abbr. RT) instability can be completely inhibited by the quantum effect stabilization in proper circumstances leading to a cutoff wavelength in the \emph{linear} motion equations. Motivated by the linear theory, we further investigate the {stability} for the \emph{nonlinear} RT problem of quantum Navier--Stokes equations in a slab with Navier boundary condition, and rigorously prove the inhibition of RT instability by the quantum effect under a proper setting. More precisely, if the RT density profile $\bar\rho$ satisfies an additional stabilizing condition, then there is a threshold ${\varepsilon_{{c}}}$ of the scaled Planck constant, such that if the scaled Planck constant is bigger than ${\varepsilon_{{c}}}$, the small perturbation solutions around an RT equilibrium state are algebraically stable in time. The mathematical proof is realized by a complicated multi-layer energy method with anisotropic norms of spacial derivatives.

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arXiv2025

Global Solvability for the Compressible Hookean Viscoelastic Fluids with a Free Boundary in Some Classes of Large Data

Recently Jiang-Jiang established a global (in time) existence result for unique strong solutions of the two-dimensional (2D) free-boundary problem of an incompressible Hookean viscoelastic fluid, the rest state of which is defined in a slab, in some classes of large data [28]. In particular, Jiang-Jiang's mathematical result shows that, if the initial free boundary is flat, the way the elastic deformation under the large elasticity coefficient $\kappa$ acts on the free boundary prevents the natural tendency of the fluid to form singularities, even when the initial velocity is properly large. However it is not clear whether their result can be extended to the corresponding 3D case. In this paper, we further find a similar result in the 3D stratified (immiscible) compressible Hookean viscoelastic fluids in an infinite slab with two restrictive conditions: that the elasticity coefficients of two fluids are equal, and that the initial density functions satisfy the asymptotic stability condition in Lagrangian coordinates. These two restrictive conditions in the compressible case contribute us to avoid the essential obstacles that would be faced in the extension of Jiang-Jiang's result from two dimensions to our 3D case. In addition, we can further obtain a new result regarding the vanishing phenomena of the nonlinear interactions of solutions with the fixed initial velocity and the initial zero perturbation deformation. Such a new result roughly presents that the solutions of the problem considered by us can be approximated by the ones of a linear problem for sufficiently large $\kappa$.

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arXiv2024

Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations

We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by $m$, is large compared to the perturbations. It was proved by Jiang--Jiang that the highest-order derivatives of the velocity increase with $m$, and the convergence rate of the nonlinear system towards a linearized problem is of $m^{-1/2}$ in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy estimate. This strategy prevents the appearance of terms that grow with $m$, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to $m$ does not appear. Additionally, we use the vorticity estimate to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or $m$ approaches infinity. Notably, our analysis reveals that the convergence rate in $m$ is faster compared to the finding of Jiang--Jiang. Finally, a key contribution of our work is the identification of an integrable time-decay of the lower dissipation, which can replace the time-decay of lower energy in closing the highest-order energy estimate. This finding significantly relaxes the regularity requirements for the initial perturbations.

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arXiv2022

On Inhibition of Rayleigh--Taylor Instability by a Horizontal Magnetic Field in Non-resistive MHD Fluids: the Viscous Case

It is still open whether the phenomenon of inhibition of Rayleigh--Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive \emph{viscous} magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it was roughly proved in the linearized case by Wang in \cite{WYC}. In this paper, we prove such inhibition phenomenon by the (nonlinear) inhomogeneous, incompressible, \emph{viscous case} with \emph{Navier (slip) boundary condition}. More precisely, we show that there is a critical number of field strength $m_{\mm{C}}$, such that if the strength $|m|$ of a horizontal magnetic field is bigger than $m_{\mm{C}}$, then the small perturbation solution around the magnetic RT equilibrium state is {algebraically} stable in time. In addition, we also provide a nonlinear instability result for the case $|m|\in[0, m_{\mm{C}})$. The instability result presents that a horizontal magnetic field can not inhibit the RT instability, if it's strength is too small.

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arXiv2021

Global Solutions of Three-dimensional Inviscid MHD Fluids with Velocity Damping in Horizontally Periodic Domains

The \emph{two-dimensional} (2D) existence result of global(-in-time) solutions for the motion equations of incompressible, inviscid, non-resistive magnetohydrodynamic (MHD) fluids with velocity damping had been established in [Wu--Wu--Xu, SIAM J. Math. Anal. 47 (2013), 2630--2656]. This paper further studies the existence of global solutions for the \emph{three-dimensional} (a dimension of real world) initial-boundary value problem in a horizontally periodic domain with finite height. Motivated by the multi-layers energy method introduced in [Guo--Tice, Arch. Ration. Mech. Anal. 207 (2013), 459--531], we develop a new type of two-layer energy structure to overcome the difficulty arising from three-dimensional nonlinear terms in the MHD equations, and thus prove the initial-boundary value problem admits a unique global solution. Moreover the solution has the exponential decay-in-time around some rest state. Our two-layer energy structure enjoys two features: (1) the lower-order energy (functional) can not be controlled by the higher-order energy. (2) under the \emph{a priori} smallness assumption of lower-order energy, we first close the higher-order energy estimates, and then further close the lower-energy estimates in turn.

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arXiv2021

On Inhibition of Rayleigh--Taylor Instability by Horizontal Magnetic Field in an Inviscid MHD Fluid with Velocity Damping

It is still an open problem whether the inhibition phenomenon of Rayleigh--Taylor (RT) instability by horizontal magnetic field can be mathematically proved in a non-resistive magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it had been roughly verified by a 2D linearized motion equations in 2012 \cite{WYC}. In this paper, we find that this inhibition phenomenon can be rigorously verified in the inhomogeneous, incompressible, inviscid case with velocity damping. More precisely, there exists a critical number $m_{\rm{C}}$ such that if the strength $|m|$ of horizontal magnetic field is bigger than $m_{\rm{C}}$, then the small perturbation solution around the magnetic RT equilibrium state is exponentially stable in time. Our result is also the first mathematical one based on the nonlinear motion equations for the proof of inhibition of flow instabilities by a horizontal magnetic field in a horizontal slab domain. In addition, we also provide a nonlinear instability result for the case $|m|\in [0,m_{\rm{C}})$. Our instability result presents that horizontal magnetic field can not inhibit the RT instability, if it's strength is to small.

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arXiv2019

On Classical Solutions of Rayleigh--Taylor Instability in Inhomogeneous Incompressible Viscous Fluids in Bounded Domains

We study the existence of unstable classical solutions of the Rayleigh--Taylor instability problem (abbr. RT problem) of an inhomogeneous incompressible viscous fluid in a bounded domain. We find that, by using an existence theory of (steady) Stokes problem and an iterative technique, the initial data of classical solutions of the linearized RT problem can be modified to new initial data, which can generate local-in-time classical solutions of the RT problem, and are close to the original initial data. Thus, we can use a classical bootstrap instability method to further obtain classical solutions of (nonlinear) RT instability based on the ones of linear RT instability.

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