arXiv subjects

Mingwen Fei

Publications and source records attributed to Mingwen Fei.

At least 19 recordsLinked to original sources

arXiv2026

Stabilizing effect of a background magnetic field on the 2D damped wave-type MHD equations

The stabilizing effect of a background magnetic field on electrically conducting fluids has been rigorously established for the standard MHD equations. This paper extends this theory to the more physically accurate damped wave-type MHD equations, where the induction equation is hyperbolic-parabolic and the velocity field has only vertical damping with no dissipation. These two features make the stability analysis harder than in the standard MHD setting. To overcome these difficulties, we design an energy functional exploiting the anisotropic structure, and discover a remarkable cancellation between the two most dangerous nonlinear terms by exploiting the full algebraic structure of the coupled system. As a consequence, we prove that any small perturbation near the background magnetic field is globally stable and establish optimal decay rates consistent with the 2D heat equation. To the best of our knowledge, this is the first rigorous stability result for the damped wave-type MHD equations near a background magnetic field.

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arXiv2026

Higher Order Convergence for the Sharp Interface Limit of 3D Navier--Stokes/Allen--Cahn Systems

We show convergence of solutions to a Navier--Stokes/Allen--Cahn system as the interfacial thickness $\varepsilon>0$ tends to zero for well-prepared initial data as long as the limit system possesses a sufficiently smooth solution. The limit system consists of a two-phase Navier--Stokes system separated by a sharp interface in the presence of surface tension coupled to a convective mean curvature flow equation. In comparison to previous results we obtain improved convergence estimates for higher-order norms. These enable us to prove convergence in the case of three space dimensions and non-constant viscosity, which was unknown before. The convergence results relies crucially on uniform higher-order estimates for the associated linearized Navier--Stokes/Allen--Cahn system in suitably weighted $L^2$-Sobolev spaces. Here a novel problem-adapted weight proportional to the sum of $\varepsilon$ and the distance to the sharp interface of the limit, which gives improved and sharp estimates, is an important new ingredient. This approach can be potentially adapted to other sharp interface limits as well.

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arXiv2026

Sharp decay characterization for the incompressible Oldroyd-B model in critical $L^p$ spaces

This paper establishes a sharp characterization of temporal decay rates for the incompressible Oldroyd-B model in a critical $L^p$ framework, covering the physically relevant and mathematically delicate case where both the fluid viscosity and the stress tensor damping are absent. We prove that an $L^2$-type condition on the low-frequencies part of the initial data $(u_0, \tau_0)$ is almost both necessary and sufficient for obtaining optimal upper and lower bounds on the temporal decay of solutions in critical Besov spaces. A key contribution is a new decomposition of the stress tensor into its incompressible and compressible parts, combined with the introduction of an effective tensor to handle the loss of regularity in the high-frequencies velocity field. This is the first result to reveal such precise two-sided asymptotics for the incompressible Oldroyd-B model without viscosity or damping.

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arXiv2026

Global weak solutions to a diffuse-interface model for quasi-incompressible two-phase flows with unmatched densities and singular potential

We study a thermodynamically consistent diffuse-interface model that describes the motion of two macroscopically immiscible, incompressible, and viscous Newtonian fluids with unmatched densities. This model is compatible with continuum mixture theory. It adopts a mass-averaged (barycentric) velocity so that the two-phase flow is quasi-incompressible: the velocity is no longer divergence-free, and the pressure enters the equation of the chemical potential. For the initial-boundary value problem in $\mathbb{T}^3$ with a class of physically relevant singular free energy densities, we prove the existence of global-in-time weak solutions. The proof relies on a suitable reduction of the original system to a Korteweg-type fluid model combined with a two-layer approximation, together with delicate estimates for the mass density and the phase-field variable inspired by the celebrated Bresch-Desjardins entropy. A key observation is that capillarity at the free interface provides a damping effect on the density evolution. For the limiting procedure, we derive delicate tail estimates to exclude possible concentrations of the singular potential, since no integrability of the pressure is available \textit{a priori}. This work appears to be the first existence result for the Navier-Stokes/Cahn-Hilliard type system with unmatched densities and mass-averaged velocity without spatial regularization.

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arXiv2025

Relaxation limit and asymptotic stability for the Euler-Navier-Stokes equations

The Euler-Navier-Stokes (E-NS) system arises as a macroscopic description of kinetic-fluid interactions, derived from the local-Maxwellian closure of the Vlasov-Fokker-Planck-Navier-Stokes flow. In this paper, we investigate the singular limit of the system in $\mathbb{R}^d$ ($d\ge2$) when the relaxation parameter $\varepsilon>0$ tends to zero. In contrast to the Euler system with velocity damping, the E-NS model features only a weaker relaxation of the relative velocity, which makes it challenging to analyze its dynamics as $\varepsilon\rightarrow 0$. We develop an energy argument to show global-in-time error estimates between the E-NS system and its limit system, the so-called Kramers-Smoluchowski-Navier-Stokes (KS-NS) system. These error estimates enable us to prove the global existence and uniform-in-$\varepsilon$ regularity of the strong solution to the E-NS system in a hybrid critical Besov space with a sharp frequency threshold of order $\mathcal{O}(\varepsilon^{-1})$ separating the low- and high-frequency regimes. Moreover, the large-time asymptotic stability of the global solution to the E-NS system is established. More precisely, we derive the optimal decay rates of the solution uniformly in $\varepsilon$, and the enhanced decay rates for the difference between the densities of the E-NS system and the KS-NS system.

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arXiv2025

Weak solutions and incompressible limit of a quasi-incompressible Navier--Stokes/Cahn--Hilliard model for viscous two-phase flows

We study a quasi-incompressible Navier--Stokes/Cahn--Hilliard coupled system which describes the motion of two macroscopically immiscible incompressible viscous fluids with partial mixing in a small interfacial region and long-range interactions. The case of unmatched densities with mass-averaged velocity is considered so that the velocity field is no longer divergence-free, and the pressure enters the equation of the chemical potential. We first prove the existence of global weak solutions to the model in a three-dimensional periodic domain, for which the implicit time discretization together with a fixed-point argument to the approximate system is employed. In particular, we obtain a new regularity estimate of the order parameter by exploiting the partial damping effect of the capillary force. Then utilizing the relative entropy method, we establish the incompressible limit -- the quasi-incompressible two-phase model converges to model H as the density difference tends to zero. Crucial to the passage of the incompressible limit, due to the lack of regularity of the pressure, are some non-standard uniform-in-density difference controls of the pressure, which are derived from the structure of the momentum equations and the improved regularity of the order parameter.

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arXiv2024

Local-in-time existence of strong solutions to a quasi-incompressible Cahn--Hilliard--Navier--Stokes system

We analyze a quasi-incompressible Cahn--Hilliard--Navier--Stokes system (qCHNS) for two-phase flows with unmatched densities. The order parameter is the volume fraction difference of the two fluids, while mass-averaged velocity is adopted. This leads to a quasi-incompressible model where the pressure also enters the equation of the chemical potential. We establish local existence and uniqueness of strong solutions by the Banach fixed point theorem and the maximal regularity theory.

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arXiv2024

Prandtl-Batchelor flows with a point vortex on a disk

We study steady two-dimensional incompressible Navier-Stokes flow in the unit disk in a high Reynolds number regime, driven by a localized central forcing and a nearly rigid rotating boundary. The forcing models a compact source of circulation that sustains a vortex core in a viscous flow. For sufficiently small viscosity and boundary perturbation, we construct a very weak steady solution on the whole disk and identify its inviscid limit: a point vortex embedded in a constant vorticity background and joined to the boundary motion by a thin boundary layer. The background vorticity is selected by the Batchelor-Wood formula. The singular core introduces non-coercive interactions in the linearized problem that are absent for regular Prandtl-Batchelor flows. To overcome this difficulty, we map the punctured disk to a semi-infinite cylinder by a logarithmic radial transformation and develop a Fourier-mode analysis that isolates the delicate first Fourier mode, yielding uniform stability and vorticity estimates in the presence of the point-vortex singularity.

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arXiv2024

Vanishing viscosity limit of the Navier-Stokes equations in a horizontally periodic strip

In this paper, we establish vanishing viscosity limit of the 2D Navier-Stokes equations in a horizontally periodic strip. On the vertical direction, the horizontal component of the velocity is subjected to two different types of boundary conditions: at the lower boundary, we give the degenerate zero boundary condition, while at the upper boundary, a small smooth perturbation of non-zero constant is prescribed. Due to this different boundary condition setting, the boundary layer effects are different near the lower and upper boundaries, which result in different thickness of boundary layer and different leading order boundary layer equations. We will construct an approximate solution to this 2D Navier-Stokes equations by using higher order asymptotic approximation and show the validity of the boundary layer expansion. The leading order of the Euler solution is the Couette flow $(Ay,0)$ for some suitable constant $A$, which is determined by using the principle of the Prandtl-Batchelor theory.

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arXiv2023

Prandtl-Batchelor flows on an annulus

For steady two-dimensional Navier-Stokes flows with a single eddy (i.e. nested closed streamlines) in a simply connected domain, Prandtl (1905) and Batchelor (1956) found that in the inviscid limit, the vorticity is constant inside the eddy. In this paper, we consider the generalized Prandtl-Batchelor theory for the forced steady Navier-Stokes equations on an annulus. First, we observe that in the limit of infinite Reynolds number, if forced steady Navier-Stokes solutions has nested closed streamlines on an annulus, then the inviscid limit is a rotating shear flow uniquely determined by the external force and boundary conditions. We call solutions of steady Navier-Stokes equations with the above property Prandtl-Batchelor flows. Then, by constructing higher order approximate solutions of the forced steady Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on an annulus with the wall velocities slightly different from the rigid-rotations along the same direction.

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arXiv2023

Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System in the Case of a Vanishing Mobility

We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility $m_\varepsilon=\sqrt{\varepsilon}$, where the small parameter $\varepsilon>0$ related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier-Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen-Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable $\varepsilon$-scaled and coupled model problem. Moreover, we apply the novel idea of introducing $\varepsilon$-dependent coordinates.

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arXiv2023

Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System with Different Viscosities

We discuss the sharp interface limit of a coupled Navier-Stokes/Allen-Cahn system in a two dimensional, bounded and smooth domain, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero rigorously. We prove convergence of the solutions of the Navier-Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature flow with an additional convection term coupled to a two-phase Navier-Stokes system with surface tension. This is done by constructing an approximate solution from the limiting system via matched asymptotic expansions together with a novel Ansatz for the highest order term, and then estimating its difference with the real solution with the aid of a refined spectral estimate of the linearized Allen-Cahn operator near the approximate solution.

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arXiv2022

Ill-posedness of the hyperbolic Keller-Segel model in Besov spaces

In this paper, we give a new construction of $u_0\in B^σ_{p,\infty}$ such that the corresponding solution to the hyperbolic Keller-Segel model starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^σ_{p,\infty}(\R^d)$ with $d\geq1$ and $1\leq p\leq\infty$, which implies the ill-posedness for this equation in $B^σ_{p,\infty}$. Our result generalizes the recent work in \cite{Zhang01} (J. Differ. Equ. 334 (2022)) where the case $d=1$ and $p=2$ was considered.

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arXiv2021

Prandtl-Batchelor flows on a disk

For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. In this paper, by constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying Batchelor-Wood formula.

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arXiv2021

Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem

In this paper, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation, which takes the form: $$\partial_t A=ΔA-\varepsilon^{-2}( A A^{\mathrm{T}}A- A)~~\text{with}~~A:Ω\subset\mathbb{R}^m\to\mathbb{R}^{n\times n}.$$ We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in $O^+(n)$ and $O^-(n)$ (represent the sets of $n\times n$ orthogonal matrices with determinant $+1$ and $-1$ respectively); on the interface, the phase matrices in two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller-Rubinstein-Sternberg's (conjecture) problem in this $O(n)$ setting. Our proof relies on two key ingredients. First, in order to construct the approximate solution by matched asymptotic expansions as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible "degenerations" or higher dimensional kernels for the linearized operators due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which is accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

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arXiv2020

Phase-field approximation of the Willmore flow

We investigate the phase-field approximation of the Willmore flow. This is a fourth-order diffusion equation with a parameter $ε>0$ that is proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, as $ε$ trends to zero the level-set of solution will converge to motion by Willmore flow before the singularity of the later occurs. This is done by constructing an approximate solution from the limiting flow via matched asymptotic expansions, and then estimating its difference with the real solution. The crucial step and also the major contribution of this work is to show a spectrum condition of the linearized operator at the optimal profile. This is a fourth-order operator written as the sum of the squared Allen-Cahn operator and a singular linear perturbation.

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arXiv2017

Sharp interface limit of a diffuse interface model for tumor-growth

We consider the asymptotic limit of a diffuse interface model for tumor-growth when a parameter $\varepsilon$ proportional to the thickness of the diffuse interface goes to zero. An approximate solution which shows explicitly the behavior of the true solution for small $\varepsilon$ will be constructed by using matched expansion method. Based on the energy method, a spectral condition in particular, we establish a smallness estimate of the difference between the approximate solution and the true solution.

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arXiv2016

On the zero-viscosity limit of the Navier-Stokes equations in the half-space

We consider the zero viscosity limit of the incompressible Navier-Stokes equations with non-slip boundary condition in the half-space for the initial vorticity located away from the boundary. By using the vorticity formulation and Cauchy-Kowaleskaya theorem, Maekawa proved the local in time convergence of the Navier-Stokes equations in the half- plane to the Euler equations outside a boundary layer and to the Prandtl equations in the boundary layer. In this paper, we develop the direct energy method to generalize Maekawa's result to the half-space.

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