arXiv subjects

Quansen Jiu

Publications and source records attributed to Quansen Jiu.

At least 19 recordsLinked to original sources

arXiv2026

Existence of Periodic Solutions to Steady Viscous Burgers Equation with a General Force

In this paper, we will construct periodic solutions to the viscous steady Burgers equation with an external force $f(x)$, based on the following formal expansion $u^\varepsilon(x)=u_0(x)+\varepsilon u_1(x)+\cdots+\varepsilon^n u_n(x)+\cdots$, where $\varepsilon \ge 0$ represents the viscosity and $u_0(x)$ is a solution to non-viscous steady Burgers equation with the external force $f(x)$. We will focus on the solutions which are uniformly bounded with respect to the viscosity. In our previous work, starting from $u_0=-(2+\cos x)$, the authors constructed the periodic solutions to the viscous steady Burgers equation with the external force $f=u_0(x)u_{0x}=-2\sin x-\sin x\cos x$. In this paper, we will extend the main result obtained in our previous work and construct the solutions starting from general $u_0$ and $f$ satisfying the non-viscous steady Burgers equation $u_0(x)u_{0x}=f$. It will be shown that there exists a $\varepsilon_0>0$, which depends on $n$, such that for any $0<\varepsilon<\varepsilon_0$, the viscous steady Burgers equation with the external force $f$ has a periodic solution $u^\varepsilon(x) \in C^2([0,2\pi])$, satisfying $|u^\varepsilon(x)-u_0(x)-\varepsilon u_1(x)-\cdots-\varepsilon^n u_n(x)| \leq C\varepsilon^{n+1}$, where $C>0$ is a constant which depends on $n$, but is independent of $\varepsilon$. Compared with Jauslin-Kreiss-Moser's result, we present a new approach to construct periodic solutions to the viscous steady Burgers with a general external force. The constructed solutions will tend to the ones of the non-viscous Burgers equation with a sharper convergence rate (up to higher order) when the viscosity vanishes.

math.AP↗

arXiv2026

Periodic Solutions to the Steady Viscous Burgers Equation: A Constructive Example

In this paper, we consider the periodic problem to steady viscous Burgers equation with an periodic and external force. Our main aim is to construct periodic solutions to this model which are uniformly bounded with respect to the viscosity via a direct Fourier series approach. Firstly, as an example, starting from a special solution of the non-viscous Burgers equation with a specific force, we solve the approximate solutions to the viscous Burgers equation in an explicit way. Secondly, we construct the solution to the periodic problem of the viscous Burgers equation with the external force by solving the initial value problem to a second ordinary differential equations, which is uniformly bounded with respect to the viscosity. Compared with Jauslin-Kreiss-Moser's result, our approach provides an explicit constructive procedure for the periodic solutions and yields a sharper convergence rate (up to higher order) when the viscosity vanishes.

math.AP↗

arXiv2026

Stability and instability of a one-dimensional MHD model

We consider a one-dimensional magnetohydrodynamics model introduced by Dai \textit{et al.}~(2023), in a parameter regime where, in the absence of a magnetic field, the system reduces to the De Gregorio model for the Euler equations. We analyze stability and instability near the first excited state on the torus, thus generalizing the recent results obtained by Guo and Jiu~(2025) for the De Gregorio model. Specifically, we establish global well-posedness of the linearized system, local well-posedness for the nonlinear system, and demonstrate both linear and nonlinear instability for a broad class of initial data in the weighted Sobolev space introduced by Lai \textit{et al.}~(2020). We identify the principal linearized operator, which is structurally equivalent to that of the De Gregorio model, as the primary mechanism of instability. Moreover, we prove global well-posedness and stability of both linear and nonlinear systems for initial data in a particular subspace of the aforementioned weighted Sobolev space.

math.AP↗

arXiv2026

Non-uniform Continuity for the MHD equations with only Magnetic Diffusion

In this paper, we prove the non-uniform continuity of the data-to-solution map for the incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion in Sobolev spaces $H^s(\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Our results are first studies on the non-uniform continuity of the data-to-solution map for the resistive MHD equations. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\mathbf{B_0} \in \mathbb{R}^d$, which reveal that the strong magnetic background fields may provide the stabilization effect but still preserve the analytical feature of non-uniform continuity of the data-to-solution map.

math.AP↗

arXiv2026

Global Regularity for Non-resistive or Non-viscous MHD System on the Torus

In this paper, we establish the global well-posedness of the incompressible magnetohydrodynamics (MHD) system on $n-$dimensional $(n\geq 2)$ periodic boxes with either no magnetic diffusivity (non-resistive case) or no fluid viscosity (non-viscous case) under assumption that initial magnetic fields are sufficiently close to the background magnetic field ${\bf e}_n=(0,\cdots,0,1)$. In Eulerian coordinates, we develop novel time-weighted energy estimates and commutator estimates involving Riesz transforms in negative Sobolev spaces to handle two distinct dissipation cases under different initial symmetry assumptions. The analysis becomes much more difficult and delicate in three- or higher-dimensional cases. In particular, for the three-dimensional and non-resistive case, compared with the regularity requirement proposed by Pan, Zhou and Zhu {\it [Arch. Ration. Mech. Anal. 2018]}, our result relaxes it from $H^{11}(\mathbb{T}^3)$ to $H^{\frac{9}{2}+}(\mathbb{T}^3)$. And we further establish precise decay rates and growth bounds for both $u(t)$ and $\partial_n(u(t),b(t))$ in Sobolev norms. For the three-dimensional and non-viscous case, we prove the first nonlinear stability result near the background field $\mathbf{e}_3 = (0,0,1)$. This sharply contrasts with the recent blow-up results on the 3D incompressible Euler equations by Elgindi {\it [Ann. Math. 2021]}, Chen-Hou {\it [Commun. Math. Phys. 2021]} and by Chen-Hou {\it [arXiv:2210.07191]}. Our results show that, under certain symmetry assumptions, magnetic fields near the background field provide enhanced dissipations and suppress potential blow-up mechanisms in non-viscous MHD system.

math.AP↗

arXiv2026

On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition

In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution $u \in L^p(\mathbb{R}^3)$, $3 \le p \le \frac{9}{2}$, with bounded density, one obtains $u \equiv0$. This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition $\int_{\mathbb{R}^3} |\nabla u|^2 \, dx < \infty$. If $\frac{9}{2} < p < 6$, Liouville-type theorem holds under the additional oscillation condition for momentum $\rho u \in \dot{B}^{\frac{3}{p} - \frac{3}{2}}_{\infty,\infty}(\mathbb{R}^3)$. For the marginal case $u \in L^6(\mathbb{R}^3)$, the oscillation condition can be replaced by $\rho u \in BMO^{-1}(\mathbb{R}^3)$. We also present results in Morrey-type spaces: $u \in \dot{M}^{s,6}(\mathbb{R}^3)$ and $\rho u \in \dot{M}_w^{q,3}(\mathbb{R}^3)$ for $2 \le s \le 6$ and $\frac{3}{2} < q \le 3$.

math.AP↗

arXiv2025

Finite time blow-up analysis for the generalized Proudman-Johnson model

In this paper, we study the generalized Proudman-Johnson equation posed on the torus. In the critical regime where the parameter $a$ is close to and slightly greater than 1, we establish finite time blow-up of smooth solutions to the inviscid case. Moreover, we show that the blow-up is asymptotically self-similar for a class of smooth initial data. In contrast, when the parameter $a$ lies slightly below 1, we prove the global in time existence for the same initial data. In addition, we demonstrate that inviscid Proudman-Johnson equation with H\"{o}lder continuous data also develops a self-similar blow-up. Finally, for the viscous case with $a>1$, we prove that smooth initial data can still lead to finite time blow-up.

math.AP↗

arXiv2025

Stability and Instability on the De Gregorio Modification of the Constantin-Lax-Majda model

The Constantin-Lax-Majda (CLM) model and the De Gregorio model which is a modification of the CLM model are well-known for their ability to emulate the behavior of the 3D Euler equations, particularly their potential to develop finite-time singularities. The stability properties of the De Gregorio model on the torus near the ground state $-\sin\theta$ have been well studied. However, the stability analysis near excited states $-\sin k\theta$ with $k\ge 2$ remains challenging. This paper focuses on analyzing the stability and instability of the De Gregorio model on torus around the first excited state $-\sin 2\theta$. The linear and nonlinear instability are established for a broad class of initial data, while nonlinear stability is proved for another large class of initial data in this paper. Our analysis reveals that solution behavior to the De Gregorio model near excited states demonstrates different stability patterns depending on initial conditions. One of new ingredients in our instability analysis involves deriving a second-order ordinary differential equation (ODE) governing the Fourier coefficients of solutions and examining the spectral properties of a positive definite quadratic form emerging from this ODE. The approach of this paper would be applicable to other related models and problems.

math.AP↗

arXiv2024

Local well-posedness of strong solutions to the 2D nonhomogeneous primitive equations with density-dependent viscosity

In this paper, we consider the initial-boundary value problem of the nonhomogeneous primitive equations with density-dependent viscosity. Local well-posedness of strong solutions is established for this system with a natural compatibility condition. The initial density does not need to be strictly positive and may contain vacuum. Meanwhile, we also give the corresponding blow-up criterion if the maximum existence interval with respect to the time is finite.

math.AP↗

arXiv2024

Global existence and uniqueness of strong solutions to the 2D nonhomogeneous primitive equations with density-dependent viscosity

This paper is concerned with an initial-boundary value problem of the two-dimensional inhomogeneous primitive equations with density-dependent viscosity. The global well-posedness of strong solutions is established, provided the initial horizontal velocity is suitably small, that is, $\|\nabla u_{0}\|_{L^{2}}\leq \eta_{0}$ for suitably small $\eta_{0}>0$. The initial data may contain vacuum. The proof is based on the local well-posedness and the blow-up criterion proved in \cite{0}, which states that if $T^{*}$ is the maximal existence time of the local strong solutions $(\rho,u,w,P)$ and $T^{*}<\infty$, then \begin{equation*} \sup_{0\leq t<T^{*}}(\left\|\nabla \rho(t)\right\|_{L^{\infty}}+\left\|\nabla^{2}\rho(t)\right\|_{L^{2}}+\left\|\nabla u(t)\right\|_{L^{2}})=\infty. \end{equation*} To complete the proof, it is required to make an estimate on a key term $\|\nabla u_{t}\|_{L_{t}^{1}L_{\Omega}^{2}}$. We prove that it is bounded and could be as small as desired under certain smallness conditions, by making use of the regularity result of hydrostatic Stokes equations and some careful time weighted estimates.

math.AP↗

arXiv2024

Asymptotic stability for $n$-dimensional isentropic compressible MHD equations without magnetic diffusion

Whether the global well-posedness of strong solutions of $n$-dimensional compressible isentropic magnetohydrodynamic (MHD for short) equations without magnetic diffusion holds true or not remains an challenging open problem, even for the small initial data. In recent years, stared from the pioneer work by Wu and Wu [Adv. Math. 310 (2017), 759--888], much more attention has been paid to the system when the magnetic field near an equilibrium state (the background magnetic field for short). In particular, when the background magnetic field satisfies the Diophantine condition (see (1.3) for details), Wu and Zhai [Math. Models Methods Appl. Sci. 33 (2023), no. 13, 2629--2656] established the decay estimates and asymptotic stability for smooth solutions of the 3D compressible isentropic MHD system without magnetic diffusion in $H^{4r+7}(\mathbb{T}^3)$ with $r>2$ by exploiting a wave structure. In this paper, a new dissipative mechanism is found out and applied so that we can improve the spaces where the decay estimates and asymptotic stability of solutions are taking place by Wu and Zhai. More precisely, we establish the decay estimates of solutions in $H^{r+1}(\mathbb{T}^n)$ and asymptotic stability result in $H^{\left(3r+3\right)^+}(\mathbb{T}^n)$ for any dimensional periodic domain $\mathbb{T}^n$ with $n\geq 2$ and $r>n-1$. Our results provide an approach for establishing the decay estimates and asymptotic stability in the Sobolev spaces with much lower regularity and uniform dimension, which can be used to study many other related models such as the compressible non-isentropic MHD system without magnetic diffusion and so on.

math.AP↗

arXiv2024

Sharp decay estimates and asymptotic stability for incompressible MHD equations without viscosity or magnetic diffusion

Whether the global existence and uniqueness of strong solutions of $n$-dimensional incompressible magnetohydrodynamic (MHD for short) equations with only kinematic viscosity or magnetic diffusion holds true or not remains an outstanding open problem. In recent years, more attention has been paid to the case when the magnetic field close to an equilibrium state (the background magnetic field for short). Specifically, when the background magnetic field satisfies the Diophantine condition (see (1.2) for details), Chen, Zhang and Zhou [Sci. China Math. 41 (2022), pp.1-10] first studied the perturbation system and established the decay estimates and stability of its solutions in 3D periodic domain $\mathbb{T}^3$, which was then improved to $H^{(3+2\beta)r+5+(\alpha+2\beta)}(\mathbb{T}^2)$ for 2D periodic domain $\mathbb{T}^2$ and any $\alpha>0$, $\beta>0$ by Zhai [J. Differ. Equ. 374 (2023), pp.267-278]. In this paper, we seek to find the optimal decay estimates and improve the space where the global stability is taking place. Through deeply exploring and fully utilizing the structure of perturbation system, we discover a new dissipative mechanism, which enables us to establish the decay estimates in Sobolev space with much lower regularity. Based on the above discovery, we greatly reduce the initial regularity requirement of aforementioned two works from $H^{4r+7}(\mathbb{T}^3)$ and $H^{(3+2\beta)r+5+(\alpha+2\beta)}(\mathbb{T}^2)$ to $H^{(3r+3)^+}(\mathbb{T}^n)$ for $r>n-1$ when $n=3$ and $n=2$ respectively. Additionally, we first present the linear stability result via the method of spectral analysis in this paper. From which, the decay estimates obtained for the nonlinear system can be seen as sharp in the sense that they are in line with those for the linearized system.

math.AP↗

arXiv2022

A new blow-up criterion for the 2D full compressible Navier-Stokes equations without heat conduction in a bounded domain

This paper is to derive a new blow-up criterion for the 2D full compressible Navier-Stokes equations without heat conduction in terms of the density $\rho$ and the pressure $P$. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm $\|\rho||_{{L^\infty(0,t;L^{\infty})}}+||P||_{L^{p_0}(0,t;L^\infty)}<\infty$ for some constant $p_0$ satisfying $1<p_0\leq 2$. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as $\|\rho||_{{L^\infty(0,t;L^{\infty})}}+||P||_{L^{\infty}(0,t;L^\infty)}<\infty$.

math.AP↗

arXiv2021

Global well-posedness to the two-dimensional incompressible vorticity equation in the half plane

This paper is concerned with the global well-posedness of the two-dimensional incompressible vorticity equation in the half plane. Under the assumption that the initial vorticity $\omega_0\in W^{k,p}(\R^{2}_+)$ with $k\geq3$ and $1 0$. An elementary and self-contained proof is presented and delicate estimates of the velocity and its derivatives are obtained in this paper. It should be emphasized that the uniform estimate on $\int^t_0\|u(\tau)\|_{W^{1,\infty}(\R^2_+)}d\tau$ is required to complete the global regularity of the solution. To do that, the double exponential growth in time of the gradient of the vorticity in the half plane is established and applied. This is different from the proof of global well-posedness of the Euler velocity equations in the Sobolev spaces, in which a Kato-type or logarithmic-type estimate of the gradient of the velocity is enough to close the energy estimates.

math.AP↗

arXiv2021

Formation of singularities for multi-dimensional transport equations with nonlocal velocity

This paper is concerned with a class of multi-dimensional transport equations with nonlocal velocity. It is shown that the local smooth solution cannot exist globally in time via the De Giorgi iteration technique.

math.AP↗

arXiv2020

Leray's backward self-similar solutions to the 3D Navier-Stokes equations in Morrey spaces

In this paper, it is shown that there does not exist a non-trivial Leray's backward self-similar solution to the 3D Navier-Stokes equations with profiles in Morrey spaces $\dot{\mathcal{M}}^{q,1}(\mathbb{R}^{3})$ provided $3/2 3/2$, and Guevara-Phuc [11, SIAM J. Math. Anal. 12 (2018)] in $\dot{\mathcal{M}}^{q,\frac{12-2q}{3}}(\mathbb{R}^{3})$ with $12/5\leq q<3$ and in $L^{q, \infty}(\mathbb{R}^3)$ with $12/5\leq q<6$.

math.AP↗

arXiv2020

An Extension of Calder$\acute{\rm O}$n-Zygmund type singular integral

In this paper, we consider a kind of singular integral which can be viewed as an extension of the classical Calder$\acute{\rm o}$n-Zygmund type singular integral. We establish an estimate of the singular integral in the $L^q$ space for $1<q<\infty$. In particular, the Calder$\acute{\rm o}$n-Zygmund estimate can be recovered from our obtained estimate. The proof of our main result is via the so called "geometric approach", which was applied in \cite{CP} on the $L^q$ estimate of the elliptic equations and in \cite{LW,Wang} on a new proof of the the Calder$\acute{\rm o}$n-Zygmund estimate.

math.AP↗

arXiv2019

Refined blow up criteria for the full compressible Navier-Stokes equations involving temperature

In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier-Stokes equations, we improve almost all the blow up criteria involving temperature to allow the temperature in its scaling invariant space for the 3D full compressible Navier-Stokes equations. Enlightening regular criteria via pressure $\Pi=\frac{\text {divdiv}}{-\Delta}(u_{i}u_{j})$ of the 3D incompressible Navier-Stokes equations on bounded domain, we generalize Beirao da Veiga's result in [1] from the incompressible Navier-Stokes equations to the isentropic compressible Navier-Stokes system in the case away from vacuum.

math.AP↗