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Qionglei Chen

Publications and source records attributed to Qionglei Chen.

At least 19 recordsLinked to original sources
arXiv2026

Norm Inflation for Inviscid and Fully Dissipative Boussinesq Systems in Supercritical Spaces

We prove norm inflation, in the sense of strong ill-posedness, for the two-dimensional Boussinesq system in supercritical Besov spaces. For the inviscid system, norm inflation holds in \(\dot B^β_{p,q}(\mathbb R^2)\times \dot B^β_{p,r}(\mathbb R^2)\) for \(β\neq0\), \(1<p\leq\infty\), \(1\leq q,r\leq\infty\), and \(-2<β-\frac{2}{p}<1\). For the fully dissipative system, the same conclusion holds in the range \(-2<β-\frac{2}{p}<-1\). In both cases, the results cover almost all supercritical Besov spaces satisfying the local integrability condition. Norm inflation occurs in the density component \(ρ\), while the velocity component \(u\) remains bounded. In the fully dissipative case, the inflation space is supercritical for \(u\), but subcritical for \(ρ\) with respect to its own scaling. This is not a contradiction: the density is transported by a velocity field in a supercritical regime, and this transport mechanism is precisely what produces norm inflation in \(ρ\).

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arXiv2026

Improved global well-posedness for the cubic NLS on two-dimensional waveguide $\R\times\T$

In this article, we show that the solution to defocusing cubic nonlinear Schrödinger equation (NLS) posed on the two-dimensional waveguide \begin{align*} i\partial_tu+Δ_{\R\times\T}u=|u|^2u \end{align*} is globally well-posed in $H^s(\R\times\T)$ with $s>\frac{1}{2}$. The proof is based on the $I$-method. Inspired by Colliander-Keel-Staffilani-Takaoka-Tao [Discrete Contin. Dyn. Syst. 21 (2008), 665-686], we construct the modified energy to improve the energy increment. The main difficulty lies in controlling the resonant interactions caused by the modified energy. To this end, we establish refined bilinear Strichartz estimates with angular truncation on the rescaled waveguide, thereby generalizing results previously obtained by Takaoka [J. Differ. Equa. 394 (2024), 296-319]. Furthermore, we demonstrate polynomial growth of $H^s$ with $\frac{1}{2} < s < 1$. Our result extends the recent work of Deng-Fan-Yang-Zhao-Zheng [J. Func. Anal. 287 (2024), 110595].

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arXiv2026

Quantitative stability of the 2D Monotone shear flow for Boussinesq equation in a finite channel

Neither natural nor laboratory laminar flows are perfectly steady. Instead, they are frequently highly unsteady, as illustrated by experimental studies on Bénard convection. In the paper, we investigate the transition threshold of the Boussinesq equations around a time-dependent monotone shear flow $(U(t,y),0)$ with a constant background temperature $a\in\mathbb{R}$. The analysis is performed in the finite channel $\mathbb{T}\times[0,1]$ with non-slip boundary condition. By means of the sharp resolvent estimates and space-time estimates, we establish that the Boussinesq system admits a globally stable solution around the monotone shear flow, provided that the initial perturbation satisfies $\|u^{\mathrm{in}}\|_{H^2}\leq cν^{\frac12}, \|\langle D_x\rangle θ^{\mathrm{in}}\|_{L^2} \leq cν^{\frac56}$. Moreover, we derive the enhanced dissipation estimate of the vorticity and inviscid damping estimate of the velocity.

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arXiv2025

The optimal transition threshold for the 2D Couette flow in the infinite channel

We investigate the stability of the 2-D Navier-Stokes equations in the infinite channel $\mathbb{R}\times [-1,1]$ with the Navier-slip boundary condition. We show that if the initial perturbations $ω^{in}$ around the Couette flow satisfy $\|ω^{in}\|_{H^3_{x,y}\cap L^1_x H^3_y}\leq cν^{\frac13}$, the solution admits enhanced dissipation at $x$-frequencies $|k|\gg ν$ and inviscid damping effect. The key contributions lie in two parts: (1) we adopt the new decomposition of the vorticity $ω=ω_{L}+ω_e$, where $ω_L$ effectively captures a ``weak" enhanced dissipation $(1+ν^{\frac13} t)^{-\frac14}e^{-νt}$ and the corresponding velocity exhibits the inviscid damping effect; (2) we introduce the dyadic decomposition for the long time scale $t\geq ν^{-\frac16}$ and apply the ``infinite superposition principle" to the equation for $ω_e$ in order to control the growth induced by echo cascades, which appears to be novel and may hold independent significance.

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arXiv2025

Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition

In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability in the context of long wave associated with the frequency range $0\leq |k|<1$ by developing the resolvent estimate argument. The new ingredient is to discover the key division point at $10ν$ in the frequency interval $(0,1)$ by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval $(0,1)$, and then we establish the space-time estimates on the low-frequency $0\leq |k|\leq 10 ν$ and the intermediate-frequency $ 10 ν\leq |k|<1$, respectively. As an application of the space-time estimates, we obtain the nonlinear transition threshold to be $γ\leq\frac{1}{2}$.Meanwhile, we also show that when the frequencies $|k|\geq ν^{1-}$, the enhanced dissipation effect occurs for the linearized Navier-Stokes equations.

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arXiv2024

Sharp ill-posedness for the non-resistive MHD equations in Sobolev spaces

In this paper, we prove a sharp ill-posedness result for the incompressible non-resistive MHD equations. In any dimension $d\ge 2$, we show the ill-posedness of the non-resistive MHD equations in $H^{\frac{d}{2}-1}(\mathbb{R}^d)\times H^{\frac{d}{2}}(\mathbb{R}^d)$, which is sharp in view of the results of the local well-posedness in $H^{s-1}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)(s>\frac{d}{2})$ established by Fefferman et al.(Arch. Ration. Mech. Anal., \textbf{223} (2), 677-691, 2017). Furthermore, we generalize the ill-posedness results from $H^{\frac{d}{2}-1}(\mathbb{R}^d)\times H^{\frac{d}{2}}(\mathbb{R}^d)$ to Besov spaces $B^{\frac{d}{p}-1}_{p, q}(\mathbb{R}^d)\times B^{\frac{d}{p}}_{p, q}(\mathbb{R}^d)$ and $\dot B^{\frac{d}{p}-1}_{p, q}(\mathbb{R}^d)\times \dot B^{\frac{d}{p}}_{p, q}(\mathbb{R}^d)$ for $1\le p\le\infty, q>1$. Different from the ill-posedness mechanism of the incompressible Navier-Stokes equations in $\dot B^{-1}_{\infty, q}$ \cite{B,W}, we construct an initial data such that the paraproduct terms (low-high frequency interaction) of the nonlinear term make the main contribution to the norm inflation of the magnetic field.

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arXiv2023

Stress concentration for nonlinear insulated conductivity problem with adjacent inclusions

A high-contrast two-phase nonlinear composite material with adjacent inclusions of $m$-convex shapes is considered for $m>2$. The mathematical formulation consists of the insulated conductivity problem with $p$-Laplace operator in $\mathbb{R}^{d}$ for $p>1$ and $d\geq2$. The stress, which is the gradient of the solution, always blows up with respect to the distance $\varepsilon$ between two inclusions as $\varepsilon$ goes to zero. We first establish the pointwise upper bound on the gradient possessing the singularity of order $\varepsilon^{-β}$ with $β=(1-α)/m$ for some $α\geq0$, where $α=0$ if $d=2$ and $α>0$ if $d\geq3$. In particular, we give a quantitative description for the range of horizontal length of the narrow channel in the process of establishing the gradient estimates, which provides a clear understanding for the applied techniques and methods. For $d\geq2$, we further construct a supersolution to sharpen the upper bound with any $β>(d+m-2)/(m(p-1))$ when $p>d+m-1$. Finally, a subsolution is also constructed to show the almost optimality of the blow-up rate $\varepsilon^{-1/\max\{p-1,m\}}$ in the presence of curvilinear squares. This fact reveals a novel dichotomy phenomena that the singularity of the gradient is uniquely determined by one of the convexity parameter $m$ and the nonlinear exponent $p$ except for the critical case of $p=m+1$ in two dimensions.

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arXiv2022

Ill-posedness issue for the 2D viscous shallow water equations in some critical Besov spaces

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^2)$. As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^2)$ for $p<4$ and ill-posed in $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^2)$ for $p>4$. In this paper, we prove that this system is ill-posed for the critical case $p=4$ in the sense of "norm inflation". Furthermore, we also show that the system is ill-posed in $\dot B^{\frac{1}{2}}_{4,1}(\mathbb{R}^2)\times \dot B^{-\frac{1}{2}}_{4,q}(\mathbb{R}^2)$ for any $q\neq 2$

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arXiv2019

Global attractors and their upper semicontinuity for a structural damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$

The paper investigates the existence of global attractors and their upper semicontinuity for a structural damped wave equation on $\mathbb{R}^{N}: u_{tt}-Δu+(-Δ)^αu_{t}+u_{t}+u+g(u)=f(x)$, where $α\in (1/2, 1)$ is called a dissipative index. We propose a new method based on the harmonic analysis technique and the commutator estimate to exploit the dissipative effect of the structural damping $(-Δ)^αu_{t}$ and to overcome the essential difficulty: "both the unbounded domain $\mathbb{R}^N$ and the supercritical nonlinearity cause that the Sobolev embedding loses its compactness"; Meanwhile we show that there exists a supercritical index $p_α\equiv\frac{N+4α}{N-4α}$ depending on $α$ such that when the growth exponent $p$ of the nonlinearity $g(u)$ is up to the supercritical range: $1\leqslant p 0$; (ii) the related solution semigroup possesses a global attractor $\mathcal{A}_α$ in natural energy space for each $α\in (1/2, 1)$; (iii) the family of global attractors $\{\mathcal{A}_α\}_{α\in (1/2, 1) }$ is upper semicontinuous at each point $α_0\in (1/2, 1)$.

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arXiv2019

The two-dimensional Euler equations in Yudovich type space and $\mathrm{\textbf{bmo}}$-type space

We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a $\rm bmo$-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space involving an unbounded and non-decaying vorticity. Next, we establish an estimate with a logarithmic loss of regularity for the transport equation in a bmo-type space by developing classical analysis tool such as the John-Nirenberg inequality. We also optimize estimates of solutions to the vorticity-stream formulation of the two-dimensional Euler equations with a bi-Lipschitz vector field in bmo-type space by combining an observation introduced by Yodovich with the so-called "quasi-conformal property" of the incompressible.

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arXiv2018

Global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism

We prove the global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism on the stress tensor in $\mathbb{R}^d$ for the small initial data. Our proof is based on the observation that the behaviors of Green's matrix to the system of $\big(u,(-Δ)^{-\frac12}\mathbb{P}\nabla\cdotτ\big)$ as well as the effects of $τ$ change from the low frequencies to the high frequencies and the construction of the appropriate energies in different frequencies.

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arXiv2013

On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces

We prove the ill-posedness of the 3-D baratropic Navier-Stokes equation for the initial density and velocity belonging to the critical Besov space $(\dot{B}^{\f 3p}_{p,1}+\barρ,\,\dot{B}^{\f 3p-1}_{p,1})$ for $p>6$ in the sense that a ``norm inflation" happens in finite time, here $\barρ$ is a positive constant. Our argument also shows that the compressible viscous heat-conductive flows is ill-posed for the initial density, velocity and temperature belonging to the critical Besov space $(\dot{B}^{\f 3p}_{p,1}+\barρ,\,\dot{B}^{\f 3p-1}_{p,1},\,\dot{B}^{\f 3p-2}_{p,1})$ for $p>3$. These results shows that the compressible Navier-Stokes equations are ill-posed in the smaller critical spaces compared with the incompressible Navier-Stokes equations.

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arXiv2010

Global well-posedness for the micropolar fluid system in the critical Besov spaces

We prove the global well-posedness for the 3-D micropolar fluid system in the critical Besov spaces by making a suitable transformation to the solutions and using the Fourier localization method, especially combined with a new $L^p$ estimate for the Green matrix to the linear system of the transformed equation. This result allows to construct global solutions for a class of highly oscillating initial data of Cannone's type. Meanwhile, we analyze the long behavior of the solutions and get some decay estimates.

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arXiv2009

Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity

Cannone \cite{Cannone} proved the global well-posedness of the incompressible Navier-Stokes equations for a class of highly oscillating data. In this paper, we prove the global well-posedness for the compressible Navier-Stokes equations in the critical functional framework with the initial data close to a stable equilibrium. Especially, this result allows us to construct global solutions for the highly oscillating initial velocity. The proof relies on a new estimate for the hyperbolic/parabolic system with convection terms.

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