arXiv subjects

Quansen Jiu

Publications and source records attributed to Quansen Jiu.

At least 37 records · Page 2Linked to original sources

arXiv2019

Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation

This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-Δ)^αu$ and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case $α=1$. This paper discovers that there are new phenomena with the case $α<1$. The approach for $α=1$ can not be directly extended to $α<1$. We establish that, for $α<1$, any initial data $(u_0, b_0)$ in the inhomogeneous Besov space $B^σ_{2,\infty}(\mathbb R^d)$ with $σ> 1+\frac{d}{2}-α$ leads to a unique local solution. For the case $α\ge 1$, $u_0$ in the homogeneous Besov space $\mathring B^{1+\frac{d}{2}-2α}_{2,1}(\mathbb R^d)$ and $b_0$ in $ \mathring B^{1+\frac{d}{2}-α}_{2,1}(\mathbb R^d)$ guarantees the existence and uniqueness. These regularity requirements appear to be optimal.

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arXiv2018

An Extension of Riesz Transform

In this paper, we consider the following singular integral \begin{equation*} T_jf(x)=K_j*f(x), K_j(x)=\frac{x_j}{|x|^{n+1-β}}, \end{equation*} where $x\in R^n, 0\le β<n, j=1,2,\cdots, n$. When $β=0$, it corresponds to the Riesz transform. We will make an estimate the $L^q (1<q<\infty)$ norm of $T_jf$, which holds uniformly for $0\leβ<\frac{n(q-1)}{q}$. In particular, when $β=0$, the strong $(q,q)$ type estimate of the Riesz transform for $1<q<\infty$ is recovered from the obtained estimate.

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arXiv2018

Remarks on well-posedness of the generalized surface quasi-geostrophic equation

In this paper, we are concerned with the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation in which the velocity field is expressed as $u=K\astω$, where $ω=ω(x,t)$ is an unknown function and $K(x)=\frac{x^\perp}{|x|^{2+2α}}, 0\leα\le \frac12.$ When $α=0$, it is the two-dimensional Euler equations. When $α=\frac 12$, it corresponds to the inviscid SQG. We will prove that if the existence interval of the smooth solution to the generalized SQG for some $0<α_0\le\frac12$ is $[0,T]$, then under the same initial data, the existence interval of the generalized SQG with $α$ which is close to $α_0$ will keep on $[0,T]$. As a byproduct, our result implies that the construction of the possible singularity of the smooth solution of the Cauchy problem to the generalized SQG with $α>0$ will be subtle, in comparison with the singularity presented in [Kiselev et al 2016]. To prove our main results, the difference between the two solutions and meanwhile the approximation of the singular integrals will be dealt with. Some new uniform estimates with respect to $α$ on the singular integrals and commutator estimates will be shown in this paper.

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arXiv2018

On Wolf's regularity criterion of suitable weak solutions to the Navier-Stokes equations

In this paper, we consider the local regularity of suitable weak solutions to the 3D incompressible Navier-Stokes equations. By means of the local pressure projection introduced by Wolf in [15,16], we present a $\varepsilon$-regularity criterion below of suitable weak solutions $$ \iint_{Q(1)}|u|^{20/7}dxdt\leq \varepsilon, $$ which gives an improvement of previous corresponding results obtained in Chae and Wolf [3, Arch. Ration. Mech. Anal., 225: 549-572, 2017], in Guevara and Phuc [6, Calc. Var., 56:68, 2017] and in Wolf [16, Ann. Univ. Ferrara, 61: 149-171, 2015].

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arXiv2017

Global weak solutions to 3D compressible Primitive equations with density-dependent viscosity

This paper is devoted to investigating the global existence of weak solutions for the compressible primitive equations (CPE) with damping term in a three-dimensional torus for large initial data. The system takes into account density-dependent viscosity. In our proof, we represent the vertical velocity as a function of the density and the horizontal velocity which will play a role to use the Faedo-Galerkin method to obtain the global existence of the approximate solutions. Motivated by Vasseur and Yu [yucheng2016], we obtain the key estimates of lower bound of the density, the Bresch-Desjardin entropy on the approximate solutions. Based on these estimates, using compactness arguments, we prove the global existence of weak solutions of CPE by vanishing the parameters in our approximate system step by step.

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arXiv2017

Global existence of weak solutions to the 3D incompressible axisymmetric Euler equations without swirl

In this paper, we mainly investigate the tridimensional incompressible axisymmetric Euler equations without swirl in the whole space. Specifically, we prove the global existence of weak solutions if the swirl component of initial vorticity $w_0^θ$ satisfies that $\frac{w_0^θ}r\in L^1\cap L^p({\Bbb R}^3)$ for some $p>1$. To achieve this goal, we establish the $L_{\rm loc}^{2+α}({\Bbb R}^3)$ estimate of velocity fields for some $α>0$, which is innovative to the best of our knowledge. Our result extends previous work in the literature.

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arXiv2017

The limit of vanishing viscosity for the incompressible 3D Navier-Stokes equations with helical symmetry

In this paper, we are concerned with the vanishing viscosity problem for the three-dimensional Navier-Stokes equations with helical symmetry, in the whole space. We choose viscosity-dependent initial $\bu_0^ν$ with helical swirl, an analogue of the swirl component of axisymmetric flow, of magnitude $\mathcal{O}(ν)$ in the $L^2$ norm; we assume $\bu_0^ν\to \bu_0$ in $H^1$. The new ingredient in our analysis is a decomposition of helical vector fields, through which we obtain the required estimates.

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arXiv2016

On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation

This paper focuses on the initial- and boundary-value problem for the two-dimensional micropolar equations with only angular velocity dissipation in a smooth bounded domain. The aim here is to establish the global existence and uniqueness of solutions by imposing natural boundary conditions and minimal regularity assumptions on the initial data. Besides, the global solution is shown to possess higher regularity when the initial datum is more regular. To obtain these results, we overcome two main difficulties, one due to the lack of full dissipation and one due to the boundary conditions. In addition to the global regularity problem, we also examine the large-time behavior of solutions and obtain explicit decay rates.

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arXiv2015

Well-Posedness of the Limiting Equation of a Noisy Consensus Model in Opinion Dynamics

This paper establishes the global well-posedness of the nonlinear Fokker-Planck equation for a noisy version of the Hegselmann-Krause model. The equation captures the mean-field behavior of a classic multiagent system for opinion dynamics. We prove the global existence, uniqueness, nonnegativity and regularity of the weak solution. We also exhibit a global stability condition, which delineates a forbidden region for consensus formation. This is the first nonlinear stability result derived for the Hegselmann-Krause model.

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arXiv2015

Global well-posedness for axisymmetric MHD system with only vertical viscosity

In this paper, we are concerned with the global well-posedness of a tri-dimensional MHD system with only vertical viscosity in velocity equation for the large axisymmetric initial data. By making good use of the axisymmetric structure of flow and the maximal smoothing effect of vertical diffusion, we show that $\displaystyle\sup_{2\leq p<\infty}\int_0^t\frac{\|\partial_{z}u(τ)\|_{L^p}^{2}}{p^{3/4}}\,\mathrm{d}τ<\infty$. With this regularity for the vertical first derivative of velocity vector field, we further establish losing estimates for the anisotropy tri-dimensional MHD system to get the high regularity of $(u,b)$, which guarantees that $\int_0^t\|\nabla u(τ)\|_{L^\infty}\,\mathrm{d}τ<\infty$. This together with the classical commutator estimate entails the global regularity of a smooth solution.

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arXiv2015

On Liouville Type of Theorems to the 3-D Incompressible Axisymmetric Navier-Stokes Equations

Liouville type of theorems play a key role in the blow-up approach to study the global regularity of the three-dimensional Navier-Stokes equations. In this paper, we will prove Liouville type of theorems to the 3-D axisymmetric Navier-Stokes equations with swirls under some suitable assumptions on swirl component velocity $u_θ$ which are scaling invariant. It is known that $ru_θ$ satisfies the maximum principle. The assumptions on $u_θ$ will be natural and useful to make further studies on the global regularity to the three-dimensional incompressible axisymmetric Navier-Stokes equations.

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arXiv2014

Decay of solutions to the three-dimensional generalized Navier-Stokes equations

In this paper, we first obtain the temporal decay estimates for weak solutions to the three dimensional generalized Navier-Stokes equations. Then, with these estimates at disposal, we obtain the temporal decay estimates for higher order derivatives of the smooth solution with small initial data. The decay rates are optimal in the sense that they coincides with ones of the corresponding generalized heat equation. These results improve the previous known results to the classical Navier-Stokes equations.

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arXiv2014

On possible time singular points and eventual regularity of weak solutions to the fractional Navier-Stokes equations

In this paper, we intend to reveal how the fractional dissipation $(-Δ)^α$ affects the regularity of weak solutions to the 3d generalized Navier-Stokes equations. Precisely, it will be shown that the $(5-4α)/2α$ dimensional Hausdorff measure of possible time singular points of weak solutions on the interval $(0,\infty)$ is zero when $5/6\leα< 5/4$. To this end, the eventual regularity for the weak solutions is firstly established in the same range of $α$. It is worth noting that when the dissipation index $α$ varies from $5/6$ to $ 5/4$, the corresponding Hausdorff dimension is from $1$ to $0$. Hence, it seems that the Hausdorff dimension obtained is optimal. Our results rely on the fact that the space $H^α$ is the critical space or subcritical space to this system when $α\geq5/6$.

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arXiv2013

Global classical solution of the Cauchy problem to 1D compressible Navier-Stokes equations with large initial data

In this paper, we prove that the 1D Cauchy problem of the compressible Navier-Stokes equations admits a unique global classical solution $(ρ,\rm u)$ if the viscosity $μ(ρ)=1+ρ^β$ with $β\geq0$. The initial data can be arbitrarily large and may contain vacuum. Some new weighted estimates of the density and velocity are obtained when deriving higher order estimates of the solution.

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arXiv2013

Remarks on Blow-up of Smooth Solutions to the Compressible Fluid with Constant and Degenerate Viscosities

In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial data has compact support or contain vacuum in any finite regions. Moreover, a simplified and unified proof on the blow-up results to the classical solutions of the full compressible Navier-Stokes equations without heat conduction by Xin \cite{Xin} and with heat conduction by Cho-Bin \cite{CJ} will be given.

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arXiv2013

Global Regularity of 2D Generalized MHD Equations with Magnetic Diffusion

This paper is concerned with the global regularity of the 2D (two-dimensional) generalized magnetohydrodynamic equations with only magnetic diffusion $Λ^{2β} b$. It is proved that when $β>1 $ there exists a unique global regular solution for this equations. The obtained result improves the previous known one which requires that $β>\frac32$.

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arXiv2013

A Remark On Global Regularity of 2D Generalized Magnetohydrodynamic Equations

In this paper we study the global regularity of the following 2D (two-dimensional) generalized magnetohydrodynamic equations \begin{eqnarray*} \left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot \nabla b - ν(-\triangle)^α u b_t + u \cdot \nabla b & = & b \cdot \nabla u - κ(-\triangle)^β b \end{array}\right. \end{eqnarray*} and get global regular solutions when $ 0\leqslantα< 1 / 2,\,\, β\geqslant 1, \,\,3α+ 2β>3 $, which improves the results in \cite{TYZ2013}. In particular, we obtain the global regularity of the 2D generalized MHD when $α=0$ and $β>\frac 32$.

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arXiv2012

The 2D incompressible Boussinesq equations with general critical dissipation

This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to $α+β=1$ when $Λ^α\equiv (-Δ)^{\fracα{2}}$ and $Λ^β$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We establish the global regularity for the general case with $α+β=1$ and $0.9132\approx α_0<α<1$. The cases when $α=1$ and when $α=0$ were previously resolved by Hmidi, Keraani and Rousset \cite{HKR1,HKR2}. The global existence and uniqueness is achieved here by exploiting the global regularity of a generalized critical surface quasi-gesotrophic equation as well as the regularity of a combined quantity of the vorticity and the temperature.

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