arXiv subjects

Xiaoxin Zheng

Publications and source records attributed to Xiaoxin Zheng.

14 recordsLinked to original sources

arXiv2023

Asymptotic stability of homogeneous solutions to Navier-Stokes equations under $L^{p}$-perturbations

It is known that there has been classified for all $(-1)$-homogeneous axisymmetric no-swirl solutions of the three-dimensional Navier-Stokes equations with a possible singular ray. The main purpose of this paper is to show that the least singular solutions among such solutions other than Landau solutions to the Navier-Stokes equations are asymptotically stable under $L^{3}$-perturbations. Moreover, we establish the $L^{q}$ decay estimate with an explicit decay rate and a sharp constant for any $q>3$. For that purpose, we first study the global well-posedness of solutions to the perturbed equations under small initial data in $L_σ^{3}$ space and the local well-posedness with any initial data in $L_σ^{p}$ spaces for $p\geq3$.

math.AP↗

arXiv2022

Global regularity and decay behavior for Leray equations with critical-dissipation and Its Application to Self-similar Solutions

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our method is based on the maximal smoothing effect, $L^{p}$-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we shall construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations, and more importantly, prove the global regularity and the optimal decay without additional requirement of existing literatures.

math.AP↗

arXiv2022

Global-in-time Boundedness of solution for Cauchy problem to the Parabolic-Parabolic Keller-Segel system with logistic growth

We study global-in-time well-posedness and the behaviour and of the solution to Cauchy problem in the classical Keller-Segel system with logistic term \begin{equation*} \left. \aligned \partial_tn-Δn=&-χ\nabla\cdot(n\nabla c)+\la n-μn^2 τ\partial_tc-Δc=&-c+n \endaligned \right\}\quad\text{in}\,\,\,\RR^d\times\RR^+, \end{equation*} where $d\ge 1$, $τ,\, χ,\, μ>0$ and $λ\ge 0$. It's inspired by a previous result \cite[M. Winkler, Commun. Part. Diff. Eq., 35 (2010), 1516-1537]{Win10}, where the global-in-time boundedness of the above Keller-Segel system in smooth \emph{bounded }convex domains is established for large $μ$. However, his approach in bounded domain ceases to directly apply in the entire space $\RR^d$, and then they raised an interesting question whether a similar global-in-time boundedness statement remains true of Cauchy problem. In this paper, we answer this open problem by developing local-in-space estimates. More precisely, we prove that the above Keller-Segel system possesses a uniquely global-in-time bounded solution for any $τ>0$ under the assumption that $μ$ is large. The key point of our proof heavily relies on localization in space of solution caused by "local effect" of $L^\infty(\RR^d)$-norm.

math.AP↗

arXiv2020

On the $C_0$ semigroup generated by the Oseen operator around a steady flow exterior to a rotating obstacle

We consider the motion of an incompressible viscous fluid filling the whole space exterior to a moving with rotation and translation obstacle. We show that the Stokes operator around the steady flow in the exterior of this obstacle generates a $C_0$-semigroup in $L^p$ space and then develop a series of $L^p$-$L^q$ estimates of such semigroup. As an application, we give out the stability of such steady flow when the initial disturbance in $L^3$ and the steady flow are sufficiently small.

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arXiv2019

Forward self-similar solutions of the fractional Navier-Stokes Equations

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion $(-Δ)^α.$ First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with $5/6<α\leq1$ for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in $\mathbb R^3\times (0,+\infty)$. In particular, when $α=1$, we prove that the solution constructed by Korobkov-Tsai [Anal. PDE 9 (2016), 1811-1827] satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in [Jia and Šverák, Invent. Math. 196 (2014), 233-265].

math.AP↗

arXiv2019

Between homogeneous and inhomogeneous Navier-Stokes systems: the issue of stability

We construct large velocity vector solutions to the three dimensional inhomogeneous Navier-Stokes system. The result is proved via the stability of two dimensional solutions with constant density, under the assumption that initial density is point-wisely close to a constant. Key elements of our approach are estimates in the maximal regularity regime and the Lagrangian coordinates. Considerations are done in the whole $\R^3$.

math.AP↗

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.

math.AP↗

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

Minimal blow-up initial data in critical Fourier-Herz spaces for potential Navier-Stokes singularities

In this paper, we mainly prove the existence of the minimal blow-up initial data in critical Fourier-Herz space $F\dot{B}^{2-{\frac3p}}_{p,q}(\RR^3)$ with $1<p\leq\infty$ and $1\leq q<\infty$ for the three dimensional incompressible potential Navier-Stokes equations by developing techniques of "localization in space" involving the partial regularity given by the De Giorgi iteration, weak-strong uniqueness, the short-time behaviour of the kinetic energy and stability of singularity of Calderón's solution.

math.AP↗

arXiv2016

Global well-posedness for the two-dimensional Maxwell-Navier-Stokes equations

In this paper, we investigate Cauchy problem of the two-dimensional full Maxwell-Navier-Stokes system, and prove the global-in-time existence and uniqueness of solution in the borderline space which is very close to $L^2$-energy space by developing the new estimate of $\sup_{j\in\mathbb Z} 2^{2j} \int_0^t \sum_{k\in\mathbb{Z}^2} \big\| \sqrt{ϕ_{i,k}} u(τ) \big\|^2_{L^2(\mathbb{R}^2)} \text{d}τ< \infty$. This solves the open problem in the framework of borderline space purposed by Masmoudi in \cite{Masmoudi-10}.

math.AP↗

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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arXiv2014

Time-dependent singularities in the Navier-Stokes system

We show that, for a given Hölder continuous curve in $\{(γ(t),t)\,:\, t>0\} \subset R^3\times R^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $R^3$ which is smooth outside this curve and singular on it. This is a pointwise solution of the system outside the curve, however, as a distributional solution on $R^3\times R^+$, it solves an analogous Navier-Stokes system with a singular force concentrated on the curve.

math.AP↗

arXiv2013

Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity

In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*} {{array}{ll} (\partial_{t}+u\cdot\nabla)u-κΔ_{h} u+\nabla Π=ρe_{3},\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}, (\partial_{t}+u\cdot\nabla)ρ=0, \text{div}u=0. {array}. {equation*} Under the assumption that the support of the axisymmetric initial data $ρ_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\fracρr$ for large time by taking advantage of characteristic of transport equation. This growing property together with the horizontal smoothing effect enables us to establish $H^1$-estimate of the velocity via the $L^2$-energy estimate of velocity and the Maximum principle of density. Based on this, we further establish the estimate for the quantity $\|ω(t)\|_{\sqrt{\mathbb{L}}}:=\sup_{2\leq p<\infty}\frac{\norm{ω(t)}_{L^p(\mathbb{R}^3)}}{\sqrt{p}}<\infty$ which implies $\|\nabla u(t)\|_{\mathbb{L}^{3/2}}:=\sup_{2\leq p<\infty}\frac{\norm{\nabla u(t)}_{L^p(\mathbb{R}^3)}}{p\sqrt{p}}<\infty$. However, this regularity for the flow admits forbidden singularity since $ \mathbb{L}$ (see \eqref{eq-kl} for the definition) seems be the minimum space for the gradient vector field $u(x,t)$ ensuring uniqueness of flow. To bridge this gap, we exploit the space-time estimate about $ \sup_{2\leq p<\infty}\int_0^t\frac{\|\nabla u(τ)\|_{L^p(\mathbb{R}^3)}}{\sqrt{p}}\mathrm{d}τ<\infty$ by making good use of the horizontal smoothing effect and micro-local techniques. The global well-posedness for the large initial data is achieved by establishing a new type space-time logarithmic inequality.

math.AP↗

arXiv2012

On the global well-posedness for the Boussinesq system with horizontal dissipation

In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is an axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between $\frac{u^{r}}{r}$ and $\frac{ω_θ}{r}$ by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together with the structure of the coupling of \eqref{eq1.1} entails the desired regularity.

math.AP↗