Jian Zhai
An upper bound of blow up rate for the Navier-Stokes equations with small data in L^2(R^3) is obtained.
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Limit behaviors of blow up solutions for impressible Navier-Stokes equations are obtained.
We consider the interior regularity of Leray-Hopf solutions to Navier-Stokes equations on critical case L^2_w(0,T;L^\infty(R^3)). Particularly, an open problem proposed in [KK] was solved.
Global regularity of Leray-Hopf weak solutions to 3D Navier-Stokes equations
本文通过构造带稀疏逆对数权重的超临界空间并叠加高频能量估计，证明了初值属于H^{1/2}的三维Navier-Stokes方程Leray-Hopf弱解具有全局正则性。
The dyadic problem $\dot u_n + λ^{2n} u_n - λ^{βn} u_{n-1}^2 + λ^{β(n+1)} u_n u_{n+1} = 0$ with "smooth" initial data is considered. The uniqueness of the Leray-Hopf solution is proved.
Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise
本文证明了受传输噪声扰动的3D分数阶Navier--Stokes方程在无外力情况下存在无穷多个全局Leray--Hopf解，首次实现了该类方程的随机扰动非唯一性结果，其证明基于凸积分技术与流变换。
We prove a general result that implies that very weak solutions to the Cauchy problem for the Navier-Stokes equations must be, in fact, Leray-Hopf solutions if only their initial data are (solenoidal) with finite kinetic energy.
In this note we prove that each weak solution for the 3D Navier-Stokes system satisfies Leray-Hopf property. Moreover, each weak solution is rightly continuous in the standard phase space $H$ endowed with the strong convergence topology.
The dyadic model $\dot u_n + λ^{2n}u_n - λ^{βn}u_{n-1}^2 + λ^{β(n+1)}u_nu_{n+1} = f_n$, $u_n(0)=0$, is considered. It is shown that in the case of non-trivial right hand side the system can have two different Leray-Hopf solutions.
Non-unique weak solutions in Leray-Hopf class are constructed for the three dimensional magneto-hydrodynamics with Hall effect. We adapt the widely appreciated convex integration framework developed in a recent work of Buckmaster and Vicol for the Navier-Stokes equation, and with deep roots in a sequence of breakthrough papers for the Euler equation.
In present note we establish the following inequality for the the Leray-Hopf solutions of the 3-D $Ω$-periodic Navier-Stokes Equations: \[ϕ(|u(t)|^2)-ϕ(|u(t_0)|^2)\le 2\int_{t_0}^{t}ϕ'(|u(τ)|^2) [-ν|A^{1/2}u(τ)|^2+(g(τ),u(τ))]\,dτ\] for all $t_0$ Leray-Hopf points, $t\ge t_0$, and $ϕ:\mathbb{R}_{+}\to\mathbb{R}$ is an absolutely continouos non-decreasing function with bounded derivative. %with $ϕ'(ξ)\ge0$ for all $ξ>0$. Here $(\cdot,\cdot)$ and $|\cdot|$ is correspondingly the $L^2$ inner product and the $L^2$ norm on $Ω$, and $A$ is the Stokes operator.
Eventual regularity and asymptotic behavior of Leray-Hopf weak solutions for the Hall-MHD system
研究不可压缩、粘性、电阻性Hall-MHD系统，证明二维Leray-Hopf弱解在有限时间后变光滑，三维中构造出磁涡度场最终获得额外正则性的弱解，并在初始数据低频频谱假设下结合广义Fourier分裂方法导出速度场和磁场解耦的代数衰减率。
Employing the approach of A. Takeshita [Pacific J. Math., Vol. 157 (1993), 151--158], we give an elementary proof of the invalidity of the Leray-Hopf Extension Condition for certain multiply connected bounded domains of R^n, n=2,3, whenever the flow through the different components of the boundary is non-zero. Our proof is alternative to and, to an extent, more direct than the recent one proposed by J.G. Heywood [J. Math. Fluid Mech. Vol. 13 (2011), 449--457].
Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness
本文刻画了二维非齐次 Navier-Stokes 系统在正时间下变为强解的 Leray-Hopf 解，基于强能量不等式与压力正则性建立了弱-强唯一性结果，并为该领域近期进展提供了统一框架。
Nonuniqueness of Leray-Hopf solutions to the forced fractional Navier-Stokes Equations in three dimensions, up to the J. L. Lions exponent
本文证明在三维受迫分数阶Navier-Stokes方程中，当指数$\alpha\in(1/2,5/4)$时存在非唯一Leray-Hopf解，从而表明J. L. Lions指数的尖锐性。
We consider the three dimensional incompressible Navier-Stokes equations with non stationary source terms f chosen in a suitable space. We prove the existence of Leray-Hopf weak solutions and that it is possible to characterize (up to sub-sequences) their long-time averages, which satisfy the Reynolds averaged equations, involving a Reynolds stress. Moreover, we show that the turbulent dissipation is bounded by the sum of the Reynolds stress work and of the external turbulent fluxes, without any additional assumption, than that of dealing with Leray-Hopf weak solutions. Finally, in the last se…
We consider the Navier-Stokes equation on $\mathbb{H}^{2}(-a^{2})$, the two dimensional hyperbolic space with constant sectional curvature $-a^{2}$. We prove an ill-posedness result in the sense that the uniqueness of the Leray-Hopf weak solutions to the Navier-Stokes equation breaks down on $\mathbb{H}^{2}(-a^{2})$. We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions.
Existence and Non-Uniqueness of Ergodic Leray-Hopf Solutions to the Stochastic Power-Law Flows
本文研究了随机幂律流体中 Leray-Hopf 解的存在性与非唯一性，通过引入新的能量相关函数和凸积分技术，构造了无限多个概率强解，并给出了全局时间估计。
Uniqueness of Leray-Hopf weak solutions to the two-dimensional viscous Primitive Equations
本文解决了二维粘性原始方程Leray-Hopf弱解唯一性的长期公开问题，证明了周期通道中满足特定边界条件、任意L²初始数据下该类弱解的唯一性。
Non-uniqueness of Leray-Hopf Solutions to Forced Stochastic Hyperdissipative Navier-Stokes Equations up to Lions Index
本文证明了带线性乘性噪声和特定外力的随机分数阶Navier-Stokes方程局部强解的非唯一性（含外力扰动情形），并通过新方法得到了某些随机外力下光滑的全局Leray-Hopf解的非唯一性结果。
Blow-up and uniqueness of Leray-Hopf solutions to forced Navier-Stokes equations
构造外力与光滑初值，使得3D Navier-Stokes方程的Leray-Hopf解唯一、全局存在、满足能量等式，但存在爆破时刻。
We prove the ill-posedness for the Leray-Hopf weak solutions of the incompressible and ipodissipative Navier-Stokes equations, when the power of the diffusive term $(-Δ)^γ$ is $γ< \frac{1}{3}$. We construct infinitely many solutions, starting from the same initial datum, which belong to $C_{x,t}^{\frac{1}{3}-}$ and strictly dissipate their energy in small time intervals. The proof exploits the "convex integration scheme" introduced by C. De Lellis and L. Székelyhidi for the incompressible Euler equations, joining these ideas with new stability estimates for a class of non-local advection-diffu…
Non-uniqueness in the Leray-Hopf class for a dyadic Navier-Stokes model
本文在二元Obukhov壳模型中构造了临界正则性下产生两个不同Leray-Hopf解的初始数据，揭示非唯一性源于标度对称性部分破缺，并指出其非线性相互作用与凸积分一致，为完整Navier-Stokes方程的非唯一性研究提供了潜在机制。
Non-Leray-Hopf solutions to 3D stochastic hyper-viscous Navier-stokes equations: beyond the Lions exponents
研究三维随机超粘性Navier-Stokes方程，证明在超越Lions指数的高粘性区域，尽管Leray-Hopf类中解唯一，但在混合Lebesgue空间中存在无穷多个非Leray-Hopf解，并建立了噪声消失极限结果。
Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations
本文研究随机受迫三维 Navier-Stokes 方程，借助 Albritton 等人的思想证明局部 Leray-Hopf 解及分布律的非唯一性，并在确定性情形刻画非唯一力的稠密性与初值可控性。
We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba-Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.
Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equation
本文通过计算机辅助证明首次展示了拉耶-霍夫解的非唯一性，通过构造自相似解并分析线性化算子稳定性，证明了存在无限多个拉耶-霍夫解。
We construct non-unique Leray-Hopf solutions for some dyadic models for magnetohydrodynamics when the intermittency dimension $δ$ is less than 1. In contrast, uniqueness of Leray-Hopf solution is established in the case of $δ\geq 1$. Analogous results on uniqueness and non-uniqueness of Leray-Hopf solution are also obtained for dyadic models of MHD with fractional diffusion.
The projection method to solve the incompressible Navier-Stokes equations was first studied by Chorin [Math. Comp., 1969] in the framework of a finite difference method and Temam [Arch. Rational Mech. and Anal., 1969] in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with the periodic boundary condition assuming existence of a $C^5$-solution, while Temam demonstrated an abstract argument to obtain a Leray-Hopf weak solution in problems on a bounded domain with the no-slip boundary condition. In the present paper, the aut…
Partial regularity of Leray-Hopf weak solutions to the incompressible Navier-Stokes equations with hyperdissipation
本文研究具超耗散 α∈(1,5/4) 的不可压缩 Navier-Stokes 方程 Leray-Hopf 弱解的部分正则性，证明爆破点集的 Hausdorff 维数与盒计数维数具有由 α 决定的显式上界。
The main subject of this paper concerns the establishment of certain classes of initial data, which grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. In particular, our main theorem that this holds for any solenodial initial data, with finite $L_2(\mathbb{R}^3)$ norm, that also belongs to to certain subsets of $VMO^{-1}(\mathbb{R}^3)$. As a corollary of this, we obtain the same conclusion for any solenodial $u_{0}$ belonging to $L_{2}(\mathbb{R}^3)\cap \mathbb{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$, for any $3…
Leray--Hopf Type Weak Solutions for the Three-Dimensional Beris--Edwards System with Stable Landau--de Gennes Potential
本文研究了在全空间中Beris-Edwards系统在稳定bulk假设下的弱解存在性，通过超粘性近似和局部尾估计获得物理自由能不等式，并利用低阶链规则推导扩展的Leray-Hopf型能量不等式，最终证明了弱解的存在性。
It is a simple consequence of the Cafarelli-Kohn-Nirenberg theory that every possible singularity in a thin Haussdorff-measurable set of a Leray- Hopf solution of the incompressible Navier Stokes equation is on the tip of a small open cone, where the solution is smooth. Using global regularity results for weak Hopf-Leray solutions this potential singularity can be analyzed by investigation of the asymptotic behavior at infinite time of a solution of a related initial-boundary value problem posed in transformed coordinates on a cylinder. Next to some new consequences such as global regularity o…
Convergence of a Second-Order Projection Method to Leray-Hopf Solutions of the Incompressible Navier-Stokes Equations
本文分析了有界 Lipschiz 域上不可压缩 Navier-Stokes 方程的二阶投影方法，首次在极小数据假设下证明了其向 Leray-Hopf 弱解的收敛性，为高阶投影方法提供了首个严格收敛证明。
In this paper we establish a Serrin type regularity criterion on the gradient of pressure in weak spaces for the Leray-Hopf weak solutions of the Navier-Stokes equations in R3.
In the celebrated paper by Jean Leray, published in JMPA journal in 1933, the invading domains method was proposed to construct D-solutions for the stationary Navier-Stokes flow around obstacle problem. In two dimensions, whether Leray's D-solution achieves the prescribed limiting velocity at spatial infinity became a major open problem since then. In this paper, we solve this problem at small Reynolds numbers. The proof builds on a novel blow-down argument which rescales the invading domains to the unit disc, and the ideas developed in a recent paper [Korobkov-Pileckas-Russo2020], where the n…
Motivated by Kolmogorov's theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber $Λ(t)$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we also we prove that…
We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension $d\ge 2$. For the power index $q$ below the compactness threshold, i.e. $q \in (1, \frac{2d}{d+2})$, we show ill-posedness of Leray-Hopf solutions. For a wider class of indices $q \in (1, \frac{3d+2}{d+2})$ we show ill-posedness of distributional (non-Leray-Hopf) solutions, extending the seminal paper of Buckmaster and Vicol. In this wider class we also construct non-unique solutions for every datum in $L^2$.
In this paper, using the theory of critical points at infinity of Bahri, we derive an exact bubbling rate formula for the resonant prescribed Q-curvature equation on closed even-dimensional Riemannian manifolds. Using this, we derive new existence results for the resonant prescribed Q-curvature problem under a positive mass type assumption. Moreover, we derive a compactness theorem for conformal metrics with prescribed Q-curvature under a non-degeneracy assumption. Furthermore, combining the bubbling rate formula with the construction of some blowing-up solutions, we compute the Leray-Schauder…
We demonstrate that the problem of existence of Leray self-similar blow up solutions in a generalized mild Navier-Stokes system with the fractional Laplacian $(-Δ)^{γ/2}$ can be stated as a fixed point problem for a "renormalization" operator. We proceed to construct {\it a-priori} bounds, that is a renormalization invariant precompact set in an appropriate weighted $L^p$-space. As a consequence of a-priori bounds, we prove existence of renormalization fixed points for $d \ge 2$ and $d<γ<2 d+2$, and existence of non-trivial Leray self-similar mild solutions in $C^\infty([0,T),(H^k)^d \cap (L^p…
Using the concept of stationary statistical solution, which generalizes the notion of invariant measure, it is proved that, in a suitable sense, time averages of almost every Leray-Hopf weak solution of the three-dimensional incompressible Navier-Stokes equations converge as the averaging time goes to infinity. This system of equations is not known to be globally well-posed, and the above result answers a long-standing problem, extending to this system a classical result from ergodic theory. It is also showed that, from a measure-theoretic point of view, the stationary statistical solution obt…
Euler-Leray data functions of first and second order are defined by first and second order derivatives of the nonlinear spatial part of the incompressible Euler equation operator in Leray projection form applied to Cauchy data. The Lipschitz continuity of these functions for a certain class of strongly regular Cauchy data is sufficient for the existence of global regular upper bounds of incompressible Navier Stokes equation solutions. Global regular upper bounds of global solution branches of the incompressible Euler equation can be obtained for a class of strongly regular Cauchy data, if the …
In a well-known work, M. Anderson constructed a Hadamard manifold $(M^n, g)$ which carries non-zero $L^2$ harmonic $p$-forms when $p \neq n/2$, thus disproving the Dodziuk-Singer conjecture. In this paper, we use the manifold $(M^3, g)$ in order to solve another problem in geometric analysis, namely the nonuniqueness of solutions of Leray-Hopf type of the Navier-Stokes equations.
Kuroki and Soga [Numer. Math. 2020] proved that a version of Chorin's fully discrete projection method, originally introduced by A. J. Chorin [Math. Comp. 1969], is unconditionally solvable and convergent within an arbitrary fixed time interval to a Leray-Hopf weak solution of the incompressible Navier-Stokes equations on a bounded domain with an arbitrary external force. This paper is a continuation of Kuroki-Soga's work. We show time-global solvability and convergence of our scheme; $L^2$-error estimates for the scheme in the class of smooth exact solutions; application of the scheme to the …
Closed Estimates of Leray Projected Transport Noise and Strong Solutions of the Stochastic Euler Equations
本文研究了三维环面上受传输或传输-拉伸Stratonovich噪声扰动的不可压缩Euler和Navier-Stokes方程，通过控制Leray投影破坏导数抵消带来的影响，首次证明了相应随机Euler方程局部强解的存在性及光滑解在爆破前的存在性。
A new proof of the Théorème de Structure related to a weak solution to the Navier-Stokes equations
本文通过新的证明方法，利用先验估计实现了Navier-Stokes方程弱解的结构定理，并扩展到IBVP情形。
Non-Smooth Solutions of the Navier-Stokes Equation and their Means
在有限周期立方体T3中构造非光滑Leray-Hopf解，通过熵产生最大化选择湍流初始数据，并利用弱解的解析性证明有限时间爆破。
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.
We prove the equivalence of being a Leray-Hopf weak solution to the Navier-Stokes equations in $\mathbb{R}^m, m \ge 3$, to satisfying a well known integral equation. We use this equation to derive some properties of these weak solutions.
Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations
该研究针对非齐次Navier-Stokes方程，利用相对能量方法和新的密度稳定性估计，证明了在远离真空的初始密度条件下，强解与Leray-Hopf弱解的唯一性。
First, we show that if the pressure $p$ (associated to a weak Leray-Hopf solution $v$ of the Navier-Stokes equations) satisfies $\|p\|_{L^{\infty}_{t}(0,T^*; L^{\frac{3}{2},\infty}(\mathbb{R}^3))}\leq M^2$, then $v$ possesses higher integrability up to the first potential blow-up time $T^*$. Our method is concise and is based upon energy estimates applied to powers of $|v|$ and the utilization of a `small exponent'. As a consequence, we show that if a weak Leray-Hopf solution $v$ first blows up at $T^*$ and satisfies the Type I condition $\|v\|_{L^{\infty}_{t}(0,T^*; L^{3,\infty}(\mathbb{R}^3)…
We study the incompressible Navier-Stokes equations with a fractional Laplacian and prove the existence of discontinuous Leray-Hopf solutions in the largest critical space with arbitrarily small initial data.
Energy Superposition and Regularity for 3D Navier-Stokes Equations in the Largest Critical Space
该研究提出能量叠加新方法，证明了最大临界空间中三维Navier-Stokes方程Leray-Hopf弱解的正则性，简化了已知正则准则的证明，方法可推广到其他超临界非线性PDE。
In this paper we consider classes of initial data that ensure local-in-time Hadamard well-posedness of the associated weak Leray-Hopf solutions of the three-dimensional Navier-Stokes equations. In particular, for any solenodial $L_{2}$ initial data $u_{0}$ belonging to certain subsets of $VMO^{-1}(\mathbb{R}^3)$, we show that weak Leray-Hopf solutions depend continuously with respect to small divergence-free $L_{2}$ perturbations of the initial data $u_{0}$ (on some finite-time interval). Our main result is inspired and improves upon previous work of the author \cite{barker2018} and work of Je…
We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to Hölder ($C^δ$, $δ>0$), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it do…
In this work, we examine two important problems in the theory of nonlinear PDEs. In Part I, we propose and solve a more general and complete version of the celebrated Leray's problem for the incompressible Navier-Stokes equations in $ \mathbb{R}^{3} \!$, which in its simplest form was suggested by J.$\;$Leray in 1934 (and solved only in the 1980s by T.$\;$ Kato, K.$\;$Masuda and other authors). A number of related new results of clear interest to the theory of Leray's solutions are also given here. In Part II, which is independent of Part I and can be read separately, we introduce an important…
If $u$ is a Leray-Hopf solution to the Navier-Stokes equations with the initial data in $ L^2(\mathbb{R}^3)$, then $u$ is regular.
This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $ε$ regularity criterion, we show that if the mild solution $u$ with initial data in $\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d) $, $d<p,\,q<\infty$ becomes singular at a finite time $T_*$, then $$ \limsup_{t\to T_*} \|u(t)\|_{\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d)} = \infty. $$ The corresponding result in 3D case has been obtained by I.Gallagher, G.S.KochandF.Planchon. As a by-product, we also prove a regularity criterion for the Le…
On the integrability properties of Leray-Hopf solutions of the Navier-Stokes equations on $\mathbb{R}^3$
本文刻画 $\mathbb{R}^3$ 上 Navier–Stokes 弱解在齐次 Banach 空间中的时间可积性及黏性一致界，排除一般初值的若干能量等式条件，并导出全局正则性的相关推论。
On Local Regularity of Distributional Solutions to the Navier--Stokes Equations
本文研究了纳维-斯托克斯方程分布解的局部正则性，证明满足Prodi-Serrin条件的解在空间变量上正则，且无需属于莱亚-霍夫类。
Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace"
本文系统梳理并补充了Leray 1934年关于Navier-Stokes方程的基础性工作，包括强解局部存在性、弱解全局存在性、弱-强唯一性及奇异时间集合的维数上界，以现代视角揭示其方法的完整性与优雅性。
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