Jian Zhai
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.
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Limit behaviors of blow up solutions for impressible Navier-Stokes equations are obtained.
An upper bound of blow up rate for the Navier-Stokes equations with small data in L^2(R^3) is obtained.
In this paper, we establish the existence and the regularity of a unique weak solution to turbulent flows in a bounded domain $Ω\subset \mathbb R^3$ governed by the so-called Leray-α model. We consider the Navier slip boundary conditions for the velocity. Furthermore, we show that, when the filter coefficient α tends to zero, the weak solution constructed converges to a suitable weak solution to the incompressible Navier Stokes equations subject to the Navier boundary condition. Similarly, if λ tends to 1- we recover a solution to the Leray-α model with the homogeneous Dirichlet boundary condi…
In this paper we study the Leray weak solutions of the incompressible Navier Stokes equation in an exterior domain.We describe, in particular, an hyperbolic version of the so called artificial compressibility method investigated by J.L.Lions and Temam. The convergence of these type of approximation show in general a lack of strong convergence due to the presence of acoustic waves. In this paper we face this difficulty by taking care of the dispersive nature of these waves by means of the Strichartz estimates or waves equations satisfied by the pressure. We actually decompose the pressure in di…
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.
Very Weak Solutions and Asymptotic Behavior of Leray Solutions to the Stationary Navier-Stokes Equations
本文研究外区域定常Navier-Stokes方程的Leray解，提出基于极弱解新理论的方法，证明满足特定球面范数条件时解在无穷远以$|x|^{-1}$衰减，相关极弱解理论具有独立价值。
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.
We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus ${\mathbb T}^d$, assuming that the solutions have norms for Besov space $B^{σ,\infty}_3({\mathbb T}^d),$ $σ\in (0,1],$ that are bounded in the $L^3$-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form $O(ν^{(3σ-1)/(σ+1)}),$ vanishing as $ν\to0$ if $σ>1/3.$ A consequence is that Onsager-type "quasi-singularities" are required in the Leray solutions, even if the total energy dissipation vanishes in the limit $ν\to 0$, as lon…
Based on Leray's formulation of the Navier-Stokes equations and the conditions of the exact linear representation of the nonlinear problem found in this paper, a compact explicit expression for the exact operator solution of the Navier-Stokes equations is given. It is shown that the introduced linear operator for Leray's equations is the generator of one-parameter contraction semigroup. This semigroup yields the existence of a unique and smooth classical solution of the associated Cauchy problem of Navier-Stokes equations in space $\mathbb{R}^3$ under smooth initial conditions.
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].
We rule out the existence of Leray's backward self-similar solutions to the Navier--Stokes equations with profiles in $L^{12/5}(\R^3)$ or in the Marcinkiewicz space $L^{q,\infty}(\R^3)$ for $q\in(12/5, 6)$. This follows from a more general result formulated in terms of Morrey spaces and the first order Riesz's potential.
We consider Leray's problem on stationary Navier-Stokes flows with arbitrary large fluxes in an unbounded cylinder with several exits to infinity. For a stationary Navier-Stokes flow with large fluxes in the unbounded cylinder we prove that, if the difference between the pressure of the main flow and the pressure of the Poiseuille flow with the same flux in a branch of the cylinder satisfies some asymptotic boundedness condition at $|x|\rightarrow \infty$ then the flow behaves at infinity of the branch like the Poiseuille flow.
We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian $(-Δ)^α$ with exponent $α< \frac{1}{5}$. The proof follows the ''convex integration methods'' introduced by the second author and László Székelyhidi Jr. for the incomprresible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$.
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。
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…
In the seminal work [39], Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a `background' solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik in [43,44]. The second solution is a trajectory on the unstable manifold asso…
A remark on very weak suitable solutions and Leray solutions of the Navier-Stokes equations
介绍纳维 - 斯托克斯方程非常弱适解概念，利用局部能量不等式在局部莫雷空间框架下建立确保非常弱解是勒雷解的条件及相关结论。
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。
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.
Non-uniqueness of Leray solutions to the hypodissipative Navier-Stokes equations in two dimensions
该研究构造了二维带外力低耗散Navier-Stokes方程的超临界标度非唯一Leray解，采用拟线性微扰方法将Vishik的Euler方程非唯一性结果推广至低耗散情形，规避了已有研究的谱理论方法。
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。
The original Leray's problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe, which approach to the Poiseuille flow subject to the no-slip boundary condition at spacial infinity. In this paper, the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition, is addressed. Due to the complexity of the boundary condition, some new ideas, presented as follows, are introduced to handle the extra difficulties caused by boundary terms. First, the Poiseuille flow in the semi-infinite straight pip…
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.
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。
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 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…
In this small note we strengthen the classic result about the regularity time t* of arbitrary Leray solutions to the (incompressible) Navier-Stokes equations in Rn (n = 3, 4), which have the form: t* <= K_{3} nu^{-5} || u(.,0) ||_{L2}^{4} if n = 3, and t* <= K_{4} nu^{-3} || u(.,0) ||_{L2}^{2} if n = 4 (in particular, by reducing the current best known values for the constants K_{3}, K_{4}). Some related results of clear interest are also included (derived) in our discussion.
We show that t^{3/4}|| u(.,t) ||_{sup} --> 0 as t --> infty for all (global) Leray solutions of the incompressible Navier-Stokes equations in R3. It is also shown that t || u(.,t) - v(.,t) ||_{sup} --> 0 as t --> infty, where v(.,t) is the Stokes approximation, as well as other fundamental results. In spite of the difficulty of these questions, our approach is elementary and is based on standard tools like conventional Fourier and energy methods.
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.
Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of the Navier-Stokes solutions in two dimensions have been known for a long time. Leray $\cite{jL34}$ showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer and Shnirelman obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli-Kohn-Nirenberg improved Scheffer's results, and F.-H. Lin simplified the proof of the results of J. Leray. Many problems and conjectures about the behavior of solutio…
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…
We establish existence of solutions in a scale of classes weaker than the finite energy Leray class and stronger than the infinite energy Lemarié-Rieusset class. The new classes are based on the $L^2$ Wiener amalgam spaces. Solutions in the classes closer to the Leray class are shown to satisfy some properties known in the Leray class but not the Lemarié-Rieusset class, namely eventual regularity and long time estimates on the growth of the local energy. In this sense, these solutions bridge the gap between Leray's original solutions and Lemarié-Rieusset's solutions and help identify scalings …
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。
Constrained large solutions to Leray's problem in a distorted strip with the Navier-slip boundary condition
本文研究了二维无限扭曲带中带Navier-slip边界条件的定常Navier-Stokes系统Leray问题，证明了解的存在性、唯一性、正则性和渐近行为，并发现小摩擦系数下速度总通量可相对较大。
In this paper, we introduce the concept of local Leray solutions starting from a locally square-integrable initial data to the fractional Navier-Stokes equations with $s\in [3/4,1)$. Furthermore, we prove its local in time existence when $s\in (3/4, 1)$. In particular, if the locally square-integrable initial data vanishs at infinity, we show that the fractional Navier-Stokes equations admit a global-in-time local Leray solution when $s\in [5/6, 1)$. For such local Leray solutions starting from locally square-integrable initial data vanishing at infinity, the singularity only occurs in $B_R(0)…
In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution ${\mathbf u}$ of 3D-NSE and each $(s,x)\in\mathbb{R}_+\times\mathbb{R}^3$, we show the existence of weak solutions to the following SDE, which has a density $ρ_{s,x}(t,y)$ belonging to $\mathbb{H}^{1,p}_q$ provided $p,q\in[1,2)$ with $\frac{3}{p}+\frac{2}{q}>4$: $$ \mathrm{d} X_{s,t}={\mathbf u} (s,X_{s,t})\mathrm{d} t+\sqrt{2ν}\mathrm{d} W_t,\ \ X_{s,s}=x,\ \ t\geq s, $$ where $W$ is a three dimensional standard Brownian …
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 $3…
It has recently become common to study many different approximating equations of the Navier-Stokes equation. One of these is the Leray-$α$ equation, which regularizes the Navier-Stokes equation by replacing (in most locations) the solution $u$ in the equation with $(1-α^2\triangle)u$ the operator $(1-α^2\triangle)$. Another is the generalized Navier-Stokes equation, which replaces the Laplacian with a Fourier multiplier with symbol of the form $|ξ|^γ$ ($γ=2$ is the standard Navier-Stokes equation), and recently in [14] Tao also considered multipliers of the form $|ξ|^γ/g(|ξ|)$, where $g$ is (e…
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 $3/2<q<6$, or in $\dot{\mathcal{M}}^{q,l}(\mathbb{R}^{3})$ provided $6\leq q<\infty$ and $2<l\leq q$. This generalizes the corresponding results obtained by Nečas-Råužička-Šverák [19, Acta.Math. 176 (1996)] in $L^{3}(\mathbb{R}^{3})$, Tsai [25, Arch. Ration. Mech. Anal. 143 (1998)] in $L^{p}(\mathbb{R}^{3})$ with $p\geq3$,, Chae-Wolf [3, Arch. Ration. Mech. Anal. 225 (2017)] …
A refined uniqueness result of Leray's problem in an infinite-long pipe with the Navier-slip boundary
该研究针对带Navier滑移边界的无限长管道广义Leray问题，证明通量不大于与摩擦比无关的临界值时解为平行Poiseuille流，给出单位圆盘下临界通量下界，改进了相关唯一性结论。
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。
This paper aims to study a family of Leray-$α$ models with periodic bounbary conditions. These models are good approximations for the Navier-Stokes equations. We focus our attention on the critical value of regularization "$θ$" that garantees the global well-posedness for these models. We conjecture that $θ= 1/4$ is the critical value to obtain such results. When alpha goes to zero, we prove that the Leray-$α$ solution, with critical regularization, gives rise to a suitable solution to the Navier-Stokes equations. We also introduce an interpolating deconvolution operator that depends on "$θ$".…
We study the uniqueness, the continuity in $L^2$ and the large time decay for the Leray solutions of the $3D$ incompressible Navier-Stokes equations with nonlinear exponential damping term $a (e^{b |u|^{\bf 4}}-1)u$, ($a,b>0$).
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.
Homogenization of Leray's flux problem for the steady-state Navier-Stokes equations in a multiply-connected planar domain
本文研究带小孔多连通平面域内粘性不可压缩流体稳态运动的Navier-Stokes方程均匀化问题，通过能量法和基于伯努利定律的反证法得到一致界，证明孔洞趋于零时等效方程不变。
In this paper, we establish the global well-posedness of stochastic 3D Leray-$α$ model with general fractional dissipation driven by multiplicative noise. This model is the stochastic 3D Navier-Stokes equation regularized through a smoothing kernel of order $θ_1$ in the nonlinear term and a $θ_2$-fractional Laplacian. In the case of $θ_1 \ge 0, θ_2 > 0$ and $θ_1+θ_2 \geq\frac{5}{4}$, we prove the global existence and uniqueness of strong solutions. The main results cover many existing works in the deterministic cases, and also generalize some known results of stochastic models as our special c…
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