Nikolai Filonov, Pavel Khodunov
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.
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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.
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.
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。
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.
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。
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-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。
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。
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 consider the solutions of the Cauchy problem for a dyadic model of Euler equations. We prove global existence and uniqueness of Leray-Hopf solutions in a rather large class K that implies in particular global existence and uniqueness in l^2 for all initial positive conditions in l^2.
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…
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.
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。
An upper bound of blow up rate for the Navier-Stokes equations with small data in L^2(R^3) is obtained.
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。
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。
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…
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。
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。
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 $L^2$.
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。
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 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)…
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。
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.
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。
The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on $\R^{2}$. In our previous work we showed the breakdown of uniqueness in a hyperbolic setting. In this article, we show how to formulate the problem in order so the uniqueness can be restored.
We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent $α$ can be larger than the Lions exponent $5/4$. It is well-known that, due to Lions [55], for any $L^2$ divergence-free initial data, there exist unique smooth Leray-Hopf solutions when $α\geq 5/4$. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces $L^γ_tW^{s,p}_x$, in view of the generalized Ladyženskaja-Prodi-Serrin condition. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints $(3/p…
A classical result of J.-L. Lions asserts that if a solution to the Navier-Stokes equations is such that: (i) it is in the Leray-Hopf class, and (ii) belongs to $L^4(0,T;L^4)$, then it must satisfy the energy equality in the time interval $[0,T]$. In this note we show that assumption (i) is not necessary.
We consider a sequence of Leray-Hopf weak solutions of the 2D Navier-Stokes equations on a bounded domain, in the vanishing viscosity limit. We provide sufficient conditions on the associated vorticity measures, away from the boundary, which ensure that as the viscosity vanishes the sequence converges to a weak solution of the Euler equations. These assumptions are consistent with vortex sheet solutions of the Euler equations.
A variational approach to the Navier-Stokes equations with shear-dependent viscosity
提出变分方法构造非牛顿Navier-Stokes系统的Leray-Hopf解，通过最小化WIDE泛函并取极限，将幂律指数结果扩展至弱解不满足能量等式的范围，并提供椭圆-抛物线螺线Lipschitz截断处理非线性粘度项极限。
We show that if a Leray-Hopf solution $u$ to the 3D Navier-Stokes equation belongs to $C((0,T]; B^{-1}_{\infty,\infty})$ or its jumps in the $B^{-1}_{\infty,\infty}$-norm do not exceed a constant multiple of viscosity, then $u$ is regular on $(0,T]$. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya-Prodi-Serrin criterion.
Layer separation of the 3D incompressible Navier-Stokes equation in a bounded domain
本文研究了光滑有界区域内三维不可压Navier-Stokes方程的边界层分离，为低粘性Leray-Hopf解与正则Euler解间的差异提供了无条件的$L^2$上界，并推广了先前有限通道中的结果。
When a Leray--Hopf weak solution to the NSE has a singularity set $S$ of dimension $d$ less than $3$---for example, a suitable weak solution---we find a family of new $L^q L^p$ conditions that guarantee validity of the energy equality. Our conditions surpass the classical Lions--Ladyženskaja $L^4 L^4$ result in the case $d<1$. Additionally, we establish energy equality in certain cases of Type-I blowup. The results are also extended to the NSE with fractional power of the Laplacian below $1$.
We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in small cylinders. As a special case, our results imply Caccioppoli's type estimates for the Stokes systems with variable coefficients. A new $ε$-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations is also obtained as a consequence of our regularity results, which in turn implies some borderline cases of the well-known Serrin's regular…
An important open problem in the theory of the Navier-Stokes equations is the uniqueness of the Leray-Hopf weak solutions with $L^2$ initial data. In this paper we give sufficient conditions for non-uniqueness in terms of spectral properties of a natural linear operator associated to scale-invariant solutions recently constructed in \cite{JiaSverak}. If the spectral conditions are satisfied, non-uniqueness and ill-posedness can appear for quite benign compactly supported data, just at the borderline of applicability of the classical perturbation theory. The verification of the spectral conditi…
Consider any Leray-Hopf weak solution of the three-dimensional Navier-Stokes equations for incompressible, viscous fluid flows. We prove that any Lagrangian trajectory associated with such a velocity field has an asymptotic expansion, as time tends to infinity, which describes its long-time behavior very precisely.
Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound \[ |v(x, t)| \le \frac{C}{r^2} |\ln r|^{1/2}, \]where $r \in (0, 1/2)$ is the distance from $x$ to the z axis, and $C$ is a constant depending only on the initial value. This provides a pointwise upper bound (worst case scenario) for possible singularities while the recent papers \cite{CSTY2} and \cite{KNSS} gave a lower bound. The gap is polynomial order 1 modulo a half log term.
It is shown both locally and globally that $L_t^{\infty}(L_x^{3,q})$ solutions to the three-dimensional Navier-Stokes equations are regular provided $q\not=\infty$. Here $L_x^{3,q}$, $0<q\leq\infty$, is an increasing scale of Lorentz spaces containing $L^3_x$. Thus the result provides an improvement of a result by Escauriaza, Seregin and {\v S}verák ((Russian) Uspekhi Mat. Nauk {\bf 58} (2003), 3--44; translation in Russian Math. Surveys {\bf 58} (2003), 211--250), which treated the case $q=3$. A new local energy bound and a new $ε$-regularity criterion are combined with the backward uniquenes…
In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the three-dimensional magneto-hydrodynamic (MHD) system. More precisely, we show that any weak solution $(v,b)\in L^p_tL^{\infty}_x$ is non-unique in $L^p_tL^{\infty}_x$ with $1\le p<2$, which reveals the strong non-uniqueness, and the sharpness in terms of the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint $(2, \infty)$. Moreover, for any $1\le p<2$ and $ε>0$, we construct non-Leray-Hopf weak solutions in $L^p_tL^{\infty}_x\cap L^1_tC^{1-ε}$. The results of Navier-Stokes equations in \cite{1C…
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.
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。
Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier-Stokes equations perturbed by transport noise
本文证明了受Stratonovich输运噪声扰动的三维Navier-Stokes方程柯西问题存在全局的概率强解析弱解，且解不唯一；可指定任意无散度平方可积初值或以高概率任意大的停时前动能，解具有Sobolev正则性但非Leray-Hopf。
Non-uniqueness of three-dimensional Euler equations and Navier-Stokes equations forced by random noise, path-wise and more recently even in law, have been proven by various authors. We prove non-uniqueness in law of the three-dimensional Navier-Stokes equations forced by random noise and diffused via a fractional Laplacian that has power between zero and one half. The solution we construct has H$\ddot{\mathrm{o}}$lder regularity with a small exponent rather than Sobolev regularity with a small exponent. For the power sufficiently small, the non-uniqueness in law holds at the level of Leray-Hop…
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