Duy Phan, Sérgio S. Rodrigues
The Navier--Stokes system is considered in a compact Riemannian manifold. Gevrey class regularity is proven under Lions boundary conditions: in 2D for the Rectangle, Cylinder, and Hemisphere, and in 3D for the Rectangle. The cases of the 2D Sphere and 2D and 3D Torus are also revisited.
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In this paper we consider the inviscid limit for the periodic solutions to Navier-Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier-Stokes equation is independent of viscosity, and that the solutions of the Navier-Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover the convergence rate in Gevrey class is presented.
In this paper, we justify the limit from the Navier-Stokes system in a thin domain to the hydrostatic Navier-Stokes/Prandtl system for the convex initial data with Gevrey 9/8 regualrity in $x$.
Global Gevrey solution of 3D anisotropic Navier-Stokes system in a strip domain
针对条形区域中仅水平耗散的三维各向异性Navier-Stokes系统，通过在垂直方向施加Gevrey类正则性条件，证明解在其余方向具时空解析或Gevrey正则化，且强扩散方向有增强Gevrey正则性，克服边界项与垂直耗散缺失难题。
Optimal Gevrey stability of hydrostatic approximation for the Navier-Stokes equations in a thin domain
本文研究了薄区域中 Navier-Stokes 系统的流体静力近似，利用独立于 ε 的方法验证了在最优 Gevrey 指标 3/2 下的极限，并证明了系统的适定性，改进了已有结果。
In this paper, we study the asymptotic behavior of solutions to the three-dimensional incompressible Navier-Stokes equations (NSE) with periodic boundary conditions and potential body forces. In particular, we prove that the Foias-Saut asymptotic expansion for the regular solutions of the NSE in fact holds in {\textit{all Gevrey classes}}. This strengthens the previous result obtained in Sobolev spaces by Foias-Saut. By using the Gevrey-norm technique of Foias-Temam, the proof of our improved result simplifies the original argument of Foias-Saut, thereby, increasing its adaptability to other d…
On solutions for damped fractional Navier-Stokes equations in a critical Lei-Lin-Gevrey space
研究临界Lei-Lin-Gevrey空间中带三次阻尼的分数阶Navier-Stokes方程，建立了局部与小数据全局适定性、爆破准则、Gevrey半径损失估计及临界端点情形下的扰动稳定性。
We consider a bi-dimensional viscous incompressible fluid in interaction with a beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result where we supposed that the initial deformation of the beam was small. The main point of the proof consists in the study of the linearized system and in particular in proving that the corresponding semigroup is of Gevrey class.
On the decay and Gevrey regularity of the solutions to the Navier-Stokes equations in general two-dimensional domains
本文证明了一般二维区域中Navier-Stokes方程有限能量整体解任意阶导数的时间衰减估计，该估计仅依赖于导数阶数和初始数据的L2范数，并基于能量估计和Ladyzhenskaya不等式推导出Gevrey正则性结果。
We consider the hyperbolic version of three-dimensional anisotropic Naver-Stokes equations in a thin strip and its hydrostatic limit that is a hyperbolic Prandtl type equations. We prove the global-in-time existence and uniqueness for the two systems and the hydrostatic limit when the initial data belong to the Gevrey function space with index 2. The proof is based on a direct energy method by observing the damping effect in the systems.
Using the convex integration technique for the three-dimensional Navier-Stokes equations introduced by T. Buckmaster and V. Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier-Stokes equations with fractional hyperviscosity $(-Δ)^θ$, whenever the exponent $θ$ is less than J.-L. Lions' exponent $5/4$, i.e., when $θ< 5/4$.
For barotropic compressible Navier-Stokes equations in three-dimensional (3D) bounded domains, we prove that any finite energy weak solution obtained by Lions [Mathematical topics in fluid mechanics, Vol. 2. Compressible models(1998)] and Feireisl-Novotný-Petzeltová [J. Math. Fluid Mech. 3(2001), 358-392] decays exponentially to the equilibrium state. This result is established by both using the extra integrability of the density due to Lions and constructing a suitable Lyapunov functional just under the framework of Lions-Feireisl's weak solutions.
We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebr…
We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^ν(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrtν})\,, \, 0 \big )\, , \quad 0<ν\ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrtν$ then $u^ν$ is stable over some time interval $(0,T)$, $T$ independent of $ν$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic c…
Lions (1959, Bull. Soc. Math. France, \textbf{87}, 245--273) introduced the Navier-Stokes equations with a viscous diffusion in the form of a fractional Laplacian; subsequently, he (1969, Dunod, Gauthiers-Villars, Paris) claimed the uniqueness of its solution when its exponent is not less than five quarters in case the spatial dimension is three. Following the work of Hofmanov$\acute{\mathrm{a}}$, Zhu and Zhu (2019, arXiv:1912.11841 [math.PR]), we prove the non-uniqueness in law for the three-dimensional stochastic Navier-Stokes equations with the viscous diffusion in the form of a fractional …
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 investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations, which is obtained by using Cattaneo type law instead of Fourier law, evolving in a thin strip $\R\times (0,\varepsilon)$. The formal limit of these equations is a hyperbolic Prandtl type equation. We first prove the global existence of solutions to these equations under a uniform smallness assumption on the data in Gevrey $2$ class. Then we justify the limit globally-in-time from the anisotropic hyperbolic Navier-Stokes system to the hyperbolic Prandtl system with such Gevrey $2$ class data. Compar…
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…
In this paper we prove, if $u$ is a global solution to Navier-Stokes equations in the Sobolev-Gevrey spaces $H^1_{a,σ}(\mathbb R^3)$, then $\|u(t)\|_{H^1_{a,σ}}$ decays to zero as time goes to infinity. Fourier analysis is used.
An explicit saturating set consisting of eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The existence of saturating sets implies the approximate controllability for Navier--Stokes equations in $\rm 3D$ Cylinders under Lions boundary conditions.
Gevrey regularity and analyticity for the solutions of the Vlasov-Navier-Stokes system
本文采用傅里叶空间方法，证明了Vlasov-Navier-Stokes系统解的$\frac{1}{s}$-Gevrey正则性与解析性的传播性，给出定量估计并推导得出对应解的存在性及全局Gevrey解的存在条件。
The existence of dissipative solutions to the compressible isentropic Navier-Stokes equations was established in this paper. This notion was inspired by the concept of dissipative solutions to the incompressible Euler equations of Lions (\cite{Lions-1996}, Section 4.4). Our method is to recover such solutions by passing to the limits from approximated solutions, thanks to compactness argument.
Nonlinearly Exponential Stability for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations with Large Potential External Forces
本文针对三维有界区域中带大位势外力的正压可压缩Navier-Stokes方程，通过构造Lyapunov泛函并利用密度额外可积性与Taylor展开，证明了Lions-Feireisl有限能量弱解随时间指数衰减到平衡态。
In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global …
The set of nonzero external forces for which the zero function is in the global attractor of the 2D Navier-Stokes equations is shown to be meagre in a Fréchet topology. A criterion in terms of a Taylor expansion in complex time is used to characterize the forces in this set. This leads to several relations between certain Gevrey subclasses of $C^{\infty}$ and a new upper bound for a Gevrey norm of solutions in the attractor, valid in the strip of analyticity in time.
Remarks on the three-dimensional Navier-Stokes equations with Lions' exponent forced by space-time white noise
本文研究受时空白噪声驱动且处于能量临界情形的三维Navier-Stokes方程，沿用Hairer和Rosati的方法证明了其全局解理论。
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。
In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in "critical" Sobolev spaces. Moreover, the Gevrey class that we obtain is "near optimal" and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier-Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the…
This paper is about Lions' open problem on density patches \cite{LIONS}: whether inhomogeneous incompressible Navier-Stokes equations preserve the initial regularity of the free boundary given by density patches. Using classical Sobolev spaces for the velocity, we first establish the propagation of $C^{1+γ}$ regularity with $0<γ<1$ in the case of positive density. Furthermore, we go beyond to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for $C^{2+γ}$ regularity.
This work addresses an extension of the Aubin-Lions-Simon compactness result to generalized Bochner spaces $L^2(0,T;H(t))$, where $H(t)$ is a family of Hilbert spaces, parameterized by $t$. A compactness result of this type is needed in the study of the existence of weak solutions to nonlinear evolution problems governed by partial differential equations defined on moving domains. We identify the conditions on the regularity of the domain motion in time under which our extension of the Aubin-Lions-Simon compactness result holds. Concrete examples of the application of the compactness theorem a…
In this paper, we derive an energy conservation criterion based on a combination of velocity and its gradient for the weak solutions of both the homogeneous incompressible Navier-Stokes equations and the general compressible Navier-Stokes equations. For the incompressible case, this class implies most known corresponding results on periodic domain via either the velocity or its gradient including the famous Lions' energy conservation criterion obtained in \cite{[Lions]}. For the compressible case, this helps us to extend the previously known criteria for the energy conservation of weak solutio…
On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations
本文在初始密度仅有界、初始速度属于 $H^1(\mathbb{R}^2)$ 的一般条件下构造了二维非齐次Navier-Stokes方程的全局强解，并证明Lions弱解与强解一致，从而完整解决了Lions提出的密度斑块问题。
In this paper we give a new proof to the energy conservation for the weak solutions of the incompressible Navier-Stokes equations. This result was first proved by Shinbrot. The new proof relies on a lemma introduced by Lions.
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。
Recently, the Navier-Stokes-Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier-Stokes equations for the purpose of direct numerical simulations. In this work we prove that the global attractor of the $D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one…
Gevrey well-posedness of quasi-linear hyperbolic Prandtl equations
针对双曲型 Prandtl 方程拟线性项引发的导数损失与不稳定性，本文引入新辅助函数，证明其在切向 Gevrey 3/2、法向解析的各向异性空间中适定。
It is well-known that a Leray-Hopf weak solution in $L^4 (0,T; L^4(Ω))$ for the incompressible Navier-Stokes system is persistence of energy due to Lions [19]. In this paper, it is shown that Lions's condition for energy balance is also valid for the weak solutions of the isentropic compressible Navier-Stokes equations allowing vacuum under suitable integrability conditions on the density and its derivative. This allows us to establish various sufficient conditions implying energy equality for the compressible flow as well as the non-homogenous incompressible Navier-Stokes equations. This is a…
We show that solutions to a large class of inviscid equations, in Eulerian variables, extend as holomorphic functions of time, with values in a Gevrey class (thus space-analytic), and are solutions of complexified versions of the said equations. The class of equations we consider includes those of fluid dynamics such as the Euler, surface quasi-geostrophic, Boussinesq and magnetohydrodynamic equations, as well as other equations with analytic nonlinearities. The initial data are assumed to belong to a $\mathit{Gevrey\:class}$, i.e., analytic in the space variable. Our technique follows that of…
We present a technique for derivation of a priori bounds for Gevrey-Sobolev norms of space-periodic three-dimensional solutions to evolutionary partial differential equations of hydrodynamic type. It involves a transformation of the flow velocity in the Fourier space, which introduces a feedback between the index of the norm and the norm of the transformed solution, and results in emergence of a mildly dissipative term. To illustrate the technique, we derive finite-time bounds for Gevrey-Sobolev norms of solutions to the Euler and inviscid Burgers equations, and global in time bounds for the V…
We address the local well-posedness of the hydrostatic Navier-Stokes equations. These equations, sometimes called reduced Navier-Stokes/Prandtl, appear as a formal limit of the Navier-Stokes system in thin domains, under certain constraints on the aspect ratio and the Reynolds number. It is known that without any structural assumption on the initial data, real-analyticity is both necessary and sufficient for the local well-posedness of the system. In this paper we prove that for convex initial data, local well-posedness holds under simple Gevrey regularity.
Gevrey regularity solution for initial data in Triebel-Lizorkin-Lorentz spaces via single norm defined by nonlinearity of frequency
本文通过引入单一范数迭代空间并利用 Fefferman-Stein 不等式，在临界 Triebel-Lizorkin-Lorentz 空间中建立了 Navier-Stokes 方程的全局适定性与 mild 解的 Gevrey 正则性，推广了已有的 Besov 空间结果。
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.
Blow-Up of Modified Navier--Stokes Equations in Nonhomogeneous Fourier Spaces
本文研究带非线性阻尼项的三维不可压缩Navier--Stokes方程，在Fourier和Fourier--Gevrey空间中建立局部适定性，推导爆破准则，并给出解在极大时间附近的定量下界。
On Lions' density patch problem at a critical level of regularity
本文在临界正则性下研究了Lions密度斑问题，证明了流体在特定初始条件下的全局存在性、唯一性与稳定性，并展示了斑的Lipschitz正则性及长时间动力学行为。
In this paper, we consider the zero-viscosity limit of the Navier-Stokes equations in a half space with the Navier friction boundary condition $$ (βu^{\varepsilon}-\varepsilon^γ\partial_y u^{\varepsilon})|_{y=0}=0, $$ where $β$ is a constant and $γ\in (0,1]$. In the case of $γ=1$, the convergence to the Euler equations and the Prandtl equation with the Robin boundary condition is justified for the analytic data. In the case of $γ\in (0,1)$, the convergence to the Euler equations and the linearized Prandtl equation is justified for the data in the Gevrey class $\frac 1 γ$.
Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density
本文解决了Lions关于2-D非齐次Navier-Stokes方程弱解唯一性的开放问题，证明了初始密度有界条件下的全局存在性与唯一性，并将Fujita-Kato的3-D结果推广至INS方程。
Pseudo-Gevrey Smoothing for the Passive Scalar Equations near Couette
本文研究接近 Couette 流的被动标量方程和 Poisson 方程，在退化伪 Gevrey 空间中建立了关于扩散系数一致的光滑化估计，为非线性 Navier-Stokes 稳定性分析提供关键技术工具。
On Stationary Gevrey Solutions to the Gravitational Boussinesq System and Applications to Uniqueness
本文研究了引力Boussinesq系统在一般重力加速度项下的平稳解，证明了弱解的存在性，并展示Gevrey类正则性可用于研究解的唯一性，同时获得Navier-Stokes方程的新正则性结果和Liouville型结论。
We show the $H^1$ stability of shear flows of Prandtl type: $U^ν= (U_s(y/\sqrtν),0)$, in the steady two-dimensional Navier-Stokes equations, under the natural assumptions that $U_s(Y) > 0$ for $Y > 0$, $U_s(0) = 0$, and $U_s'(0) > 0$. Our result is in sharp contrast with the unsteady ones, in which at most Gevrey stability can be obtained, even under global monotonicity and concavity hypotheses. It provides the first positive answer to the inviscid limit problem in Sobolev regularity for a non-trivial class of steady Navier-Stokes flows with no-slip boundary condition.
The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore,…
In this paper, we study the global well-posedness of a coupled system of kinetic and fluid equations. More precisely, we establish the global existence of weak solutions for Navier-Stokes-BGK system consisting of the BGK model of Boltzmann equation and incompressible Navier-Stokes equations coupled through a drag forcing term. This is achieved by combining weak compactness of the particle interaction operator based on Dunford-Pettis theorem, strong compactness of macroscopic fields of the kinetic part relied on velocity averaging lemma and a high order moment estimate, and strong compactness o…
In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation. The main contribution of this paper is to derive the Mellet-Vasseur type inequality for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible Navier-Stokes equations, for any $γ>1$, in three dimensional space, with large initial data possibly vanishing on the vacuum. This solves an open pro…
We approximate a two--phase model by the compressible Navier-Stokes equations with a singular pressure term. Up to a subsequence, these solutions are shown to converge to a global weak solution of the compressible system with the congestion constraint studied for instance by P.L. Lions and N. Masmoudi [ Annales I.H.P., 1999]. The paper is an extension of the previous result obtained in one-dimensional setting by D. Bresch et al. [ C. R. Acad. Sciences Paris, 2014] to the multi-dimensional case with heterogeneous barrier for the density.
In this paper, we investigate the time decay behavior to Lions weak solution of 2D incompressible density-dependent Navier-Stokes equations with variable viscosity.
Smooth solutions of the stationary Navier-Stokes equations in an infinitely long pipe, equipped with the Navier-slip or Navier-Hodge-Lions boundary condition, are considered in this paper. Three main results are presented. First, when equipped with the Navier-slip boundary condition, it is shown that, $W^{1,\infty}$ axially symmetric solutions with zero flux at one cross section, must be swirling solutions: $u=(- C x_2, C x_1,0)$, and $x_3-$periodic solutions must be helical solutions: $u=(-C_1x_2,C_1x_1,C_2)$. Second, also equipped with the Navier-slip boundary condition, if the swirl or vert…
Space-time analyticity and refined analyticity radius of the Navier-Stokes equations in the critical Besov spaces
本文证明了临界Besov空间中小初始数据的不可压缩Navier-Stokes方程全局解的时空解析性，结合Gevrey类估计与迭代导数技术推导了高阶导数衰减率与解析半径下界，并获得了任意初始数据下空间解析半径下界的对数改进。
Classical Sobolev approach for a critical fourth-order Leray-Lions type problem: existence and multiplicity of solutions
本文研究Navier和Dirichlet边界条件下具Sobolev临界增长的Leray-Lions型四阶椭圆问题，结合变分方法和临界点理论，在经典Sobolev空间中证明次线性扰动下解的存在性与超线性扰动下至少一个非平凡解的存在性，并推广至哈密顿系统。
In this paper we study how to approximate the Leray weak solutions of the incompressible Navier Stokes equation. In particular we describe an hyperbolic version of the so called artificial compressibility method investigated by J.L.Lions and Temam. By exploiting the wave equation structure of the pressure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a Leray weak soluti…
The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier-Stokes system. The Marcinkiewicz space $L^{3,\infty}$ is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical ``regularized'' Navier-Stokes systems. The first one was introduced by J. Leray and consists in ``mollifying'' the nonlinearity. The second one was proposed by J.L. Lions, who added the artificial hyper-viscosity $(-Δ)^{\ell/2}$, $\ell>2$, to the mode…
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 $1$.
In this paper, we draw on the ideas of [5] to extend the standard Serrin criterion [17] to an anisotropic version thereof. Because we work on weak solutions instead of strong ones, the functions involved have low regularity. Our method summarizes in a joint use of a uniqueness lemma in low regularity and the existence of stronger solutions. The uniqueness part uses duality in a way quite similar to the DiPerna-Lions theory, first developed in [7]. The existence part relies on L p energy estimates, whose proof may be found in [5], along with an approximation procedure.
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