Alexei A. Ilyin, Edriss S. Titi
We obtain a logarithmically sharp estimate for the space-analyticity radius of the solutions of the damped-driven 2D Navier-Stokes equations with periodic boundary conditions and relate this to the small scales in this system. This system is inspired by the Stommel--Charney barotropic ocean circulation model.
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In this paper, we show the optimal decay rate estimates of the space-time derivatives and the joint space-time analyticity of solutions to the Navier-Stokes equations. As it is known from the Hartogs's theorem, for a complex function with two complex variables, the joint analyticity with respect to two variables can be derived from combining of analyticity with respect to each variable. However, as a function of two real variables for space and time, the joint space-time analyticity of solutions to the Navier-Stokes equations cannot be directly obtained from the combination of space analyticit…
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。
By using a new bilinear estimate, a pointwise estimate of the generalized Oseen kernel and an idea of fractional bootstrap, we show in this note that solutions to the Navier-Stokes equations with fractional dissipation are analytic in space variables.
We study spatial analyticity properties of solutions of the Navier-Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier-Stokes equations with data in H^r, r greater or equal 1/2, and prove a stability result for the analyticity radius.
We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that the solution and the source term have the exponential growth of order $2$ with respect to the space variables. We also obtain the time analyticity for bounded mild solutions of the incompressible Navier-Stokes equations in the half space with the Dirichlet boundary condition. Our work is an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (20…
Lei and Lin have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae, and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case.
On the space-time analyticity of the Keller-Segel-Navier-Stokes system
本文研究周期条带上耦合Keller-Segel-Navier-Stokes系统，通过能量方法证明了解直到边界的同步时空解析性。
Improved regularity and analyticity of Cannone-Karch solutions of the three-dimensional Navier-Stokes equations on the torus
该研究针对初值二阶导数在伪测度空间的三维环面Navier-Stokes方程，改进了Cannone-Karch解的正时刻正则性增益，还证明了解在所有正时刻解析并给出解析半径下界。
On persistence of spatial analyticity in the hyper-dissipative Navier-Stokes models
本研究针对超耗散Navier-Stokes模型，证明当超扩散指数大于1时可排除一类有限时间爆破情形，论证基于Grujic与Xu的湍流正则性理论思路。
We study the Cauchy problem for the (generalized) incompressible Navier-Stokes equations \begin{align} u_t+(-Δ)^αu+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0. \nonumber \end{align} We show the analyticity of the local solutions of the Navier-Stokes equation ($α=1$) with any initial data in critical Besov spaces $\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$ with $1< p<\infty, \ 1\le q\le \infty $ and the solution is global if $u_0$ is sufficiently small in $\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$. In the case $p=\infty$, the analyticity for the local solutions of the Navier-Stokes equ…
In this article, we study the solutions of the damped Navier--Stokes equation with Navier boundary condition in a bounded domain $Ω$ in $\mathbb{R}^3$ with smooth boundary. The existence of the solutions is global with the damped term $\vartheta |u|^{β-1}u, \vartheta >0.$ The regularity and uniqueness of solutions with Navier boundary condition is also studied. This extends the existing results in literature.
It is proved that there exists a local-in-time solution $u\in C([0,T),bmo(\mathbb{R}^d)^d)$ of the Navier-Stokes equations such that every $u(t)$ has an analytic extension on a complex domain whose size only depends on $t$ (and increases with $t$) and the external force $f$, assuming only that the initial velocity $u_0$ is a local $BMO$ function. Our method for proving is a combination and refinement of the work by Grujić and Kukavica [13], Guberović [15] and Kozono et al. [18]. One challenging step is the estimation of the heat and Stokes semigroups from $BMO$-type spaces to $L^\infty$; a res…
Analyticity in space and time for global solutions to the anisotropic Navier--Stokes equations in the critical $L^p(\mathbb{R}^3)$ framework
本文在临界 $L^p(\mathbb{R}^3) $ 框架下研究了各向异性纳维-斯托克斯方程全局解的空间和时间分析性，证明了在特定初始条件下的唯一性和解析性。
We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order $2$ on $\M$. Here $\M$ is $\R^d$ or a complete noncompact manifold with Ricci curvature bounded from below by a constant. An implication is a sharp solvability condition for the Cauchy problem of the backward heat equation, which is a well known ill-posed problem. Another implication is a sharp criteria for time analyticity of solutions down to the initial time. The other pe…
We consider the damped and driven Navier--Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain $Ω\subset\mathbf{R}^2$. We show that the damped Euler system has a (strong) global attractor in~$H^1(Ω)$. We also show that in the vanishing viscosity limit the global attractors of the Navier--Stokes system converge in the non-symmetric Hausdorff distance in $H^1(Ω)$ to the the strong global attractor of the limiting damped Euler system (whose solutions are not necessarily unique).
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 consider the inviscid limit of the stochastic damped 2D Navier- Stokes equations. We prove that, when the viscosity vanishes, the stationary solution of the stochastic damped Navier-Stokes equations converges to a stationary solution of the stochastic damped Euler equation and that the rate of dissipation of enstrophy converges to zero. In particular, this limit obeys an enstrophy balance. The rates are computed with respect to a limit measure of the unique invariant measure of the stochastic damped Navier-Stokes equations.
Spatial analyticity and exponential decay of Fourier modes for the stochastic Navier-Stokes equation
该研究构造了仅需初始涡量有界的二维、三维随机Navier-Stokes方程局部时间空间实解析解，还证明了解全局存在时有限维Galerkin近似的指数衰减性，且该衰减在时间、初始涡量、噪声系数上均一致。
We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when the forcing term belongs to the whole scale of homogeneous Sobolev spaces from -1 to 1
Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces
本文研究零声速下可压缩Navier-Stokes-Korteweg方程初值问题，得到常平衡态附近的全局时间解析解，建立标度临界空间中的Lp-L1型衰减估计，并推导带高阶导数的一阶渐近公式。
We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in ${\mathbb R}^2$. We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstrophy balance.
We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier--Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier--Stokes system with damping our estimates for the number of the determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. Our investigation of the damped-driven 2D Navier--Stokes sys…
The paper studies the issue of stability of solutions to the Navier-Stokes and damped Euler systems in periodic boxes. We show that under action of fast oscillating-in- time external forces all two dimensional regular solutions converge to a time periodic flow. Unexpectedly, effects of stabilization can be also obtained for systems with stationary forces with large total momentum (average of the velocity). Thanks to the Galilean transformation and space boundary conditions, the stationary force changes into one with time oscillations. In the three dimensional case we show an analogical result …
On Bridging Analyticity and Sparseness in Hyperdissipative Navier-Stokes Systems
该研究通过引入时间加权桥梁不等式和聚焦极值假设，弥合超耗散纳维-斯托克斯系统中的解析性与稀疏性，排除了爆破并延展了解的解析性。
In two and three space dimensions, and under suitable assumptions on the initial data, we show global existence for a damped wave equation which approaches, in some sense, the Navier-Stokes problem. The proofs are based on a refined energy method. In this paper, we improve the results in two papers by Y. Brenier, R. Natalini and M. Puel and by M. Paicu and G. Raugel. We relax the regularity of the initial data of the former, even though we still use energy methods as a principal tool. Regarding the second paper, the improvement consists in the simplicity of the proofs since we do not use any S…
Analyticity of the data-to-solution map for a stationary Navier-Stokes fluid-structure interaction problem
该研究针对定常纳维-斯托克斯与欧拉-伯努利梁耦合的流固问题，通过复化方法和全纯隐函数定理证明数据到解映射的实解析性，给出小数据存在性与局部唯一性结果，相关结论可用于参数偏微分方程的数据驱动降阶建模。
We address the global persistence of analyticity and Gevrey-class regularity of solutions to the two and three-dimensional visco-elastic second-grade fluid equations. We obtain an explicit novel lower bound on the radius of analyticity of the solutions to the second-grade fluid equations that does not vanish as $t\to \infty$. Applications to the damped Euler equations are given.
We consider here the Navier-Stokes equations in $\mathbb{R}^{3}$ with a stationary, divergence-free external force and with an additional damping term that depends on two parameters. We first study the well-posedness of weak solutions for these equations and then, for a particular set of the damping parameters, we will obtain an upper and lower control for the energy dissipation rate $\varepsilon$ according to the Kolmogorov K41 theory. However, although the behavior of weak solutions corresponds to the K41 theory, we will show that in some specific cases the damping term introduced in the Nav…
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。
Vanishing of long time average p-enstrophy dissipation rate in the inviscid limit of the 2D damped Navier-Stokes equations
本文将二维阻尼 Navier-Stokes 方程无粘极限下长时间平均 enstrophy 耗散率消失的结果推广至 p-enstrophy，并借助全局吸引子不变测度与涡量强收敛结论简化证明。
In this paper, we consider a damped Navier-Stokes-Bardina model posed on the whole three-dimensional. These equations have an important physical motivation and they arise from some oceanic model. From the mathematical point of view, they write down as the well-know Navier-Stokes equations with an additional nonlocal operator in their nonlinear transport term, and moreover, with an additional damping term depending of a parameter $β>0$. We study first the existence and uniqueness of global in time weak solutions in the energy space. Thereafter, our main objective is to describe the long time be…
A turbulent study for a damped Navier-Stokes equation: turbulence and problems
本文研究三维空间中带无散度外力的阻尼不可压缩 Navier-Stokes 方程，在 Grashof 数足够大时，根据 Kolmogorov 耗散定律证明了流体特征速度与能量耗散率的上下界估计，主要贡献在于下界估计，并探讨了湍流理论的有效性。
In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation. This algorithm capitalizes on the fact that such infinite-dimensional dissipative dynamical systems posses finite-dimensional long-time behavior which is represented by, for instance, the finitely many determining parameters of their long-time dynamics, such as determining Fourier mo…
Through Borel summation methods, we analyze two different variations of the Navier-Stokes equation --the Boussinesq equation and the magnetic Benard equation. This method has previously been applied to the Navier-Stokes equation. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p in R^{+}, p being the Laplace dual variable of 1/t. This implies the local existence of a classical solution in a complex t-region that includes a real positive time (t)-axis segment. In this formalism, global existence of PDE solutions bec…
The article is devoted to the mathematical analysis of a fluid-structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier-Stokes-Fourier system and the structure displacement is described by a structurally damped plate equation. Our main results are the existence of strong solutions in an $L^p-L^q$ setting for small time or for small data. Through a change of variables and a fixed point argument, the proof of the main resu…
We study the so-called damped Navier-Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field $u_0\in L^\infty(\mathbb R^2)$ is allowed and no assumptions on the spatial decay of solutions as $|x|\to\infty$ are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in R2, we show that th…
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。
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 consider the wellposedness of the fractional Navier-Stokes as a generalization of the wellposedness result in Koch-Tataru's paper. An interesting remark is that our result does not contradict to the well-known ill-posedness result for Navier-Stokes with initial data in $\dot B^{-1, \infty}_\infty$ by Bourgain-Pavlovic. In the end, we also discuss the regularity and analyticity in the space variable.
In this paper, we shall establish the global well-posedness, the space-time analyticity of the Navier-Stokes equations for a class of large periodic data $u_0 \in BMO^{-1}(\mathbb{R}^3)$. This improves the classical result of Koch \& Tataru \cite{koch-tataru}, for the global well-posedness with small initial data $u_0 \in BMO^{-1}(\mathbb{R}^n)$.
Uniqueness of invariant measures for stochastic damped anisotropic Navier--Stokes equations
本文研究了随机阻尼各向异性纳维-斯托克斯方程的不变测度唯一性，通过渐近耦合方法和能量估计证明了当阻尼系数足够大时不变测度的唯一性。
We discuss several classes of linear second order initial-boundary value problems, where damping terms appear in the main wave equation as well as in the dynamic boundary condition. We investigate their well-posedness and describe some qualitative properties of their solutions, including boundedness, stability, or almost periodicity. In particular, we are able to characterize the analyticity of certain $C_0$-semigroups associated to such problems. Applications to several problems on domains and networks are shown.
On the refined analyticity radius of 3-D generalized Navier-Stokes equations
本文研究三维广义Navier-Stokes方程解析半径的瞬时增长，在次临界情形给出带对数修正的逐时下界，改进已有结果并解决一个十年公开问题，同时在临界情形证明解析半径下界随时间趋于零而趋于无穷。
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.
We study the radius of analyticity~$R(t)$ in space, of strong solutions to systems of scale-invariant semi-linear parabolic equations. It is well-known that near the initial time,~$R(t)t^{-\frac12}$ is bounded from below by a positive constant. In this paper we prove that~$\displaystyle\liminf_{t\rightarrow 0} R(t)t^{-\frac12}= \infty$, and assuming higher regularity for the initial data, we obtain an improved lower bound near time zero. As an application, we prove that for any global solution~$u\in C([0,\infty); H^{\frac12}(\R^3))$ of the Navier-Stokes equations, there holds~$\displaystyle\li…
In this paper, we consider the 3D Navier-Stokes-Voigt (NSV) equations with nonlinear damping $|u|^{r-1}u, r\in[1,\infty)$ in bounded and space-periodic domains. We formulate an optimal control problem of minimizing the curl of the velocity field in the energy norm subject to the flow velocity satisfying the damped NSV equation with a distributed control force. The control also needs to obey box-type constraints. For any $r\geq 1,$ the existence and uniqueness of a weak solution is discussed when the domain $Ω$ is periodic/bounded in $\mathbb R^3$ while a unique strong solution is obtained in t…
The goal of this paper is twofold. First, we give a simple proof that sufficiently sparse Navier--Stokes solutions do not develop singularities. This provides an alternative to the approach of \cite{Grujic2013}, which is based on analyticity and the `harmonic measure maximum principle'. Second, we analyze the claims in \cite{algebraicreduction,grujic2019asymptotic} that \emph{a priori} estimates on the sparseness of the vorticity and higher velocity derivatives reduce the 'scaling gap' in the regularity problem.
We discuss a Hilbert space method that allows to prove analytical well-posedness of a class of linear strongly damped wave equations. The main technical tool is a perturbation lemma for sesquilinear forms, which seems to be new. In most common linear cases we can furthermore apply a recent result due to Crouzeix--Haase, thus extending several known results and obtaining optimal analyticity angle.
We establish the validity of the Euler$+$Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an $L^{1}$-type norm for the vorticity of the error term in the expansion Navier-Stokes$-($Euler$+$Prandtl$)$. An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a re…
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.
Well-posedness of three-dimensional Damped Cahn-Hilliard-Navier-Stokes Equations
本文分析了三维有界区域内多孔介质中流体混合物的演化，通过耦合convective Brinkman Forchheimer方程和Cahn Hilliard方程，证明了弱解的存在性、能量等式及唯一性，并扩展至退化迁移率和奇异势情形。
Based on some elementary estimates for the space-time derivatives of the heat kernel, we use a bootstrapping approach to establish the optimal decay rates for the $L^q(\mathbb{R}^d)$ ($1\leq q\leq\infty$, $d\in\mathbb{N}$) norm of the space-time derivatives of solutions to the (modified) Patlak-Keller-Segel equations with initial data in $L^1(\mathbb{R}^d)$, which implies the joint space-time analyticity of solutions. When the $L^1(\mathbb{R}^d)$ norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the r…
In 1981, Foias, Guillopé and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier-Stokes equation. Such bounds are instructive in the numerical investigation of intermittency often observed in simulations, e.g., numerical study of vorticity moments by Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible for smoothness of solutions to the Navier-Stokes equation. We employ an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of …
We develop, analyze, and test an approximate, global data assimilation/synchronization algorithm based on purely local observations for the two-dimensional Navier-Stokes equations on the torus. We prove that, for any error threshold, if the reference flow is analytic with sufficiently large analyticity radius, then it can be recovered within that threshold. Numerical computations are included to demonstrate the effectiveness of this approach, as well as variants with data on moving subdomains. In particular, we demonstrate numerically that machine precision synchronization is achieved for mobi…
In 2001, H. Koch and D. Tataru proved the existence of global in time solutions to the incompressible Navier-Stokes equations in ${\mathbb{R}}^d$ for initial data small enough in $BMO^{-1}$. We show in this article that the Koch and Tataru solution has higher regularity. As a consequence, we get a decay estimate in time for any space derivative, and space analyticity of the solution. Also as an application of our regularity theorem, we prove a regularity result for self-similar solutions.
Sub-Riemannian Navier-Stokes system on the Heisenberg group: Weak solutions, well-posedness and smoothing effects
该研究推导了海森堡群上的次黎曼Navier-Stokes系统，证明了L^2框架下弱解全局存在、临界空间小初值解全局唯一且具竖直方向瞬时解析性，还得到大时间解析半径下界优于欧氏经典系统的结论，并恢复其余方向的无穷阶光滑性。
We consider damped wave (resp. Schr{ö}dinger and plate) equations driven by a hypoelliptic "sum of squares" operator L on a compact manifold and a damping function b(x). We assume the Chow-Rashevski-H{ö}rmander condition at rank k (at most k Lie brackets needed to span the tangent space) together with analyticity of M and the coefficients of L. We prove decay of the energy at rate $log(t)^{-1/k}$ (resp. $log(t)^{-2/k}$ ) for data in the domain of the generator of the associated group. We show that this decay is optimal on a family of Grushin-type operators. This result follows from a perturbat…
Maximal regularity for a compressible fluid-structure interaction system with Navier-slip boundary conditions
本文研究了在Navier-slip边界条件下的可压缩流体-结构相互作用系统，通过解耦线性化系统和级联策略结合Tikhonov固定点定理，证明了强解在L^p-L^q框架下的局部时间存在性和唯一性。
For incompressible Navier-Stokes equations, Necas-Ruzicka-Sverak proved that self-similar solution has to be zero in 1996. Further, Yang-Yang-Wu find symmetry property plays an important role in the study of ill-posedness. In this paper, we consider two types of symmetry property. We search special symmetric and uniform analytic functions to approach the solution and establish global uniform analytic and symmetric solution with initial value in general symmetric Fourier-Herz space. For two kinds of symmetry of initial data, we prove that the solution has also the same symmetric structure. Furt…
The purpose of this work is to study the existence and analytic smoothing effect for the compressible Navier- Stokes system with quantum pressure in pseudo-measure spaces. This system has been considered by B. Haspot and an analytic smoothing effect for a Korteweg type system was considered by F. Charve, R. Danchin and J. Xu, both of them in Besov spaces. Here we give a better lower bound of the radius of analyticity near zero. This work is an opportunity to deepen the study of partial differential equations in pseudo-measure spaces by introducing a new functional setting to deal with non-line…
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