We investigate the long-term dynamics of the three-dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view we consider the Navier-Stokes-Voight model as a non-viscous (inviscid) regularization of the three-dimensional Navier-Stokes equations. Furthermore, we also show that the weak solutions of the Navier-Stokes-Voight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient $ν\to 0$.
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…
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…
We consider two-dimensional nonstationary Navier-Stokes shear flow with multivalued and nonmonotone boundary conditions on a part of the boundary of the flow domain. We prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion with a multivalued term in the form of Clarke subdifferential. Then we prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow. This research is motivated by control problems for fluid flows in domains with semipermeable walls and membranes.
We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representi…
The three-dimensional Navier-Stokes-$α$ model for fast rotating geophysical fluids is considered. The Navier-Stokes-$α$ model is a nonlinear dispersive regularization of the exact Navier-Stokes equations obtained by Lagrangian averaging and tend to the Navier-Stokes equations as $α\rightarrow 0^+$. We estimate upper bounds for the dimensions of global attractors and study the dependence of the dimensions on the parameter $α$. All the estimates are uniform in $α$, and our estimate of attractor dimensions remain finite when $α\rightarrow 0^+$.
As in our previous paper, the 3D Navier-Stokes equations with a translationally bounded force contain pullback attractors in a weak sense. Moreover, those attractors consist of complete bounded trajectories. In this paper, we present a sufficient condition under which the pullback attractors are degenerate. That is, if the Grashof constant is small enough, the pullback attractor will be a single point on a unique, complete, bounded, strong solution. We then apply our results to provide a new proof of the existence of a unique, strong, periodic solution to the 3D Navier-Stokes with a small, per…
This article covers the construction of exponential attractors in two different functional space settings; one is in Hilbert's space, and the other is in the Banach space. The former relies on the squeezing properties of solution trajectories, but the latter does not. We present these different methods for constructing exponential attractors using the three-dimensional Rotating Navier-Stokes-$α$ equations.
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 give a comprehensive study of the 3D Navier-Stokes-Brinkman-Forchheimer equations in a bounded domain endowed with the Dirichlet boundary conditions and non-autonomous external forces. This study includes the questions related with the regularity of weak solutions, their dissipativity in higher energy spaces and the existence of the corresponding uniform attractors
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
In this paper we prove the existence of random attractors for the Navier--Stokes equations on 2 dimensional sphere under random forcing irregular in space and time. We also deduce the existence of an invariant measure.
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.
The article is devoted to the study of non-autonomous Navier-Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier-Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic,almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier-Stokes equations is given. Finally, the authors have derived a global averaging principle f…
In this article, we study the long time behavior of solutions of a variant of the Boussinesq system in which the equation for the velocity is parabolic while the equation for the temperature is hyperbolic. We prove that the system has a global attractor which retains some of the properties of the global attractors for the 2D and 3D Navier-Stokes equations. Moreover, this attractor contains infinitely many invariant manifolds in which several universal properties of the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.
In view of the possibility that the 3D Navier-Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies--weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all s…
This paper is concerned with the asymptotic behavior of solutions of the two-dimensional Navier-Stokes equations with both non-autonomous deterministic and stochastic terms defined on unbounded domains. We first introduce a continuous cocycle for the equations and then prove the existence and uniqueness of tempered random attractors. We also characterize the structures of the random attractors by complete solutions. When deterministic forcing terms are periodic, we show that the tempered random attractors are also periodic. Since the Sobolev embeddings on unbounded domains are not compact, we …
The Cahn-Hilliard-Navier-Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier-Stokes equations suitably coupled with a nonlocal Cahn-Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn-Hilliard-Navier-Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This p…
We consider a model for the evolution of a mixture of two incompressible and partially immiscible Newtonian fluids in two dimensional bounded domain. More precisely, we address the well-known model H consisting of the Navier-Stokes equation with non-autonomous external forcing term for the (average) fluid velocity, coupled with a convective Cahn-Hilliard equation with polynomial double-well potential describing the evolution of the relative density of atoms of one of the fluids. We study the long term behavior of solutions and prove that the system possesses a pullback exponential attractor. I…
The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions or on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak omega-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields) are achieved also in the strong topology of H. In particular, if a weak omega-limit set is bounded in the space V of velocity fields with square-integrable vorticity then t…
We regularized the 3D Navier-Stokes equations by adding a fourth-order viscosity term. We study the upper semicontinuity, of the global attractors of the Leray-Hopf weak solutions of the regularized 3D Navier-Stokes equations, as the artificial dissipation goes to 0. We also consider applications of obtained results to the regularized problem by allowing the family of forcing functions to vary with epsilon, for epsilon >0.
This paper is concerned with pullback dynamics of 3D Navier-Stokes equations with variable viscosity and subject to time-dependent external forces. Our main result establishes the existence of finite-dimensional pullback attractors in a general setting involving tempered universes. We also present a sufficient condition on the viscosity coefficients that guarantees the attractors are nontrivial. We end the paper by showing the upper semi-continuity of pullback attractors as the non-autonomous perturbation vanishes.
We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator. It is known that there exists a unique…
Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains
Hanoi Pedagogical University（河内师范大學）；Electric Power University（電力大學）；Tata Institute of Fundamental Research - Centre For Applicable Mathematics (TIFR-CAM)（塔塔基礎研究所應用數學中心）；NOVA School of Science and Technology (NOVA FCT)（新里斯本科技學院）；Indian Institute of Technology Roorkee-IIT Roorkee（魯爾凱印度理工學院）
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…
This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier-Stokes equation with non-homogeneous boundary condition on Lipschitz-like domain. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimate on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is t…
Multi-vortices and lower bounds on the attractor dimensions for 2D Navier--Stokes equations
Zhejiang Normal University（浙江师范大学）；Keldysh Institute of Applied Mathematics（凯尔迪什应用数学研究所）；University of Surrey（萨里大学）；Institute for Information Transmission Problems（信息传输问题研究所）；HSE University（俄罗斯国立高等经济学院）
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 stationary version of a modified definition of statistical solution for the three-dimensional incompressible Navier-Stokes equations introduced in a previous work is investigated. Particular types of such stationary statistical solutions and their analytical properties are addressed. Results on the support and carriers of these stationary statistical solutions are also given, showing in particular that they are supported on the weak global attractor and are carried by a more regular part of the weak global attractor containing Leray-Hopf weak solutions which are locally strong solutions. T…
Here we consider a Cahn-Hilliard-Navier-Stokes system characterized by a nonlocal Cahn-Hilliard equation with a singular (e.g., logarithmic) potential. This system originates from a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids. We have already analyzed the case of smooth potentials with arbitrary polynomial growth. Here, taking advantage of the previous results, we study this more challenging (and physically relevant) case. We first establish the existence of a global weak solution with no-slip and no-flux boundary conditions. Then we prove the existe…
We study the coupled Navier-Stokes Ginzburg-Landau model of nematic liquid crystals introduced by F.H. Lin, which is a simplified version of the Ericksen-Leslie system. We generalize the model to compact n-dimensional Riemannian manifolds, and show that the system comes from a variational principle. We present a new simple proof for the local well-posedness of this coupled system without using the higher-order energy law. We then prove that this system is globally well-posed and has compact global attractors when the dimension of the manifold M is two.Finally, we introduce the Lagrangian avera…
We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis…
We construct pullback attractors to the weak solutions of the three-dimensional Dirichlet problem for the incompressible Navier-Stokes equations in the case when the external force may become unbounded as time goes to plus or minus infinity.
We introduce a determining wavenumber for the forced 3D Navier-Stokes equations (NSE) defined for each individual solution. Even though this wavenumber blows up if the solution blows up, its time average is uniformly bounded for all solutions on the global attractor. The bound is compared to Kolmogorov's dissipation wavenumber and the Grashof constant.
We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlović. They showed a finite time blow-up in the case where the dissipation degree $α$ is less than 1/4. In this paper we prove the existence of weak solutions for all $α$, energy inequality for every weak solution with nonnegative initial datum starting from any time, local regularity for $α> 1/3$, and global regularity for $α\geq 1/2$. In addition, we prove a finite time blow-up in the case where $α<1/3$. It is remarkable that the model with $α=1/3$ enjoys the same estimates on the nonlinear term as the 4D Navi…
We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.
We consider a stochastic model which describes the motion of a 2D incompressible fluid in a unbounded domain with viscosity and memory effects. This model is different from the classical stochastic Navier-Stokes-Voigt equations due to the absence of the Voigt term $ -αΔu_{t}$, and has a much weaker dissipation than the usual Navier-Stokes-Voigt model since only the memory viscoelasticity is present. We are interested in the global well-posedness and long-time behaviors of this model. We first investigate the well-posedness by using the classical Faedo-Galerkin method. Unlike the general method…
We extend our earlier β-plane results [al-Jaboori and Wirosoetisno, 2011, DCDS-B 16:687--701] to a rotating sphere. Specifically, we show that the solution of the Navier--Stokes equations on a sphere rotating with angular velocity 1/ε becomes zonal in the long time limit, in the sense that the non-zonal component of the energy becomes bounded by εM. We also show that the global attractor reduces to a single stable steady state when the rotation is fast enough.
We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier--Stokes equations on the periodic $β$-plane (i.e.\ with the Coriolis force varying as $f_0+βy$) will become nearly zonal: with the vorticity $ω(x,y,t)=\wb(y,t)+\wt(x,y,t)$, one has $|\wt|_{H^s}^2\leβ^{-1} M_s(\...)$ as $t\to\infty$. We use this show that, for sufficiently large $β$, the global attractor of this system reduces to a point.
We consider a 2D system that models the nematic liquid crystal flow through the Navier--Stokes equations suitably coupled with a transport-reaction-diffusion equation for the averaged molecular orientations. This system has been proposed as a reasonable approximation of the well-known Ericksen--Leslie system. Taking advantage of previous well-posedness results and proving suitable dissipative estimates, here we show that the system endowed with periodic boundary conditions is a dissipative dynamical system with a smooth global attractor of finite fractal dimension.
A dyadic shell model for the Navier-Stokes equations is studied in the context of turbulence. The model is an infinite nonlinearly coupled system of ODEs. It is proved that the unique fixed point is a global attractor, which converges to the global attractor of the inviscid system as viscosity goes to zero. This implies that the average dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system (which is positive) as viscosity goes to zero. This phenomenon is called the dissipation anomaly predicted by Kolmogorov's theory for the actual Navier-S…
In this paper, we analyze a tamed 3D Navier-Stokes equation in uniform $C^2$-domains (not necessarily bounded), which obeys the scaling invariance principle, and prove the existence and uniqueness of strong solutions to this tamed equation. In particular, if there exists a bounded solution to the classical 3D Navier-Stokes equation, then this solution satisfies our tamed equation. Moreover, the existence of a global attractor for the tamed equation in bounded domains is also proved. As simple applications, some well known results for the classical Navier-Stokes equations in unbounded domains a…
We give an abstract framework for studying nonautonomous PDEs, called a generalized evolutionary system. In this setting, we define the notion of a pullback attractor. Moreover, we show that the pullback attractor, in the weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply our framework to both autonomous and nonautonomous evolutionary systems as they first appeared in earlier works by Cheskidov, Foias, and Lu. In this context, we compare the pullback attractor to both the global attractor (in the autono…
