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 the attraction to the set holds also in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented.
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.
We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.
In this paper, we rigorously derive the compressible one-fluid Navier-Stokes equation from the scaled compressible two-fluid Navier-Stokes-Maxwell equations locally in time under the assumption that the initial data are well prepared. We justify the singular limit by proving the uniform decay of the error system, which is obtained by elaborate energy estimates.
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.
We study Caccioppoli's inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli's inequalities of the Stokes equations and the Navier-Stokes equations, in general, fail near boundary.
This paper studies two well-known models for two-phase fluid flow at constant temperature, the isothermal Navier-Stokes-Allen-Cahn and the isothermal Navier-Stokes-Cahn-Hilliard equations, both of which consist of equations for the (total) fluid density rho, the (mass-averaged)velocity u and the concentration (of one of the phases,) c. Assuming in either case that both phases are incompressible with different densities, each of the models is shown to reduce to a system of evolution equations in rho and u alone. In the case of the Navier-Stokes-Allen-Cahn model, this reduced system is the class…
An algebraic upper bound for the decay rate of solutions to the Navier-Stokes and Navier-Stokes-Coriolis equations in the critical space $\dot{H} ^{\frac{1}{2}} (\mathbb{R} ^3)$ is derived using the Fourier Splitting Method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts.
In this paper, we first prove the global well-posedness of a scaled anisotropic Navier-Stokes system and the hydrostatic Navier-Stokes system in a 2-D striped domain with small analytic data in the tangential variable. Then we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes system with analytic data.
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 $u_0 \in BMO^{-1}(\mathbb{R}^n)$.
We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $Ω\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove existence, uniqueness of weak and strong solutions in $\mathbf{W}^{1,p}(Ω)$ and $\mathbf{W}^{2,p}(Ω)$ for all $1<p<\infty$ considering minimal regularity on the friction coefficient $α$. Moreover, we deduce uniform estimates on the solution with respect to $α$ which enables us to analyze the behavior of the solution when $α\rightarrow \infty$.
In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.
We introduce a framework to justify hydrodynamic limits of the Vlasov-Navier-Stokes system. We specifically study high friction regimes, which take into account the fact that particles of the dispersed phase are light (resp. small) compared to the fluid part, and lead to the derivation of Transport-Navier-Stokes (resp. Inhomogeneous Navier-Stokes) systems.
In the note, a local regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It reads that axially symmetric energy solutions to the Navier-Stokes equations have no Type I blowups.
We introduce a concept of space-time holomorphic solutions of partial differential equations and construct a meromorphic solution of Navier-Stokes equations.
Fundamental solutions to the time-periodic Stokes and Oseen linearizations of the Navier-Stokes equations in dimension $n\geq 2$ are investigated. Integrability properties and pointwise estimates are established.
Multiple spatial zero formations for Stokes and Navier-Stokes equations in three dimensions are shown to occur according to nodal sets of solenoidal Hermite polynomials. Extensions to well-posed Burnett equations with the bi-harmonic viscosity operator are also discussed.
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 $x$.
These notes are dedicated to the analysis of the one-dimensional free-congested Navier-Stokes equations. After a brief synthesis of the results obtained in [4] related to the existence and the asymptotic stability of partially congested profiles associated to the soft congestion Navier-Stokes system, we present a first local well-posedness result for the one-dimensional free-congested Navier-Stokes equations.
An enstrophy cascade is exhibited for the Navier-Stokes equations in physical scales independently of boundary conditions under physically reasonable assumptions on the flow.
In this paper, we prove existence of smooth solutions of the Navier-Stokes equations that gives a positive answer to the problem proposed by Fefferman [3].
In this paper, we study the upper bound of the time decay rate of solutions to the Navier-Stokes equations and generalized Navier-Stokes equations with damping term $|u|^{β-1}u$ ($β>1$) in $\mathbb{R}^3$.
We study 2D Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\R^2$ and $\T$, by a fixed point argument. We also show that the solution of constrained Navier-Stokes converges to the solution of Euler equation as viscosity $ν$ vanishes.
We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear Fokker-Planck equation describing microscopic corpora embedded in the fluid. This leads naturally to bounded added stress and hence to $W^{-1,\infty}$ forcing of the Navier-Stokes equations.
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 address the issue of existence of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions allowing the presence of vacuum zones and assuming rough conditions on the data. We also study the convergence, as the viscosity goes to zero, of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions to the strong solution of the Euler equations with variable density, provided that the initial data converge in $L^{2}$ to a smooth enough limit.
We prove in this paper a decay estimate for scaling invariant local energy solutions for some toy-models related to the incompressible Navier-Stokes system.
The global well-posedness for the incompressible Navier-Stokes-Vlasov equations in two spatial dimensions is established by a priori estimates, the characteristic method and the semigroup analysis.
A Liouville type theorem is proven for the steady-state Navier-Stokes equations. It follows from the corresponding theorem on the Stokes equations with the drift. The drift is supposed to belong to a certain Morrey space.
In the note, a new regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It is slightly supercritical.
In this paper, we study the low Mach number limit of the full compressible Navier-Stokes equations with revised Maxwell law. By applying the uniform estimation of the error system, we prove that the solutions of the full compressible Navier-Stokes equations with time relaxation converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.
We extend Barker's weak-strong uniqueness results for the Navier--Stokes equations and consider a criterion involving Besov spaces and weighted Lebesgue spaces.
It is shown that any smooth solution to the stationary Navier-Stokes system in $R^3$ with the velocity field, belonging globally to $L_6$ and $BM0^{-1}$, must be zero.
A proof is given of the global existence and uniqueness of a weak solution to Navier-Stokes boundary problem. The proof is short and essentially self-contained.
Extending the work of Jia and Sverak on self-similar solutions of the Navier-Stokes equations, we show the existence of large, forward, discretely self-similar solutions.
Applying the DeGiorgi Method the Holder continuity of the swirl is proved for solutions to the Navier-Stokes equations from Lq(0,T;Lp) with 3/p+2/q < 1.
We consider ionic electrodiffusion in fluids, described by the Nernst-Planck-Navier-Stokes system in bounded domains, in two dimensions, with Dirichlet boundary conditions for the Navier-Stokes and Poisson equations, and blocking (vanishing normal flux) or selective (Dirichlet) boundary conditions for the ionic concentrations. We prove global existence and stability results for large data.
We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier-Stokes equation with Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in $L^\infty$. This allows to get the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.
We study cavitating self-similar solutions to compressible Navier-Stokes equations with degenerate density-dependent viscosity. We prove both existence of expanders and non-existence of small shrinkers.
