On the Global Regularity of a Helical-Decimated Version of the 3D Navier-Stokes Equations

We study the global regularity, for all time and all initial data in H ^1/2, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution of Navier-Stokes (NS) equations into the subspace where helicity (the L ²-scalar product of velocity and vorticity) is sign-definite. The presence of a second (beside energy) sign-definite inviscid conserved quadratic quantity, which is equivalent to the H ^1/2-Sobolev norm, allows us to demonstrate global existence and uniqueness, of space-periodic solutions, together with continuity with respect to the initial conditions, for this decimated 3D model. This is achieved thanks to the establishment of two new estimates, for this 3D model, which show that the H ^1/2 and the time average of the square of the H ^3/2 norms of the velocity field remain finite. Such two additional bounds are known, in the spirit of the work of H. Fujita and T. Kato (Arch. Ration. Mech. Anal. 16:269–315, 1964; Rend. Semin. Mat. Univ. Padova 32:243–260, 1962), to be sufficient for showing well-posedness for the 3D NS equations. Furthermore, they are directly linked to the helicity evolution for the dNS model, and therefore with a clear physical meaning and consequences.     

Partial Regularity of Solutions to the Four-Dimensional Navier-Stokes Equations at the First Blow-up Time

The solutions of incompressible Navier-Stokes equations in four spatial dimensions are considered. We prove that the two-dimensional Hausdorff measure of the set of singular points at the first blow-up time is equal to zero. Hongjie Dong was partially supported by the National Science Foundation under agreement No. DMS-0111298. Dapeng Du was partially supported by a postdoctoral grant from School of Mathematical Sciences at Fudan University.     

On Depletion of the Vortex-Stretching Term in the 3D Navier-Stokes Equations

Certain new cancellation properties in the vortex-stretching term are detected leading to new geometric criteria for preventing finite-time blow-up in the 3D Navier-Stokes equations.     

On the Regularity Criterion of Weak Solution for the 3D Viscous Magneto-Hydrodynamics Equations

We improve and extend some known regularity criterion of the weak solution for the 3D viscous Magneto-hydrodynamics equations by means of the Fourier localization technique and Bony’s para-product decomposition.     

Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point z if either the scaled -norm of the velocity with 3/p + 2/q ≤ 2, 1 ≤ q ≤ ∞, or the -norm of the vorticity with 3/p + 2/q ≤ 3, 1 ≤ q < ∞, or the -norm of the gradient of the vorticity with 3/p + 2/q ≤ 4, 1 ≤ q, 1 ≤ p, is sufficiently small near z.     

Universal Bounds for the Littlewood-Paley First-Order Moments of the 3D Navier-Stokes Equations

We derive upper bounds for the infinite-time and space average of the L ¹-norm of the Littlewood-Paley decomposition of weak solutions of the 3D periodic Navier-Stokes equations. The result suggests that the Kolmogorov characteristic velocity scaling, U_k ~ e^1/3 k^-1/3{\mathbf{U}_\kappa\sim\epsilon^{1/3} \kappa^{-1/3}} , holds as an upper bound for a region of wavenumbers near the dissipative cutoff.     

On the Vanishing Viscosity Limit of 3D Navier-Stokes Equations under Slip Boundary Conditions in General Domains

We consider the vanishing-viscosity limit for the Navier-Stokes equations with certain slip-without-friction boundary conditions in a bounded domain with non-flat boundary. In particular, we are able to show convergence in strong norms for a solution starting with initial data belonging to the special subclass of data with vanishing vorticity on the boundary. The proof is obtained by smoothing the initial data and by a perturbation argument with quite precise estimates for the equations of the vorticity and for that of the curl of the vorticity.     

On Lie Reduction of the Navier-Stokes Equations

Abstract

Lie reduction of the Navier-Stokes equations to systems of partial differential equations in three and two independent variables and to ordinary differential equations is described.     

On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NS _ν) with initial data in the scaling invariant Besov space, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANS _ν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, and Then with initial data in the scaling invariant space we prove the global wellposedness for (ANS _ν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANS _ν) with high oscillatory initial data (1.2).     

Homogeneous Statistical Solutions and Local Energy Inequality for 3D Navier-Stokes Equations

We are interested in space-time spatially homogeneous statistical solutions of Navier-Stokes equations in space dimension three. We first review the construction of such solutions, and introduce convenient tools to study the pressure gradient. Then we show that given a spatially homogeneous initial measure with finite energy density, one can construct a homogeneous statistical solution concentrated on weak solutions which satisfy the local energy inequality.     

Energy Cascades and Flux Locality in Physical Scales of the 3D Navier-Stokes Equations

Rigorous estimates for the total – (kinetic) energy plus pressure – flux in \mathbbR³{\mathbb{R}^3} are obtained from the three dimensional Navier-Stokes equations. The bounds are used to establish a condition – involving Taylor length scale and the size of the domain – sufficient for existence of the inertial range and the energy cascade in decaying turbulence (zero driving force, non-increasing global energy). Several manifestations of the locality of the flux under this condition are obtained. All the scales involved are actual physical scales in \mathbbR³{\mathbb{R}^3} and no regularity or homogeneity/scaling assumptions are made.     

On the Dynamics of Navier-Stokes and Euler Equations

This is a detailed study on certain dynamics of Navier-Stokes and Euler equations via a combination of analysis and numerics. We focus upon two main aspects: (a) zero viscosity limit of the spectra of linear Navier-Stokes operator, (b) heteroclinics conjecture for Euler equation, its numerical verification, Melnikov integral, and simulation and control of chaos. Due to the difficulty of the problem for the full Navier-Stokes and Euler equations, we also propose and study two simpler models of them.     

Weighted Regularity Criteria of Weak Solutions to the Incompressible 3D MHD Equations

In this paper, we investigate regularity conditions of the weighted type for weak solutions to the incompressible 3D MHD equations.     

抛物化Navier-Stokes方程的降维仿真模型

利用特征投影分解技术和奇值分解方法,建立抛物化Navier-Stokes方程的降维仿真模型,讨论降维仿真模型解的稳定性和误差,并利用误差估计指导特征投影分解基函数的选取及特征投影分解基函数的更新.最后,用数值试验验证降维仿真模型的有效性和可行性.     

On the Structure of Stationary Solutions of the Navier-Stokes Equations

 This paper is a supplementary section to [1]. We show that without any additional hypothesis the main result in [1] (Theorem 1) can be considerably strengthened. Note. This paper cannot be read independently of [1]. The numbering of equations, theorems and propositions as well as cross-references used here have to be understood as if this paper were an additional section to [1]. Received: 7 May 2002 / Accepted: 15 October 2002 Published online: 10 February 2003 RID="*" ID="*" Supported in part by the Fonds National Suisse. Communicated by A. Kupiainen     

A Criterion for Uniqueness of Lagrangian Trajectories for Weak Solutions of the 3D Navier-Stokes Equations

Foias, Guillopé, & Temam showed in 1985 that for a given weak solution of the three-dimensional Navier-Stokes equations on a domain Ω, one can define a ‘trajectory mapping’ that gives a consistent choice of trajectory through each initial condition , and that respects the volume-preserving property one would expect for smooth flows. The uniqueness of this mapping is guaranteed by the theory of renormalised solutions of non-smooth ODEs due to DiPerna & Lions. However, this is a distinct question from the uniqueness of individual particle trajectories. We show here that if one assumes a little more regularity for u than is known to be the case, namely that , then the particle trajectories are unique and C ¹ in time for almost every choice of initial condition in Ω. This degree of regularity is more than can currently be guaranteed for weak solutions () but significantly less than that known to ensure that u is regular ( . We rely heavily on partial regularity results due to Caffarelli, Kohn, & Nirenberg and Ladyzhenskaya & Seregin.     

On Blowup of Classical Solutions to the Compressible Navier-Stokes Equations

In this paper, we study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see Definition 2.2). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (Comm Pure Appl Math 51:229–240, 1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy. Furthermore, the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time, if the initial data have an isolated mass group satisfying some suitable conditions.     

Global Well-Posedness for a Smoluchowski Equation Coupled with Navier-Stokes Equations in 2D

We prove global existence for a nonlinear Smoluchowski equation (a nonlinear Fokker-Planck equation) coupled with Navier-Stokes equations in 2d. The proof uses a deteriorating regularity estimate in the spirit of [5] (see also [1]).     

Controllability of 2D Euler and Navier-Stokes Equations by Degenerate Forcing

We study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe the motion of the homogeneous ideal or viscous incompressible fluid on a two-dimensional torus . We assume the system to be controlled by a degenerate forcing applied to a fixed number of modes.In our previous work [3,5,4] we studied global controllability by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing viscosity (ν > 0). Methods of differential geometric/Lie algebraic control theory have been used for that study. In [3] criteria for global controllability of finite-dimensional Galerkin approximations of 2D and 3D NS systems have been established. It is almost immediate to see that these criteria are also valid for the Galerkin approximations of the Euler systems. In [5,4] we established a much more intricate sufficient criteria for global controllability in a finite-dimensional observed component and for L ₂-approximate controllability for the 2D NS system. The justification of these criteria was based on a Lyapunov-Schmidt reduction to a finite-dimensional system. Possibility of such a reduction rested upon the dissipativity of the NS system, and hence the previous approach can not be adapted for the Euler system.In the present contribution we improve and extend the controllability results in several aspects : 1) we obtain a stronger sufficient condition for controllability of the 2D NS system in an observed component and for L ₂-approximate controllability; 2) we prove that these criteria are valid for the case of an ideal incompressible fluid (ν=0); 3) we study solid controllability in projection on any finite-dimensional subspace and establish a sufficient criterion for such controllability.The authors have been partially supported by MIUR, Italy, the COFIN grant 2004015409-003.