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.     

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.     

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.     

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.     

The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations

We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely

A Note on Similarity Solutions of Navier-Stokes Equations

This note looks at the two similarity solutions of the Navier Stokes equations in polar coordinates. In the second solution an initial value problem is reduced into generalized stationary KDV and hence integrable.     

On the Lagrangian Dynamics for the 3D Incompressible Euler Equations

In this paper we study the dynamical behaviors along the particle trajectories for some quantities of the 3D inviscid incompressible fluids. We construct evolution equations satisfied by scalar quantities composed of spectrum of the deformation tensor, the hessian of the pressure and the direction field of the vorticity, and study the dichotomy between the finite time singularity and the long time behaviors of the various scalar quantities.The work was supported partially by the KOSEF Grant no. R01-2005-000-10077-0.     

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 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.     

Concentrations of Solutions to Time-Discretizied Compressible Navier-Stokes Equations

The compactness properties of solutions to time-discretization of compressible Navier-Stokes equations are investigated in three dimensions. The existence of generalized solutions is established.     

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     

On the Partial Regularity of a 3D Model of the Navier-Stokes Equations

We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric flows with swirl using a set of new variables. It preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected in the model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. In [11], we presented numerical evidence which seems to support that the 3D model develops finite time singularities while the corresponding solution of the 3D Navier-Stokes equations remains smooth. This suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term. In this paper, we prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The partial regularity result of this paper is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations.     

Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations

We prove uniqueness of solutions to the Maxwell-Dirac system in the energy space, namely . We also give a proof for uniqueness of finite energy solutions to the Maxwell-Klein-Gordon equations, which is simpler than that given in [16]. The first author is partially supported by an NSF grant and an Alfred Sloan fellowship. The second author is supported by JSPS Postdoctoral Fellowships for Research Abroad (2001–2003)     

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.     

Uniqueness for Autonomous Planar Differential Equations and the Lagrangian Formulation of Water Flows with Vorticity

Abstract

We prove a uniqueness result for autonomous divergence-free systems of ODE’s in the plane and give an application to the study of water flows with vorticity.     

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.     

On the Conserved Quantities for the Weak Solutions of the Euler Equations and the Quasi-geostrophic Equations

In this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible Euler equations. The regularity of the weak solutions are measured in terms of the Triebel-Lizorkin type of norms, and the Besov norms, . In particular, in the Besov space case, our results refine the previous ones due to Constantin-E-Titi (energy) and the author of this paper (helicity), where the regularity is measured by a special class of the Besov space norm , which is the Nikolskii space. We also obtain a sufficient regularity condition for the conservation of the L ^p -norm of the temperature function in the weak solutions of the quasi-geostrophic equation.