Interior regularity of weak solutions to the perturbed Navier-Stokes equations

In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69 (1990), 237–254.     

On the regularity of weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity of weak solution to the incompressible magnetohydrodynamic equations. We obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations, which is similar to that of incompressible Navier-Stokes equations. Moreover, our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto-hydrodynamic equations.     

Refined a priori estimates for the axisymmetric Navier-Stokes equations

In this paper, we consider the axisymmetric Navier-Stokes equations, and provide a refined a priori estimate for the swirl component of the vorticity. This extends Theorem 2 of [D. Chae, J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645--671].     

Bilinear estimates in BMO and the Navier-Stokes equations

We prove that the BMO norm of the velocity and the vorticity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations. Our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class. Received February 15, 1999; in final form October 11, 1999 / Published online July 3, 2000     

On a theorem by Sohr for the Navier-Stokes equations

In this paper we study some criteria for the full (space-time) regularity of weak solutions to the Navier-Stokes equations. In particular, we generalize some classical and very recent criteria involving the velocity, or its derivatives. More specifically, we show with elementary tools that if a weak solution, or its vorticity, is small in appropriate Marcinkiewicz spaces, then it is regular.     

A regularity class for the Navier-Stokes equations in Lorentz spaces

We extend Serrin's regularity class for weak solutions of the Navier-Stokes equations to a larger class replacing the Lebesgue spaces by Lorentz spaces. Received November 30, 2000; accepted January 16, 2001.     

Local weak solution of the isentropic compressible Navier-Stokes equations in a half-space

In this paper, we establish the local existence of weak solutions with higher regularity of the three-dimensional half-space compressible isentropic Navier-Stokes equations with the adiabatic exponent γ > 1 in the presence of vacuum. Here we do not need any smallness of the initial data.

     

On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure

We study the Cauchy problem for the n-dimensional Navier-Stokes equations (n?3), and prove some regularity criteria involving the integrability of the pressure or the pressure gradient for weak solutions in the Morrey, Besov and multiplier spaces.     

Remark on the regularity for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic (MHD) equations. Some regularity criteria, which are related only with u+B or u?B, are obtained for weak solutions to the MHD equations. Copyright © 2008 John Wiley & Sons, Ltd.     

Regularity for 3D Navier-Stokes Equations Based on Energy Distribution at Wavenumber Bands

We present some regularity criteria for the Leray-Hopf weak solutions to the Cauchy problem for 3D Navier-Stokes equations based on energy distribution at wavenumber bands. We show sufficient conditions for regularity based on the ratios of energy dissipation rates at sufficiently high wavenumber bands and neighboring medium wavenumber bands. Moreover, we give a regularity criterion based on relations between energy at high and low wavenumber bands.     

Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain

We consider the 3-D Navier-Stokes equations in the half-space ₊³, or a bounded domain with smooth boundary, or else an exterior domain with smooth boundary. Some new sufficient conditions on pressure or the gradient of pressure for the regularity of weak solutions to the Navier-Stokes equations are obtained.Mathematics Subject Classification (2000):35B45, 35B65, 76D05     

Global existence in critical spaces for compressible Navier-Stokes equations

We investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium. We obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations. More precisely, the initial velocity has the same critical regularity index as for the incompressible homogeneous Navier-Stokes equations, and one more derivative is needed for the density. We point out a smoothing effect on the velocity and a L ¹-decay on the difference between the density and the constant reference state. The proof lies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term. Oblatum 9-II-1999 & 6-I-2000?Published online: 29 March 2000     

Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations

We consider the regularity of axisymmetric weak solutions to the Navier-Stokes equations in R³. Let u be an axisymmetric weak solution in R³×(0,T), w=curlu, and wθ be the azimuthal component of w in the cylindrical coordinates. Chae-Lee [D. Chae, J. Lee, On the regularity of axisymmetric solutions of the Navier-Stokes equations, Math. Z. 239 (2002) 645-671] proved the regularity of weak solutions under the condition wθ∈Lq(0,T;Lr), with , . We deal with the marginal case r=∞ which they excluded. It is proved that u becomes a regular solution if .     

Regularity of Pressure in the Neighbourhood of Regular Points of Weak Solutions of the Navier-Stokes Equations

In the context of the weak solutions of the Navier-Stokes equations we study the regularity of the pressure and its derivatives in the space-time neighbourhood of regular points. We present some global and local conditions under which the regularity is further improved.     

On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations

We prove two sufficient conditions for local regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. One of these conditions implies the smoothness of L_3,∞-solutions as a particular case. Bibliography: 12 titles.Dedicated to Vsevolod Alekseevich Solonnikov__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 186–198.     

The Generalized Caffarelli-Kohn-Nirenberg Theorem for the Hyperdissipative Navier-Stokes System

We introduce a notion of suitable weak solutions of the hyperdissipative Navier-Stokes equations, and we achieve a corresponding extension of the regularity theory of Caffarelli-Kohn-Nirenberg. © 2019 Wiley Periodicals, Inc.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.     

A remark on regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with the free boundary. The viscosity coefficient μ is proportional to ρθ with θ>0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H¹([0,1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.     

Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001