Bilinear estimates in BMO and the Navier-Stokes equations

Kozono, Hideo; Taniuchi, Yasushi

doi:10.1007/s002090000130

Cited by 252 publications

(167 citation statements)

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“…For this special case, just like in 2D we confirm that A L ∞ → ∞ as t → t * . Note that the prefactor 0.23 in (21) is smaller than the two-dimensional counterpart 0.34 in (17). This means that diffusion has more significant regularization effects in 3D than in 2D, which is consistent with intuition.…”

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confidence: 75%

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Characterization of blowup for the Navier-Stokes equations using vector potentials

Ohkitani

2017

AIP Advances

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We characterize a possible blowup for the 3D Navier-Stokes on the basis of dynamical equations for vector potentials 𝑨. This is motivated by a known interpolation ∥𝑨∥BMO≤∥𝒖∥L3, together with recent mathematical results. First, by working out an inversion formula for singular integrals that appear in the governing equations, we derive a criterion using the nonlinear term of 𝑨 as ∫0t∗∥∂𝑨∂t−ν△𝑨∥L∞dt=∞ for a blowup at t∗. Second, for a particular form of a scale-invariant singularity of the nonlinear term we show that the vector potential becomes unbounded in its L∞ and BMO norms. Using the stream function, we also consider the 2D Navier-Stokes equations to seek an alternative proof of their known global regularity. It is not yet proven that the BMO norm of vector potentials in 3D (or, the stream function in 2D) serve as a blow up criterion in more general cases.

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confidence: 75%

“…The Beale-Kato-Majda blowup criterion 1 for the Euler equations is also valid for the Navier-Stokes equations, in fact even with a weaker BMO norm. 17 Combining this fact with the definition − ψ = ω, the viscous term becomes singular in the sense that…”

Section: D Navier-stokes Equationsmentioning

confidence: 99%

Characterization of blowup for the Navier-Stokes equations using vector potentials

Ohkitani

2017

AIP Advances

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“…Original estimate of the Jacobian term (essentially H −1 norm) goes back to Wente [54]. L 2 norm of the Jacobian in the case with homogeneous Dirichlet boundary condition can be found in Kozono and Taniuchi [28], as well as Kim [27].…”

Section: Conclusion and Remarksmentioning

confidence: 99%

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations

Wang

2012

Numer. Math.

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Abstract. We investigate the long time behavior of the following efficient second order in time scheme for the 2D Navier-Stokes equation in a periodic box:The scheme is a combination of a 2nd order in time backward-differentiation (BDF) and a special explicit Adams-Bashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson type problem needs to be solved at each time step. We prove uniform in time bounds on this scheme inL 2 ,Ḣ 1 per andḢ 2 per provided that the time-step is sufficiently small. These time uniform estimates further lead to the convergence of long time statistics (stationary statistical properties) of the scheme to that of the NSE itself at vanishing time-step. Fully discrete schemes with either Galerkin Fourier or collocation Fourier spectral method are also discussed.Key words. Navier-Stokes equations, invariant measures, long time statistical properties, second order in time scheme, long time global stability AMS subject classifications. . If the the long time statistics of the system, i.e., the climate, is the subject of study, we then need to investigate the invariant measures of the system since it is the invariant measure (or stationary statistical solutions) that describes the long time statistics of the underlying dynamical system. Since most of these chaotic and/or turbulent systems are not amenable to analytical techniques at the present and in the near future, the issue of the development of numerical methods that are able to capture the long time statistics becomes very important. Higher order efficient schemes are apparently preferred due to the long time integration needed.In this paper, we will focus on the development and analysis of an efficient second order two step numerical method that is able to capture the long time statistics for the following two dimensional Navier-Stokes system for homogeneous incompressible Newtonian fluids in the vorticity-streamfunction formulation (see for instance [36])

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“…In the marginal case s = ∞, H. Kozono and Y. Taniuchi [7] proved the regularity of weak solutions under the condition…”

Section: Introductionmentioning

confidence: 99%

Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations

Chen

Zhang

2006

Proc. Amer. Math. Soc.

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Bilinear estimates in BMO and the Navier-Stokes equations

Abstract: We prove that the BM O 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.

Cited by 252 publications

References 15 publications

Characterization of blowup for the Navier-Stokes equations using vector potentials

Characterization of blowup for the Navier-Stokes equations using vector potentials

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations

Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations

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