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Existence of a Hölder Continuous Extension on Embedded Balls of the 3-Torus for the Periodic navier stokes Equations

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Advances in Pure Mathematics 2024年第2期14卷 118-138页

作者： Terry E. Moschandreou Intermediate Science and Mathematics TVDSB London Ontario Canada

This article gives a general model using specific periodic special functions, which is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the periodic navier stokes equations (PNS) at the centers of cells of the 3-Torus. Satisfying a divergence-free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term which is smooth and spatially periodic, the existence of solutions which have finite time singularities can occur starting with the first derivative and higher with respect to time. The existence of a subspace of the solution space where v3 is continuous and {C, y1, y12}, is linearly independent in the additive argument of the solution in terms of the Lambert W function, (y12=y2, C∈R) together with the condition v2=-2y1v1. On this subspace, the Biot Savart Law holds exactly [see Section 2 (Equation (13))]. Also on this subspace, an expression X (part of PNS equations) vanishes which contains all the expressions in derivatives of v1 and v2 and the forcing terms in the plane which are related as with the cancellation of all such terms in governing PDE. The y3 component forcing term is arbitrarily small in ε ball where Weierstrass P functions touch the center of the ball both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3 only governing PDE resulting. With viscosity present as v changes from zero to the fully viscous case at v =1 the solution for v3 reaches a peak in the third component y3. Consequently, there exists a dipole which is not centered at the center of the cell of the Lattice. Hence since the dipole by definition has an equal in magnitude positive and negative peak in y3, then the dipole Riemann cut-off surface is c

关键词： navier-stokes PNS 3-Torus Periodic Ball Sphere Hölder Continuous Riemann-Surface Uniqueness

Analysis of a Linearized Energy Stable Numerical Scheme for a Modified Incompressible Cahn-Hilliard-navier-stokes System

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Acta Mathematicae Applicatae Sinica 2023年第3期39卷 605-622页

作者： Xue WANG Hong-en JIA Ming LI Kai-tai LI School of Mathematics and Statistics Wuhan UniversityWuhan 430072China College of Mathematics Taiyuan University of TechnologyTaiyuan 030024China College of Data Science Taiyuan University of TechnologyTaiyuan 030024China School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'an 710049China

In this paper,a linearized energy stable numerical scheme is used to solve the modified Cahn-Hilliard-navier-stokes model,which is a phase-field model for two-phase incompressible *** time discretization is based on the convex splitting of the energy functional,which leads to a linearized *** order to maintain the energy stability,the definition domain of energy function is extended to *** stability of the scheme is proved and the error estimate is *** experiments are done to demonstrate the effectiveness for the proposed scheme.

关键词： convex splitting two-phase incompressible flow energy stability mixed finite element navier-stokes

On Some Properties of the Curl Operator and Their Consequences for the navier-stokes System

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Communications in Mathematical Research 2022年第4期38卷 449-497页

作者： Nicolas Lerner François Vigneron Sorbonne Universite Institut de Mathématiques de JussieuUMR 7586Campus Pierre et Marie Curie4 Place Jussieu75252 Paris cedex 05France Universitéde Reims Champagne-Ardenne Laboratoire de Mathématiques de ReimsUMR 9008Moulin de la HousseBP 103951687 Reims cedex 2France

We investigate some geometric properties of the curl operator,based on its diagonalization and its expression as a non-local symmetry of the pseudoderivative(-△)^(1/2) among divergence-free vector fields with finite *** this context,we introduce the notion of spin-definite fields,*** of(-△)^(-1/2)*** two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed *** observed that the non-linearity of navier-stokes has the structure of a crossproduct and its weak(distributional)form is a determinant that involves the vorticity,the velocity and a test function,we revisit the conservation of energy and the balance of helicity in a geometrical *** show that in the case of a finite-time blow-up,both spin-definite components of the flow will explode simultaneously and with equal rates,*** in 3D are the result of a conflict of spin,which is impossible in the poorer geometry of 2D *** investigate the role of the local and non-local determinants ∫_(0)^(T) ∫_(R^(3))det(curlu,u,(-△)^(θ)u) and their spin-definite counterparts,which drive the enstrophy and,more generally,are responsible for the regularity of the flow and the emergence of singularities or *** such,they are at the core of turbulence phenomena.

关键词： navier-stokes vorticity hydrodynamic spin critical determinants turbulence

Limiting case for the regularity criterion of the navier-stokes equations and the magnetohydrodynamic equations

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Science China Mathematics 2010年第7期53卷 1764-1771页

作者： He Cheng Wang Yun Natl Nat Sci Fdn China Dept Math & Phys Sci Div Math Beijing 100085 Peoples R China Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China Chinese Univ Hong Kong Inst Math Sci Shatin Hong Kong Peoples R China

Any weak solution u to the navier-stokes equations is showed to be regular under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small, which is a limiting case of the regularity criteria derived by Kim and Kozono. Our result gives a positive answer to the question proposed by Kim and Kozono. For the incompressible magnetohydrodynamic equations, we also show the regularity of weak solution only under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small.

关键词： navier-stokes equations magnetohydrodynamics equations regularity criterion

Comparison of Reynolds Averaged navier-stokes Based Simulation and Large-eddy Simulation for One Isothermal Swirling Flow

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Journal of Thermal Science 2012年第2期21卷 154-161页

作者： Sφren Knudsen Kr Department of Energy Technology Pontoppidanstrde 101 9220 Aalborg Denmark

The flow structure of one isothermal swirling case in the Sydney swirl flame database was studied using two numerical methods. Results from the Reynolds-averaged navier-stokes (RANS) approach and large eddy simulation (LES) were compared with experimental measurements. The simulations were applied in two different Cartesian grids which were investigated by a grid independence study for RANS and a post-estimator for LES. The RNG k-ε turbulence model was used in RANS and dynamic Smagorinsky-Lilly model was used as the sub-grid scale model in LES. A validation study and cross comparison of ensemble average and root mean square (RMS) results showed LES outperforms RANS statistic results. Flow field results indicated that both approaches could capture dominant flow structures, like vortex breakdown (VB), and precessing vortex core (PVC). Streamlines indicate that the formation mechanisms of VB deducted from the two methods were different. The vorticity field was also studied using a velocity gradient based method. This research gained in-depth understanding of isothermal swirling flow.

关键词： navier-stokes 雷诺平均大涡模拟旋流等温流场结构湍流模型数值方法

Optimal dynamical systems of navier-stokes equations based on generalized helical-wave bases and the fundamental elements of turbulence

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Science China(Physics,Mechanics & Astronomy) 2016年第11期59卷 61-69页

作者： NaiFu Peng Hui Guan ChuiJie Wu State Key Laboratory of Structural Analysis for Industrial Equipment School of Aeronautics and AstronauticsDalian University of Technology State Key Laboratory for Turbulence and Complex Systems Center for Applied Physics and TechnologyCollege of Engineering Peking University College of Meteorology and Oceanography PLA University of Science and Technology

In this paper, we present the theory of constructing optimal generalized helical-wave coupling dynamical systems. Applying the helical-wave decomposition method to navier-stokes equations, we derive a pair of coupling dynamical systems based on optimal generalized helical-wave bases. Then with the method of multi-scale global optimization based on coarse graining analysis, a set of global optimal generalized helical-wave bases is obtained. Optimal generalized helical-wave bases retain the good properties of classical helical-wave bases. Moreover, they are optimal for the dynamical systems of navier-stokes equations, and suitable for complex physical and geometric boundary conditions. Then we find that the optimal generalized helical-wave vortexes fitted by a finite number of optimal generalized helical-wave bases can be used as the fundamental elements of turbulence, and have important significance for studying physical properties of complex flows and turbulent vortex structures in a deeper level.

关键词： navier-stokes 动力系统螺旋波斯托克斯方程广义基本元素湍流基础

On scaling invariance and type-Ⅰ singularities for the compressible navier-stokes equations

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Science China Mathematics 2019年第11期62卷 2271-2286页

作者： Zhen Lei Zhouping Xin School of Mathematical Sciences LMNS and Shanghai Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghai 200433China The Institute of Mathematical Sciences and Department of Mathematics The Chinese University of Hong KongHong KongChina

We find a new scaling invariance of the barotropic compressible navier-stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here κ > 0 does not depend on the initial *** is achieved by proving the regularity of solutions under■ This new scaling invariance also motivates us to construct an explicit type-Ⅱ blowup solution for γ > 1.

关键词： type-Ⅰ singularity compressible navier-stokes equations scaling invariance blowup rate

Nash Embedding,Shape Operator and navier-stokes Equation on a Riemannian Manifold

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Acta Mathematicae Applicatae Sinica 2020年第2期36卷 237-252页

作者： Shizan FANG Institut de Mathematiques de Bourgogne UMR 5584 CNRSUniversite de Bourgogne Franche-ComteF-21000 DijonFrance

What is the suitable Laplace operator on vector fields for the navier-stokes equation on a Riemannian manifold?In this note,by considering Nash embedding,we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden’s Laplacian.A probabilistic representation formula for navier-stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved.

关键词： Nash embedding shape operator vector valued Laplacians navier-stokes equations stochastic representation

Solving navier-stokes with Maclaurin Series

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Journal of Applied Mathematics and Physics 2022年第4期10卷 1362-1374页

作者： Gabriele Martino Rome Italy

In this paper I propose a method for founding solutions of navier-stokes equations. Purpose of the research is to solve equations giving form to relations between pressure, velocity and stream. Starting from the fact we do not know the form of functions we give a general representation in Maclaurin Series and prove that with reasonable values of parameters, representation holds and therefore has meaning in continuum. Then we solve the system of equations with respect to the pressure and match equations relation between parameters: matches of equations are possible because of the physical dimensions of equations. Then values of Continuity Equation are verified. The result is a polynomial finite and that coincides with the function in continuum, or is anyway one of its representation. The result under hydrostatic condition returns Stevino formula.

关键词： navier-stokes Momentum Continuity Differential Equation Maclaurin Polynomial