Decay and Vanishing of some D-Solutions of the Navier–Stokes Equations

Carrillo, Bryan; Pan, Xinghong; Zhang, Qi S.; Zhao, Ning

doi:10.1007/s00205-020-01533-3

Cited by 26 publications

(20 citation statements)

References 16 publications

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“…Then the vanishing of u with zero force follows easily. This was extended by Carrillo et al [1,Theorem 1.1], showing that any D-solution satisfying (2) in a slab with zero BC is zero.…”

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confidence: 79%

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Liouville type theorems for stationary Navier–Stokes equations

Tsai

2021

SN Partial Differ. Equ. Appl.

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We show that any smooth stationary solution of the 3D incompressible Navier-Stokes equations in the whole space, the half space, or a periodic slab must vanish under the condition that for some 0 d 1\L and q ¼ 6ð3 À dÞ=ð6 À dÞ,We also prove sufficient conditions allowing shrinking radii ratioSimilar results hold on a slab with zero boundary condition by assuming stronger decay rates. We do not assume global bound of the velocity. The key is to estimate the pressure locally in the annuli with radii ratio L arbitrarily close to 1. IntroductionConsider the Liouville problem of 3D stationary incompressible Navier-Stokes equationswhere the domain X is either the whole space R 3 , the half space R 3 þ with zero boundary condition, or the slab X ¼ R 2 Â ð0; 1Þ with zero or periodic boundary condition (BC). In the classical setting X ¼ R 3 , one asks if the only H 1 loc solution satisfying Dedicated to Hideo Kozono on the occasion of his 60th birthday.

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“…Then the vanishing of u with zero force follows easily. This was extended by Carrillo et al [1,Theorem 1.1], showing that any D-solution satisfying (2) in a slab with zero BC is zero.…”

mentioning

confidence: 79%

“…There is also a rich literature on the Liouville problem for the subclass of axisymmetric solutions. As we will not discuss it here, we only refer to [1,10,19] and their references.…”

mentioning

confidence: 99%

Liouville type theorems for stationary Navier–Stokes equations

Tsai

2021

SN Partial Differ. Equ. Appl.

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“…There has already been much literature studying Liouville-type results on the Navier-Stokes equations subject to various boundary conditions in various unbounded domains. Readers can refer to [5,6,25,26,4,23] and references therein for more Liouville-type results on the stationary Navier-Stokes equations. Moreover, our results in the above Theorems can be extended from the stationary case to the case of ancient solutions (backward global solutions) under suitable assumptions.…”

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confidence: 99%

Characterization of smooth solutions to the Navier-Stokes equations in a pipe with two types of slip boundary conditions

Li¹,

Pan²,

Yang³

2021

Preprint

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Bounded smooth solutions of the stationary axially symmetric Navier-Stokes equations in an infinite pipe, equipped with the Navier-slip boundary condition, are considered in this paper.Here "smooth" means the velocity is continuous up to second-order derivatives, and "bounded" means the velocity itself and its gradient field are bounded. It is shown that such solutions with zero flux at one cross section, must be swirling solutions: u = (−Cx 2 , Cx 1 , 0). A slight modification of the proof will show that for an alternative slip boundary condition, solutions will be identically zero.Meanwhile, if the horizontal swirl component of the axially symmetric solution, u θ , is independent of the vertical variable z, it is proven that such solutions must be helical solutions:In this case, boundedness assumptions on solutions can be relaxed extensively to the following growing conditions:With respect to the distance to the origin, the vertical component of the velocity, u z , is sublinearly growing, the horizontal radial component of the velocity, u r , is exponentially growing, and the swirl component of the vorticity, ω θ , is polynomially growing at any order.Also, by constructing a counterexample, we show that the growing assumption on u r is optimal.

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“…(3) On the basis of the coarse grid, the normal direction grid points of the object surface are kept unchanged, the chord and spanwise grid points are appropriately increased, the length-width ratio of the grid is improved, and the hypercube medium density grid with ten thousand grid points is generated. On the basis of the medium grid [19,20], the distance from the first grid to the object surface is reduced from fire to fire. When the grid is smaller to other two directions, increase the number of normal grid points on the object surface, improve the length-width ratio of the mesh to avoid excessive stretching of the mesh near the object surface [21], and then form a hypercube fine grid, referred to as fine grid, with a scale of 10000 grid points.…”

Section: Turbulence Model-based Simulationmentioning

confidence: 99%

Simulation of J-Solution Solving Process of Navier–Stokes Equation

Wang

Ayana

2021

Mathematical Problems in Engineering

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To avoid grid degradation, the numerical analysis of the j-solution of the Navier–Stokes equation has been studied. The Navier–Stokes equations describe the motion of viscous fluid substances. On the basis of the advantages and disadvantages of the Navier–Stokes equations, the incompressible terms and the nonlinear terms are separated, and the original boundary conditions satisfying the j-solution of the Navier–Stokes equation are analyzed. Secondly, the development of a computational grid has been introduced; the turbulence model has also been described. The fluid form and the initial value of the j-solution of the Navier–Stokes equation are combined. The original boundary conditions are solved by a computer, and the nonlinear turbulence equations are derived, which control the fluid flow. The simulation of the fine grid is comprehended to analyze the research outcome. Simulation analysis is carried out to generate multiblock-structured grids with high quality. The j-solution on the grid points is the j-solution that can be used with a fewer number of meshes under the same conditions. The proposed work is easy to implement, and it consumes lesser memory. The results obtained are able to avoid mesh degradation skillfully, and the generated mesh exhibits the characteristics of smoothness, orthogonality, and controllability, which eventually improves the calculation accuracy.

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Decay and Vanishing of some D-Solutions of the Navier–Stokes Equations

Cited by 26 publications

References 16 publications

Liouville type theorems for stationary Navier–Stokes equations

Liouville type theorems for stationary Navier–Stokes equations

Characterization of smooth solutions to the Navier-Stokes equations in a pipe with two types of slip boundary conditions

Simulation of J-Solution Solving Process of Navier–Stokes Equation

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