No Finite Time Blowup for Incompressible Navier Stokes Equations via Scaling Invariance (original) (raw)
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A closely related problem to The Clay Math Institute "Navier-Stokes, breakdown of smooth solutions here on an arbitrary cube subset of three dimensional space with periodic boundary conditions is examined. The incompressible Navier-Stokes Equations are presented in a new and conventionally different way here, by naturally reducing them to an operator form which is then further analyzed. It is shown that a reduction to a general 2D N-S system decoupled from a 1D non-linear partial differential equation is possible to obtain. This is executed using integration over n-dimensional compact intervals which allows decoupling. Here we extract the measure-zero points in the domain where singularities may occur and are left with a pde that exhibits finite time singularity. The operator form is considered in a physical geometric vorticity case, and a more general case. In the general case, the solution is revealed to have smooth solutions which exhibit finite-time blowup on a fine measure zero set using the Poincaré and Gagliardo-Nirenberg inequalities and it is shown that for any non zero sufficiently large measure set in the form of cube subset of 3D there is no finite time blowup for the starred velocity for large dimension of cube and small δ. In particular vortices are shown to exist.
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The problem to The Clay Math Institute "Navier-Stokes, breakdown of smooth solutions here on an arbitrary cube subset of three dimensional space with periodic boundary conditions is examined. The incompressible Navier-Stokes Equations are presented in a new and conventionally different way here, by naturally reducing them to an operator form which is then further analyzed. It is shown that a reduction to a general 2D N-S system decoupled from a 1D non-linear partial differential equation is possible to obtain. This is executed using integration over n-dimensional compact intervals which allows decoupling. The operator form is considered in a physical geometric vorticity case, and a more general case. In the general case, the solution is revealed to have smooth solutions which exhibit finite-time blowup on a fine measure zero set and using the Prékopa-Leindler and Gagliardo-Nirenberg inequalities it is shown that for any non zero measure set in the form of cube subset of 3D there is no finite time blowup for the starred velocity for large dimension of cube and small δ. In particular vortices are shown to exist and it is shown that zero is in the attractor of the 3D Navier-Stokes equations.
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A closely related problem to The Clay Math Institute "Navier-Stokes, breakdown of smooth solutions here on an arbitrary cube subset of three dimensional space with periodic boundary conditions is examined. The incompressible Navier-Stokes Equations are presented in a new and conventionally different way here, by naturally reducing them to an operator form which is then further analyzed. It is shown that a reduction to a general 2D N-S system decoupled from a 1D non-linear partial differential equation is possible to obtain. This is executed using integration over n-dimensional compact intervals which allows decoupling. Here we extract the measure-zero points in the domain where singularities may occur and are left with a pde that exhibits finite time singularity. The operator form is considered in a physical geometric vorticity case, and a more general case. In the general case, the solution is revealed to have smooth solutions which exhibit finite-time blowup on a fine measure...
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A. Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in the companion paper [ ]. We present numerical evidence that the 3D Navier-Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 10 7. We have applied several blow-up criteria to study the potentially singular behavior of the Navier-Stokes equations. The Beale-Kato-Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier-Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya-Prodi-Serrin regularity criteria [ , , ] that are based on the growth rate of norm of the velocity with 3/ + 2/ ≤ 1. Our numerical results for the cases of (,) = (4, 8), (6, 4), (9, 3) and (,) = (∞, 2) provide strong evidence for the potentially singular behavior of the Navier-Stokes equations. The critical case of (,) = (3, ∞) is more difficult to verify numerically due to the extremely slow growth rate in the 3 norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global 3 norm of the velocity grows very slowly, the localized version of the 3 norm of the velocity experiences rapid dynamic growth relative to the localized 3 norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier-Stokes equations.
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ArXiv, 2021
Whether the 3D incompressible Navier–Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D incompressible axisymmetric Navier–Stokes equations with smooth initial data of finite energy develop nearly singular solutions at the origin. This nearly singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in [20]. One important feature of the potential Euler singularity is that the solution develops nearly self-similar scaling properties that are compatible with those of the 3D Navier–Stokes equations. We will present numerical evidence that the 3D Navier–Stokes equations develop nearly singular scaling properties with maximum vorticity increased by a factor of 107. Moreover, the nearly self-similar profiles seem to be very stable to the small perturbation of the initial data. However,...
On singularity formation of a 3D model for incompressible Navier–Stokes equations
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We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei in [16] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier-Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.
Cornell University - arXiv, 2020
A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R 3 /Z 3 has been shown by corresponding author of the present work [1]. Smooth solutions for the z−component momentum equation u z assuming the x and y component equations have vortex smooth solutions have been proven to exist, however the Clay Institute Millennium problem on the Navier Stokes equations was not proven for a general enough vorticity form and [1], [3] and references therein do not prove this as previously thought. The idea was to show that Geometric Algebra can be applied to all three momentum equations by adding any two of the three equations and thus combinatorially producing either u x , u y or u z as smooth solutions at a time. It was shown that using the Gagliardo-Nirenberg and Prékopa-Leindler inequalities together with Debreu's theorem and some auxiliary theorems proven in [1] that there is no finite time blowup for 3D Navier Stokes equations for a constant vorticity in the z direction. In part I of the present work it is shown that using Hardy's inequality for u 2 z term in the Navier Stokes Equations that a resulting PDE emerges which can be coupled to auxiliary pde's which give us wave equations in each of the three principal directions of flow. The present work is extended to all spatial directions of flow for the most general flow conditions. In Part II it is shown for the first time that the full system of 3D Incompressible Navier Stokes equations without the above mentioned coupling consists of non-smooth solutions. In particular if u x , u y satisfy a non-constant zvorticity for 3D vorticity ω, then higher order derivatives blowup in finite time but u z remains regular. So a counterexample of the Navier Stokes equations having smooth solutions is shown. A specific time dependent vorticity is also considered.
Finite Time Blow Up for a Navier-Stokes Like Equation
Proceedings of the American Mathematical Society, 2001
Abstract. We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or ...