Global Regularity Solution of the 4th Clay Millennium Problem for the Periodic Navier Stokes Equations (original) (raw)
Related papers
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.
No Finite Time Blowup for 3D Incompressible Navier Stokes Equations via Scaling Invariance
Mathematics and Statistics, 2021
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.
No Finite Time Blowup for Incompressible Navier Stokes Equations via Scaling Invariance a Preprint
2021
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.
No Finite Time Blowup for Incompressible Navier Stokes Equations via Scaling Invariance
arXiv (Cornell University), 2020
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 and using the Gagliardo-Nirenberg inequalities it is shown that for any non zero and sufficiently large measure set in the form of cube subset of 3D there is finite time blowup for the non-starred parametrized velocity in the z-direction of flow. It is proposed that for the non-dimensional quantity δ used in the reparametrization of the Navier-Stokes equations, the starred velocity in the z * direction does not blowup and this corresponds to δ approaching infinity.
A shorter solution to the Clay millennium problem about regularity of the Navier-Stokes equations
2022
The Clay millennium problem regarding the Navier-Stokes equations is one of the seven famous difficult and significant mathematical problems. Although it is known that the set of Navier-Stokes equations has a unique smooth local time solution under the assumptions of the millennium problem, it is not known whether this solution can always be extended for all times smoothly, which is called the regularity (no blow-up) of the Navier-Stokes equations in 3 dimensions. Of course, the natural outcome would be that the regularity also holds for 3 dimensions since we know that it holds in 2 dimensions. Compared to the older solution proposed by Kyritsis (2021a) for the non-periodic setting without external forcing, this paper solves it also for the case with the periodic setting without external forcing. The strategy is based again in discovering new momentum density invariants derived from the well-known Helmholtz-Kelvin-Stokes theorem of the velocity circulation.
On the Partial Regularity of a 3D Model of the Navier-Stokes Equations
Communications in Mathematical Physics, 2009
We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric flows with swirl using a set of new variables. It preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected in the model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. In [11], we presented numerical evidence which seems to support that the 3D model develops finite time singularities while the corresponding solution of the 3D Navier-Stokes equations remains smooth. This suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term. In this paper, we prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The partial regularity result of this paper is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations.
Journal of Mathematical Analysis and Applications, 2018
We prove, among others, the following regularity criterion for the solutions to the Navier-Stokes equations: If u is a global weak solution satisfying the energy inequality and ω = ∇ × u, then u is regular on (0, T), T > 0, if two components of ω belong to the space L q (0, T ; Ḃ −3/p ∞,∞) for p ∈ (3, ∞) and 2/q + 3/p = 2. This result is an improvement of the results presented by Chae and Choe (1999) [7] or Zhang and Chen (2005) [38]. Our method of the proof uses a suitable application of the Bony decomposition and can also be used for the proofs of some other kin criteria. One such example is presented in Appendix.
2008
This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. In this paper, we study the 3D axisymmetric Navier–Stokes equations with swirl. We prove the global regularity of the 3D Navier–Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms ...
Incompressible Navier Stokes Equations
arXiv: Analysis of PDEs, 2020
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...