Periodic Structure of 2-D Navier-Stokes Equations (original) (raw)
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On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations
2019
This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier-Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasiliner systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 000, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal LpL_pLp-$L_q$ regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, wh...
2018
Solution of Navier_Stokes equations is derived by transforming partial differential equation in to ordinary second order differential equations. Transforming three variable periodic initial velocity vector fields in to one variable vector field can be achieved by writing the vector field as the sum of sine and cosine series. Fourier series representation of vector fields as foundations for periodic smooth solutions. The dot product of velocity vector field with period vector is constant in space and varying in time which is consequence of divergence free nature of the vector field under study. The solutions discovered here is not unique, which implies Navier_Stokes equation has no unique periodic solution. The non uniqueness of the periodic solution implies the equation is incomplete. The existence of non unique smooth periodic solution for a given periodic smooth initial velocity vector shows the solution is counter example for Navier_Stokes equation.
Asymptotic structure for solutions of the Navier--Stokes equations
Discrete and Continuous Dynamical Systems, 2004
We study in this article the large time asymptotic structural stability and structural evolution in the physical space for the solutions of the 2-D Navier-Stokes equations with the periodic boundary conditions. Both the Hamiltonian and block structural stabilities and structural evolutions are considered, and connections to the Lyapunov stability are also given.
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.
arXiv: Analysis of PDEs, 2021
A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R 3 /T 3 is hereby shown as well as results on the velocity-pressure distribution. 1 Introduction The global regularity of the Navier-Stokes equations remains to be an outstanding unsolved problem in fluid mechanics. The Clay Institute is offering a significant prize for those who are successful in solving either one of four proposed problems, that is either a periodic or non-periodic regular or finite time blowup problem for the full 3D Navier Stokes equations.[1] The motivation behind the author's present contribution and work was to pursue the periodic Navier Stokes equations problem and it was shown by corresponding author [2](using Geometric Algebra) and then the authors in [3] (also see references contained therein) who confirmed in an applied setting that there is no finite time blowup for the special case of the theoretical class of solutions G being multiplicative between spatial variable y and coordinates (x, y, t). (see[2] and Eq.(13,15) of [3]) In addition the plots shown in [3] (Figure 1 there) were obtained for the special case of F (x, y) = 1 in [3](Eq13); Saddle surface generation and saddle orbits were obtained where the initial condition was a min-max type function with a min-max point at the center of a typical cube in the periodic Lattice of R 3 /T 3. This was plotted in [3] in Fig 1 (a,c) there. These saddle orbits indicate that there are instabilities leading to turbulence or possibly even STC. [4] Also the form of the solution obtained showed no finite time blowup. However the problem remained unsolved as to the general class of the form of solutions G due to the arbitrariness of the function itself. It is the purpose of the following article to provide a rigorous mathematical proof that G cannot be a blowup itself with respect to t except at a finite number of points in any given bounded subset of R 3 /T 3 if and only if in the initial data the energy is infinite. It is hypothesized that energy "cascades" from large-scale structures to smaller scale structures by an inertial and inviscid mechanism. This process does continue, and there is the creation of smaller and smaller structures which produce a hierarchy of eddies. It is inevitable that this process produces structures that are so small that molecular diffusion becomes important and viscous dissipation of energy eventually takes place. The scale at which this happens is the Kolmogorov length scale. [5] At these fine scales there is potential finite time blowup. This is also addressed above and beyond what was done in [2] and [3]. The issue of stability of solutions to the forced Navier-Stokes and damped Euler systems in periodic boxes has been studied in [6].There it is shown that for large, but fixed, Grashoff (Reynolds) number the turbulent behavior of all Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations, in a periodic box, is suppressed, when viewed in the right frame of reference, by large enough average flow of the initial data. In the present work there exists such a frame of reference in the z direction where the average flow of the initial data is arbitrarily large. For the form of solution G which is multiplicative, a unique time-periodic solution exists when the average flow of the initial data is large or a fast oscillating forcing term occurs, with no blowup. Here there is a suppression of turbulence. However for a general form of G, it is shown that there exists infinite type singularities on sets of measure zero corresponding to the centres of periodic cells of the
Bifurcation of periodic solutions of the Navier-Stokes equation in a thin domain
Aim of this paper is to provide conditions in order to guarantee that the periodic solutions in time and in the space variables of the Navier-Stokes equations bifurcate. Specifically, we study this problem when the considered state domain has one dimension which is small with respect to the others which we let to tend to zero. The thinness of the domain represents the bifurcation parameter in our situation. ∇ · U = 0, 1991 Mathematics Subject Classification. 35B10, 35B32.
DISCRETE AND CONTINUOUS NAVIER–STOKES EQUATIONS ON THE β-PLANE
We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier-Stokes equations on the periodic β-plane (i.e. with the Coriolis force varying as f 0 + βy) will become nearly zonal: with the vorticity ω(x, y, t) =ω(y, t) +ω(x, y, t), one has |ω| 2 H s ≤ β −1 Ms(· · · ) as t → ∞. We use this show that, for sufficiently large β, the global attractor of this system reduces to a point.
2018
Due to the existence of huge number of different information on Navier_Stokes equation on internet, introduction and method used to come to the following solution is less important than the solution its self. As a result the paper shows the periodic solution for Navier_Stokes equations. All conditions for physically reasonable solution as posted by clay mathematics institute is fulfilled. The following solution is counter example for existence of smooth unique periodic solution.