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Spontaneous symmetry breaking and finite-time singularities in d-dimensional incompressible flows with fractional dissipation
| Content Provider | Semantic Scholar |
|---|---|
| Copyright Year | 2008 |
| Abstract | We investigate the formation of singularities in incompressible flows governed by Navier-Stokes equations in d 2 dimensions with a fractional Laplacian |∇|α. We derive analytically a sufficient but not necessary condition for the solutions to remain always smooth and show that finite-time singularities cannot form for α αc = 1+ d/2. Moreover, initial singularities become unstable for α>αc. The scale invariance symmetry intrinsic to the Navier-Stokes system becomes spontaneously broken, except at the critical point α= αc. Copyright c © EPLA, 2008 Scale invariance symmetry [1–4] holds approximately for the nonlinear Navier-Stokes equations for the incompressible flow of a Newtonian fluid [5–7]. The nonlinearity and the scale invariance together can create conditions for the energy to cascade down to increasingly finer spatial and temporal scales, e.g., turbulence [7]. In two dimensions, singularities cannot form [5]. However, more than a century since the discovery of these nonlinear parabolic partial differential equations, the question remains unanswered whether or not singularities can form in three dimensions, due to the crucial role played by scale invariance. This problem becomes even more difficult in higher dimensions d> 3. Incompressible flows in higher dimensions arise, for example, in the well-known Liouville description of Hamiltonian dynamics in phase space. Under what conditions does the flow remain free of singularities? How can we quantify the effects of dimensionality on the (possibly chaotic [2] and turbulent) cascades of energy to ever finer scales? Such fundamental and challenging problems remain the subject of ongoing investigations, due to their importance to a number of fields of physics and mathematics [5,7–20]. Here we adapt the concepts and methods of statistical physics to describe quantitatively the cascade process responsible for the formation of singularities. Our approach involves interpreting the cascade process as (a)E-mail: Gandhi.Viswanathan@pq.cnpq.br (b)E-mail: viswanathan.tenkasi@gmail.com equivalent to the anomalous diffusion [21–30] of energy in Fourier space. This conceptual innovation enables us to determine in any dimension d 2 a new sufficient condition for incompressible flows to remain smooth. We briefly outline the main ideas underlying our analytical methodology before providing the complete details. We begin by investigating d-dimensional generalized NavierStokes equations with a fractional Laplacian operator [9]. The motivation for using fractional Laplacians stems from the ability that they confer to parameterize the strength of the dissipation necessary to disrupt the energy cascades, which become more violent as the number of dimensions increases. We then analytically derive a hard inequality based on the fact that no singularity can form if all partial space derivatives of all orders of the velocity field remain finite for all time. We obtain from this inequality three new results concerning the formation of singularities. We first note the well known fact that the Fourier transform of a smooth function cannot decay algebraically but rather must decay rapidly. In terms of the d-dimensional Fourier transform ṽ(k, t) of the velocity field v(x, t), we can write (2π) ∣∣∣ ∂ n1 ∂x1 1 ∂2 ∂x2 2 . . . ∂d ∂xd d v(x) ∣∣∣= ∣∣∣ ∫ ∞ −∞ dk exp[ik ·x]ṽ(k)(ik1)1(ik2)2 . . . (ikd) ∣∣∣ ∫ ∞ −∞ dkk12d |ṽ(k)|, k= |k|. (1) |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://physics.unm.edu/Kenkre/pdflink/epl_84_5_50006.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Cascade Device Component Chaos theory Control theory Critical point (network science) Dimensions Flow Fractional Fourier transform Greater Than Html Link Type - copyright In-phase and quadrature components Influenza virus A N1 Ab:PrThr:Pt:Ser:Ord Laplacian matrix Navier–Stokes equations Nonlinear system Parabolic antenna Social inequality Solutions Spontaneous order Symmetry breaking Turbulence Unstable Medical Device Problem Velocity (software development) Whole Earth 'Lectronic Link orders - HL7PublishingDomain |
| Content Type | Text |
| Resource Type | Article |
