Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence (original) (raw)
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Journal of Mathematical Physics, 2012
An unusual conditional regularity proof is presented for the three-dimensional forced Navier-Stokes equations from which a realistic picture of intermittency emerges. Based on L 2m-norms of the vorticity, denoted by Ω m (t) for m ≥ 1, the time integrals t 0 Ω αm m dτ with α m = 2m/(4m−3), play a key role in bounding the dissipation from below. By imposing a lower bound on t 0 Ω αm+1 m+1 dτ it is shown that Ω m (t) cannot become singular for large initial data. By considering movement in the value of t 0 Ω αm+1 m+1 dτ across this imposed critical lower bound, it is shown how solutions behave intermittently, in analogy with a relaxation oscillator. A cascade assumption is also considered.
Time-Global Regularity of the Navier-Stokes System with Hyper-Dissipation--Turbulent Scenario
arXiv (Cornell University), 2020
The question of whether the hyper-dissipative (HD) Navier-Stokes (NS) system can exhibit spontaneous formation of singularities in the super-critical regime-the hyper-dissipation being generated by a fractional power of the Laplacian confined to interval 1, 5 4-has been a major open problem in the mathematical fluid dynamics since the foundational work of J.L. Lions in 1960s. In this work, a mathematical evidence of criticality of the Laplacian is presented. While the framework for the proof is based on the 'scale of sparseness' of the super-level sets of the positive and negative parts of the components of the higher-order derivatives of the velocity or vorticity fields recently introduced in Grujić and Xu [31], a major novelty in the current work is the classification of the HD flows near a potential spatiotemporal singularity in two main categories, 'homogeneous' (the flows exhibiting a near-steady behavior) and 'nonhomogenous' (a generic case consistent with the formation and decay of turbulence). The main result states that in the non-homogeneous case, any power of the Laplacian greater than 1 yields a contradiction, preventing a blow-up. In order to illustrate the impact of this work on the stateof-the art, a two-parameter family of the rescaled (potential) blow-up profiles is considered, and it is shown that as soon as the power of the Laplacian is greater than one, a new region in the parameter space is ruled out (as a matter of fact, the region is a neighborhood of the self-similar profile, i.e., the 'approximately self-similar' blow-up is ruled out).
Energy-Based Regularity Criteria for the Navier–Stokes Equations
Journal of Mathematical Fluid Mechanics, 2009
We present several new regularity criteria for weak solutions u of the instationary Navier-Stokes system which additionally satisfy the strong energy inequality. (i) If the kinetic energy 1 2 u(t) 2 2 is Hölder continuous as a function of time t with Hölder exponent α ∈ ( 1 2 , 1), then u is regular. (ii) If for some α ∈ ( 1 2 , 1) the dissipation energy satisfies the left-side condition lim inf δ→0 1 δ α t t−δ ∇u 2 2 dτ < ∞ for all t of the given time interval, then u is regular. The proofs use local regularity results which are based on the theory of very weak solutions, see [1], , and on uniqueness arguments for weak solutions. Finally, in the last section we mention a local space-time regularity condition. . Primary 35Q30, 76D05, 35B65.
On the regularity of the solutions of the Navier–Stokes equations via one velocity component
Nonlinearity, 2010
We consider the regularity criteria for the incompressible Navier-Stokes equations connected with one velocity component. Based on the method from [4] we prove that the weak solution is regular, provided u 3 ∈ L t (0, T ; L s (R 3)), 2 t + 3 s ≤ 3 4 + 1 2s , s > 10 3 or provided ∇u 3 ∈ L t (0, T ; L s (R 3)), 2 t + 3 s ≤ 19 12 + 1 2s if s ∈ (30 19 , 3] or 2 t + 3 s ≤ 3 2 + 3 4s if s ∈ (3, ∞]. As a corollary, we also improve the regularity criteria expressed by the regularity of ∂p ∂x 3 or ∂u 3 ∂x 3 .
Intermittency in solutions of the three-dimensional Navier-Stokes equations
2003
Dissipation-range intermittency was first observed by Batchelor & Townsend (1949) in high Reynolds number turbulent flows. It typically manifests itself in spatio-temporal binary behaviour which is characterized by long, quiescent periods in the signal which are interrupted by short, active 'events' during which there are large excursions away from the average. It is shown that Leray's weak solutions of the three-dimensional incompressible Navier–Stokes equations can have this binary character in time.
Regimes of nonlinear depletion and regularity in the 3D Navier–Stokes equations
Nonlinearity, 2014
The periodic 3D Navier-Stokes equations are analyzed in terms of dimensionless, scaled, L 2mnorms of vorticity D m (1 ≤ m < ∞). The first in this hierarchy, D 1 , is the global enstrophy. Three regimes naturally occur in the D 1 − D m plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime [1] ; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.
Potentially Singular Behavior of the 3D Navier–Stokes Equations
Foundations of Computational Mathematics
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.
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.
On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay
Journal of Mathematical Fluid Mechanics, 2014
We study the energy decay of the turbulent solutions to the Navier-Stokes equations in the whole three-dimensional space. We show as the main result that the solutions with the energy decreasing at the rate O(t −α), t → ∞, α ∈ [0, 5/2], are exactly characterized by their initial conditions belonging into the homogeneous Besov spaceḂ −α 2,∞. Similarly, for a solution u and p ∈ [1, ∞] the integral ∞ 0 t α/2 u(t) p 1 t dt is finite if and only if the initial condition of u belongs to the homogeneous Besov spaceḂ −α 2,p. For the case α ∈ (5/2, 9/2] we present analogical results for some subclasses of turbulent solutions.