Note on loss of regularity for solutions of the 3?D incompressible euler and related equations (original) (raw)
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Intermittency and Regularity Issues in 3D Navier-Stokes Turbulence
Two related open problems in the theory of 3D Navier-Stokes turbulence are discussed in this paper. The first is the phenomenon of intermittency in the dissipation field. Dissipation-range intermittency was first discovered experimentally by Batchelor and Townsend over fifty years ago. It is characterized by spatio-temporal binary behaviour in which long, quiescent periods in the velocity signal are interrupted by short, active 'events' during which there are violent fluctuations away from the average. The second and related problem is whether solutions of the 3D Navier-Stokes equations develop finite time singularities during these events. This paper shows that Leray's weak solutions of the three-dimensional incompressible Navier-Stokes equations can have a binary character in time. The time-axis is split into 'good' and 'bad' intervals: on the 'good' intervals solutions are bounded and regular, whereas singularities are still possible within the 'bad' intervals. An estimate for the width of the latter is very small and decreases with increasing Reynolds number. It also decreases relative to the lengths of the good intervals as the Reynolds number increases. Within these 'bad' intervals, lower bounds on the local energy dissipation rate and other quantities, such as u(·, t) ∞ and ∇u(·, t) ∞ , are very large, resulting in strong dynamics at sub-Kolmogorov scales. Intersections of bad intervals for n 1 are related to the potentially singular set in time. It is also proved that the Navier-Stokes equations are conditionally regular provided, in a given 'bad' interval, the energy has a lower bound that is decaying exponentially in time.
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.
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.
An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations
Communications in Mathematical Physics, 2017
We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have nonnegative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our L 3-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.
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.
The nearly singular behavior of the 3D Navier-Stokes equations
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 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 .
Numerical study of singularity formation in a class of Euler and Navier-Stokes flows
We study numerically a class of stretched solutions of the three-dimensional Euler and Navier-Stokes equations identified by Gibbon, Fokas, and Doering ͑1999͒. Pseudo-spectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover, this singularity apparently persists in the Navier-Stokes case. Independent evidence for the existence of a singularity is given by a Taylor series expansion in time. The mechanism underlying the formation of this singularity is the two-dimensionalization of the vorticity vector under strong compression; that is, the intensification of the azimuthal components associated with the diminishing of the axial component. It is suggested that the hollowing of the vortex accompanying this phenomenon may have some relevance to studies in vortex breakdown.