The Three-Dimensional Navier–Stokes Equations

Robinson, James C.; Rodrigo, José Luis; Sadowski, Witold

doi:10.1017/cbo9781139095143

Cited by 211 publications

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“…is important, because it is a critical norm in the analysis of the Navier-Stokes system (2) (Robinson et al, 2016). As we can see in figure 15, the L 3 norm decays monotonically in each case.…”

Section: Computational Resultsmentioning

confidence: 91%

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

2020

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This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system -as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy E 0 are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time T . Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of E 0 and T , we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to E 3/2 0 as E 0 becomes large. Thus, in such worst-case scenario the enstrophy still remains bounded for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.

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Section: Computational Resultsmentioning

confidence: 91%

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

2020

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“…Consequently, the literature concerning the analysis of these equations is vast. For an overview of the field we refer the reader to the classical texts [37,191] and also to the books [56,133,22,192,79,141,126,143,127,166,193]. In order to place the convex integration constructions in context, in this section we recall only a few of the rigorous mathematical results known about (1.1) and (1.2).…”

Section: Organization Of the Papermentioning

confidence: 99%

Convex integration and phenomenologies in turbulence

Buckmaster

Vicol

2020

EMS Surv. Math. Sci.

110

145

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In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Székelyhidi Jr., who extended Nash's fundamental ideas on C 1 flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies with the phenomenological theories of hydrodynamic turbulence [47,188,50,51]. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions.First, we give an elementary construction of nonconservative C 0+ x,t weak solutions of the Euler equations, first proven by De Lellis-Székelyhidi Jr. [49,48]. Second, we present Isett's [102] recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work [18] of De Lellis-Székelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class C 1 /3− x,t are constructed, attaining any energy profile. Third, we give a concise proof of the authors' recent result [20], which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity classx . We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.

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“…The constant C * (δ) comes from going from scales r n to all r ∈ (0, 1 4 ]. The proof is by iteration following the scheme of Caffarelli, Kohn and Nirenberg [9] (see also [46]). Our aim is to propagate for k ≥ 2 the following two bounds…”

Section: Propagation Of a Morrey-type Boundmentioning

confidence: 99%

Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

Barker

Prange

2020

Arch Rational Mech Anal

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This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space L 3 , then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia andŠverák, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if (0, T * ) is a singular point thenThis result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely L 3,∞ and the Besov spaceḂ −1+ 3 p p,∞ , p ∈ (3, ∞).

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The Three-Dimensional Navier–Stokes Equations

Cited by 211 publications

References 0 publications

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

Convex integration and phenomenologies in turbulence

Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

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