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1. Potentially Singular Behavior of the 3D Navier–Stokes Equations

   https://doi.org/10.1007/s10208-022-09578-4  

   Hou, Thomas Y. ( September 2022 , Foundations of Computational Mathematics)

   Whether the 3D incompressible Navier–Stokes equations can develop a finite time sin- gularity 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 incompress- ible 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 a companion paper published in the same issue, see also Hou (Poten- tial singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equa- tions develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 107. 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 neg- ative 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 (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of Lqt Lxp norm of the velocity with 3/p + 2/q ≤ 1. Our numerical results for the cases of (p,q) = (4,8), (6,4), (9,3) and (p,q) = (∞,2) provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of (p,q) = (3,∞) is more difficult to verify numerically due to the extremely slow growth rate in the L3 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 L3 norm of the velocity grows very slowly, the localized version of the L 3 norm of the velocity experiences rapid dynamic growth relative to the localized L 3 norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations. 

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2. Scaling laws and exact results in turbulence *

   https://doi.org/10.1088/1361-6544/ad6057  

   Novack, Matthew ( July 2024 , Nonlinearity)

   In this note, we address the validity of certain exact results from turbulence theory in the deterministic setting. The main tools, inspired by the work of Duchon and Robert (2000Nonlinearity13249–55) and Eyink (2003Nonlinearity16137), are a number of energy balance identities for weak solutions of the incompressible Euler and Navier–Stokes equations. As a consequence, we show that certain weak solutions of the Euler and Navier–Stokes equations satisfy deterministic versions of Kolmogorov’s$\frac{4}{5}$,$\frac{4}{3}$,$\frac{4}{15}$laws. We apply these computations to improve a recent result of Hofmanovaet al(2023 arXiv:2304.14470), which shows that a construction of solutions of forced Navier–Stokes due to Bruèet al(2023Commun. Pure Appl. Anal.) and exhibiting a form of anomalous dissipation satisfies asymptotic versions of Kolmogorov’s laws. In addition, we show that the globally dissipative 3D Euler flows recently constructed by Giriet al(2023 arXiv:2305.18509) satisfy the local versions of Kolmogorov’s laws.

    

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3. Circulation and Energy Theorem Preserving Stochastic Fluids

   https://doi.org/10.1017/prm.2019.43  

   Drivas, Theodore D. ; Holm, Darryl D. ( December 2020 , Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   null (Ed.)

   Abstract Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems. 

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4. Stochastic Lagrangian dynamics of vorticity. Part 1. General theory for viscous, incompressible fluids

   https://doi.org/10.1017/jfm.2020.491  

   Eyink, Gregory L. ; Gupta, Akshat ; Zaki, Tamer A. ( October 2020 , Journal of Fluid Mechanics)

   null (Ed.)

   Prior mathematical work of Constantin & Iyer ( Commun. Pure Appl. Maths , vol. 61, 2008, pp. 330–345; Ann. Appl. Probab. , vol. 21, 2011, pp. 1466–1492) has shown that incompressible Navier–Stokes solutions possess infinitely many stochastic Lagrangian conservation laws for vorticity, backward in time, which generalize the invariants of Cauchy ( Sciences mathématiques et physique , vol. I, 1815, pp. 33–73) for smooth Euler solutions. We reformulate this theory for the case of wall-bounded flows by appealing to the Kuz'min ( Phys. Lett. A , vol. 96, 1983, pp. 88–90)–Oseledets ( Russ. Math. Surv. , vol. 44, 1989, p. 210) representation of Navier–Stokes dynamics, in terms of the vortex-momentum density associated to a continuous distribution of infinitesimal vortex rings. The Constantin–Iyer theory provides an exact representation for vorticity at any interior point as an average over stochastic vorticity contributions transported from the wall. We point out relations of this Lagrangian formulation with the Eulerian theory of Lighthill (Boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), 1963, pp. 46–113)–Morton ( Geophys. Astrophys. Fluid Dyn. , vol. 28, 1984, pp. 277–308) for vorticity generation at solid walls, and also with a statistical result of Taylor ( Proc. R. Soc. Lond. A , vol. 135, 1932, pp. 685–702)–Huggins ( J. Low Temp. Phys. , vol. 96, 1994, pp. 317–346), which connects dissipative drag with organized cross-stream motion of vorticity and which is closely analogous to the ‘Josephson–Anderson relation’ for quantum superfluids. We elaborate a Monte Carlo numerical Lagrangian scheme to calculate the stochastic Cauchy invariants and their statistics, given the Eulerian space–time velocity field. The method is validated using an online database of a turbulent channel-flow simulation (Graham et al. , J. Turbul. , vol. 17, 2016, pp. 181–215), where conservation of the mean Cauchy invariant is verified for two selected buffer-layer events corresponding to an ‘ejection’ and a ‘sweep’. The variances of the stochastic Cauchy invariants grow exponentially backward in time, however, revealing Lagrangian chaos of the stochastic trajectories undergoing both fluid advection and viscous diffusion. 

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5. Smooth self-similar imploding profiles to 3D compressible Euler

   https://doi.org/10.1090/qam/1661  

   Buckmaster, Tristan ; Cao-Labora, Gonzalo ; Gómez-Serrano, Javier ( September 2023 , Quarterly of Applied Mathematics)

   The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of “imploding singularities” for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for all adiabatic exponents γ > 1 \gamma >1 in the case of Euler; as well as proving asymptotic self-similar blow-up for γ = 7 5 \gamma =\frac 75 in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations. 

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