 Award ID(s):
 2205590
 NSFPAR ID:
 10437508
 Date Published:
 Journal Name:
 Foundations of Computational Mathematics
 ISSN:
 16153375
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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 selfsimilar singular scaling properties with maximum vorticity increased by a factor of 107. We have applied several blowup criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blowup criterion and the blowup 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.more » « less

Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $C^{1,\alpha}$ Velocity and Boundarynull (Ed.)Inspired by the numerical evidence of a potential 3D Euler singularity by Luo Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $C^{1,\alpha}$ initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with $C^{1,\alpha}$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30,31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $C^{1,\alpha}$ initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate selfsimilar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with $C^{1,\alpha}$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time.more » « less

The aim of this note is to present the recent results by Buckmaster, CaoLabora, and GómezSerrano [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 NavierStokes 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 selfsimilar profiles for all adiabatic exponents γ > 1 \gamma >1 in the case of Euler; as well as proving asymptotic selfsimilar blowup for γ = 7 5 \gamma =\frac 75 in the case of NavierStokes. Importantly, for the NavierStokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blowup in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations.more » « less

Abstract We study the singularity formation of a quasiexact 1D model proposed by Hou and Li (2008
Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in ther direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear inr . This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990J. Stat. Phys. 59 1251–63; De Gregorio 1996Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a selfsimilar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the lowfrequency Fourier modes and extract damping in leading order estimates for the highfrequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of highorder Sobolev norms. 
Berselli, Luigi G ; Ruzicka, Michael (Ed.)The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a specific singular scenario for the threedimensional Euler equation discovered by Hou and Luo, and analyze some associated models. Finally, we will also talk about the surface quasigeostrophic (SQG) equation, and construct an example of singularity formation in the modified SQG patch solutions as well as examples of unbounded growth of derivatives for the smooth solutions.more » « less