We study the singularity formation of a quasiexact 1D model proposed by Hou and Li (2008
Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to nonfederal websites. Their policies may differ from this site.

Abstract 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.Free, publiclyaccessible full text available January 22, 2025 
We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in highfrequency wave propagation. The ExpMsFEM is built on the nonoverlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the righthand side to break the algebraic Kolmogorov nwidth barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the righthand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the righthand side, so the stiffness matrix can be used repeatedly in multiquery scenarios.more » « less

Whether the 3D incompressible Euler 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 axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blowup scenario revealed by Luo and Hou (111:12968–12973, 2014) and (12:1722–1776, 2014), which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou and Huang in (arXiv:2102.06663, 2021) and (435:133257, 2022). One important difference between these two blowup scenarios is that the solution for our initial data has a onescale structure instead of a twoscale structure reported in Hou and Huang (arXiv:2102.06663, 2021) and (435:133257, 2022). More importantly, the solution seems to develop nearly selfsimilar scaling properties that are compatible with those of the 3D Navier–Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly selfsimilar profile seems to be very stable to the small perturbation of the initial data.more » « less

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