skip to main content
Explore Scholarly Publications and Datasets in the NSF-PAR

  • NSF Public Access
  • Search Results
  • Accepted Manuscript: Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $C^{1,\alpha}$ Velocity and Boundary
Title: Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $C^{1,\alpha}$ Velocity and Boundary
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 self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations more » with $C^{1,\alpha}$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. « less
Authors:
;
Award ID(s):
1907977 1912654
Publication Date:
NSF-PAR ID:
10286493
Journal Name:
Communications in mathematical physics
Volume:
383
Issue:
3
Page Range or eLocation-ID:
1559-1667
ISSN:
1432-0916
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ( , Communications on pure and applied mathematics)
    We present a novel method of analysis and prove finite time asymptotically self- similar blowup of the De Gregorio model [13,14] for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on R or $S^1$ for Holder continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations.
  2. ; ; ( , Communications on pure and applied mathematics)
    We present a novel method of analysis and prove finite time asymptotically self- similar blowup of the De Gregorio model [13,14] for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on R or $S^1$ for Holder continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations
  3. ; ( , Stochastic Systems)
    This work concerns the asymptotic behavior of solutions to a (strictly) subcritical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidth-sharing policy. Here we consider fair bandwidth-sharing policies that are a slight generalization of the [Formula: see text]-fair policies introduced by Mo and Walrand [Mo J, Walrand J (2000) Fair end-to-end window-based congestion control. IEEE/ACM Trans. Networks 8(5):556–567.]. Since the year 2000, it has been a standing problem to prove stability of the data communications network model of Massoulié and Roberts [Massoulié L, Roberts J (2000) Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15(1):185–201.], with general file sizes and operating under fair bandwidth sharing policies, when the offered load is less than capacity (subcritical conditions). A crucial step in an approach to this problem is to prove stability of subcritical fluid model solutions. In 2012, Paganini et al. [Paganini F, Tang A, Ferragut A, Andrew LLH (2012) Network stability under alpha fair bandwidth allocation with general file size distribution. IEEE Trans. Automatic Control 57(3):579–591.] introduced a Lyapunov function for this purpose and gave an argument, assuming that fluid model solutions are sufficiently smooth in timemore »and space that they are strong solutions of a partial differential equation and assuming that no fluid level on any route touches zero before all route levels reach zero. The aim of the current paper is to prove stability of the subcritical fluid model without these strong assumptions. Starting with a slight generalization of the Lyapunov function proposed by Paganini et al., assuming that each component of the initial state of a measure-valued fluid model solution, as well as the file size distributions, have no atoms and have finite first moments, we prove absolute continuity in time of the composition of the Lyapunov function with any subcritical fluid model solution and describe the associated density. We use this to prove that the Lyapunov function composed with such a subcritical fluid model solution converges to zero as time goes to infinity. This implies that each component of the measure-valued fluid model solution converges vaguely on [Formula: see text] to the zero measure as time goes to infinity. Under the further assumption that the file size distributions have finite pth moments for some p > 1 and that each component of the initial state of the fluid model solution has finite pth moment, it is proved that the fluid model solution reaches the measure with all components equal to the zero measure in finite time and that the time to reach this zero state has a uniform bound for all fluid model solutions having a uniform bound on the initial total mass and the pth moment of each component of the initial state. In contrast to the analysis of Paganini et al., we do not need their strong smoothness assumptions on fluid model solutions and we rigorously treat the realistic, but singular situation, where the fluid level on some routes becomes zero, whereas other route levels remain positive.« less
  4. ; ; ; ; ; ( , Advances in Computational Mathematics)
    Abstract

    We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid,more »including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

    « less
  5. ; ; ( , IMA Journal of Numerical Analysis)
    Abstract

    Our aim is to approximate a reference velocity field solving the two-dimensional Navier–Stokes equations (NSE) in the absence of its initial condition by utilizing spatially discrete measurements of that field, available at a coarse scale, and continuous in time. The approximation is obtained via numerically discretizing a downscaling data assimilation algorithm. Time discretization is based on semiimplicit and fully implicit Euler schemes, while spatial discretization (which can be done at an arbitrary scale regardless of the spatial resolution of the measurements) is based on a spectral Galerkin method. The two fully discrete algorithms are shown to be unconditionally stable, with respect to the size of the time step, the number of time steps and the number of Galerkin modes. Moreover, explicit, uniform-in-time error estimates between the approximation and the reference solution are obtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensional NSE, subject to the no-slip Dirichlet or periodic boundary conditions, are used in this work as a paradigm. The complete analysis that is presented here can be extended to other two- and three-dimensional dissipative systems under the assumption of global existence and uniqueness.

  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email