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Multi-scale self-similar finite-time blowups of the Constantin-Lax-Majda model for the 3D Euler equations

Published 26 Jan 2024 in math.AP | (2401.14615v2)

Abstract: We construct a new class of asymptotically self-similar finite-time blowups that have two collapsing spatial scales for the 1D Constantin-Lax-Majda model. The larger spatial scale measures the decreasing distance between the bulk of the solution and the eventual blowup point, while the smaller scale measures the shrinking size of the bulk of the solution. Similar multi-scale blowup phenomena have recently been discovered for many higher dimensional equations. Our study may provide some understanding of the common mechanism behind these multi-scale blowups.

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  1. A. Castro and D. Córdoba. Infinite energy solutions of the surface quasi-geostrophic equation. Advances in Mathematics, 225(4):1820–1829, 2010.
  2. Formation of singularities for a transport equation with nonlocal velocity. Annals of mathematics, pages 1377–1389, 2005.
  3. Collapsing-ring blowup solutions for the keller-segel system in three dimensions and higher. Journal of Functional Analysis, page 110065, 2023.
  4. J. Chen and T. Y. Hou. Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data. arXiv preprint arXiv:2210.07191, 2022.
  5. J. Chen. Singularity formation and global well-posedness for the generalized Constantin–Lax–Majda equation with dissipation. Nonlinearity, 33(5):2502, 2020.
  6. On the finite time blowup of the De Gregorio model for the 3D Euler equations. Communications on pure and applied mathematics, 74(6):1282–1350, 2021.
  7. Asymptotically self-similar blowup of the Hou–Luo model for the 3D Euler equations. Annals of PDE, 8(2):24, 2022.
  8. On the finite-time blowup of a one-dimensional model for the three-dimensional axisymmetric Euler equations. Communications on Pure and Applied Mathematics, 70(11):2218–2243, 2017.
  9. A simple one-dimensional model for the three-dimensional vorticity equation. Communications on pure and applied mathematics, 38(6):715–724, 1985.
  10. S. De Gregorio. A partial differential equation arising in a 1d model for the 3d vorticity equation. Mathematical methods in the applied sciences, 19(15):1233–1255, 1996.
  11. On the stability of self-similar blow-up for C1,αsuperscript𝐶1𝛼{C}^{1,\alpha}italic_C start_POSTSUPERSCRIPT 1 , italic_α end_POSTSUPERSCRIPT solutions to the incompressible Euler equations on ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. arXiv preprint arXiv:1910.14071, 2019.
  12. Stable self-similar blow-up for a family of nonlocal transport equations. Analysis & PDE, 14(3):891–908, 2021.
  13. On the effects of advection and vortex stretching. Archive for Rational Mechanics and Analysis, 235(3):1763–1817, 2020.
  14. T. M. Elgindi. Finite-time singularity formation for C1,αsuperscript𝐶1𝛼{C}^{1,\alpha}italic_C start_POSTSUPERSCRIPT 1 , italic_α end_POSTSUPERSCRIPT solutions to the incompressible Euler equations on ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Annals of Mathematics, 194(3):647–727, 2021.
  15. T. Y. Hou and D. Huang. A potential two-scale traveling wave singularity for 3D incompressible Euler equations. Physica D: Nonlinear Phenomena, 435:133257, 2022.
  16. T. Y. Hou and D. Huang. Potential singularity formation of incompressible axisymmetric Euler equations with degenerate viscosity coefficients. Multiscale Modeling & Simulation, 21(1):218–268, 2023.
  17. T. Y. Hou. Potential singularity of the 3D euler equations in the interior domain. Foundations of Computational Mathematics, pages 1–47, 2022.
  18. On the exact self-similar finite-time blowup of the hou-luo model with smooth profiles. arXiv preprint arXiv:2308.01528, 2023.
  19. Self-similar finite-time blowups with smooth profiles of the generalized Constantin–Lax–Majda model. arXiv preprint arXiv:2305.05895, 2023.
  20. G. Luo and T. Y. Hou. Potentially singular solutions of the 3D axisymmetric Euler equations. Proceedings of the National Academy of Sciences, 111(36):12968–12973, 2014.
  21. P. Liu. Spatial Profiles in the Singular Solutions of the 3D Euler Equations and Simplified Models. PhD thesis, California Institute of Technology, 2017.
  22. Collapse versus blow-up and global existence in the generalized Constantin–Lax–Majda equation. Journal of Nonlinear Science, 31(5):1–56, 2021.
  23. On a generalization of the Constantin–Lax–Majda equation. Nonlinearity, 21(10):2447, 2008.

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