Regularity of euler equations for a class of three-dimensional initial data

Alex Mahalov, B. Nicolaenko, C. Bardos, F. Golse

Research output: Chapter in Book/Report/Conference proceedingChapter

3 Citations (Scopus)

Abstract

The 3D incompressible Euler equations with initial data characterized by uniformly large vorticity are investigated. We prove existence on long time intervals of regular solutions to the 3D incompressible Euler equations for a class of large initial data in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach is based on fast singular oscillating limits, nonlinear averaging and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, resonance conditions and a nonstandard small divisor problem, we obtain fully 3D limit resonant Euler equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data and bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler equations with weakly aligned uniformly large vorticity at t = 0.

Original languageEnglish (US)
Title of host publicationProgress in Nonlinear Differential Equations and Their Application
PublisherSpringer US
Pages161-185
Number of pages25
Volume61
DOIs
StatePublished - Jan 1 2005

Publication series

NameProgress in Nonlinear Differential Equations and Their Application
Volume61
ISSN (Print)1421-1750
ISSN (Electronic)2374-0280

Fingerprint

Euler equations
Euler Equations
Vorticity
Incompressible Euler Equations
Regularity
Three-dimensional
Small Divisors
Global Regularity
Almost Periodic Functions
Interval
Regular Solution
Averaging Method
Nonlinear Interaction
Cancellation
Global Solution
Bootstrap
Averaging
Oscillation
Restriction
Arbitrary

Keywords

  • Conservation laws
  • Fast singular oscillating limits
  • Three-dimensional Euler equations
  • Vorticity

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Control and Optimization
  • Mathematical Physics
  • Computational Mechanics

Cite this

Mahalov, A., Nicolaenko, B., Bardos, C., & Golse, F. (2005). Regularity of euler equations for a class of three-dimensional initial data. In Progress in Nonlinear Differential Equations and Their Application (Vol. 61, pp. 161-185). (Progress in Nonlinear Differential Equations and Their Application; Vol. 61). Springer US. https://doi.org/10.1007/3-7643-7317-2_13

Regularity of euler equations for a class of three-dimensional initial data. / Mahalov, Alex; Nicolaenko, B.; Bardos, C.; Golse, F.

Progress in Nonlinear Differential Equations and Their Application. Vol. 61 Springer US, 2005. p. 161-185 (Progress in Nonlinear Differential Equations and Their Application; Vol. 61).

Research output: Chapter in Book/Report/Conference proceedingChapter

Mahalov, A, Nicolaenko, B, Bardos, C & Golse, F 2005, Regularity of euler equations for a class of three-dimensional initial data. in Progress in Nonlinear Differential Equations and Their Application. vol. 61, Progress in Nonlinear Differential Equations and Their Application, vol. 61, Springer US, pp. 161-185. https://doi.org/10.1007/3-7643-7317-2_13
Mahalov A, Nicolaenko B, Bardos C, Golse F. Regularity of euler equations for a class of three-dimensional initial data. In Progress in Nonlinear Differential Equations and Their Application. Vol. 61. Springer US. 2005. p. 161-185. (Progress in Nonlinear Differential Equations and Their Application). https://doi.org/10.1007/3-7643-7317-2_13
Mahalov, Alex ; Nicolaenko, B. ; Bardos, C. ; Golse, F. / Regularity of euler equations for a class of three-dimensional initial data. Progress in Nonlinear Differential Equations and Their Application. Vol. 61 Springer US, 2005. pp. 161-185 (Progress in Nonlinear Differential Equations and Their Application).
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