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 language | English (US) |
|---|---|
| Title of host publication | Progress in Nonlinear Differential Equations and Their Application |
| Publisher | Springer US |
| Pages | 161-185 |
| Number of pages | 25 |
| Volume | 61 |
| DOIs | |
| State | Published - Jan 1 2005 |
Publication series
| Name | Progress in Nonlinear Differential Equations and Their Application |
|---|---|
| Volume | 61 |
| ISSN (Print) | 1421-1750 |
| ISSN (Electronic) | 2374-0280 |
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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
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 proceeding › Chapter
}
TY - CHAP
T1 - Regularity of euler equations for a class of three-dimensional initial data
AU - Mahalov, Alex
AU - Nicolaenko, B.
AU - Bardos, C.
AU - Golse, F.
PY - 2005/1/1
Y1 - 2005/1/1
N2 - 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.
AB - 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.
KW - Conservation laws
KW - Fast singular oscillating limits
KW - Three-dimensional Euler equations
KW - Vorticity
UR - http://www.scopus.com/inward/record.url?scp=67649323941&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=67649323941&partnerID=8YFLogxK
U2 - 10.1007/3-7643-7317-2_13
DO - 10.1007/3-7643-7317-2_13
M3 - Chapter
AN - SCOPUS:67649323941
VL - 61
T3 - Progress in Nonlinear Differential Equations and Their Application
SP - 161
EP - 185
BT - Progress in Nonlinear Differential Equations and Their Application
PB - Springer US
ER -