Non blow-up of the 3D Euler equations for a class of three-dimensional initial data in cylindrical domains

Non blow-up of the 3D incompressible Euler Equations is proven for a class of three- dimensional initial data characterized by uniformly large vorticity 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 of proving regularity is based on investigation of fast singular oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant Euler equations without any restriction on the size of 3D initial data. After establishing strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler Equations with weakly aligned uniformly large vorticity at $t = 0$.

2010 Mathematics Subject Classification

Primary 35Q35. Secondary 35B35, 76B03.

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Published 1 January 2004