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September  2010, 14(2): 603-627. doi: 10.3934/dcdsb.2010.14.603

On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models

Adam Larios 1, and  E. S. Titi 2,

1. 

Department of Mathematics, University of California, Irvine, Irvine CA 92697-3875, United States
2. 

Department of Mathematics and Department of Mechanics and Aerospace Engineering, University of California, Irvine, CA 92697, United States

Received  September 2009 Revised  February 2010 Published  June 2010

We prove higher-order and a Gevrey class (spatial analytic) regularity of solutions to the Euler-Voigt inviscid $\alpha$-regularization of the three-dimensional Euler equations of ideal incompressible fluids. Moreover, we establish the convergence of strong solutions of the Euler-Voigt model to the corresponding solution of the three-dimensional Euler equations for inviscid flow on the interval of existence of the latter. Furthermore, we derive a criterion for finite-time blow-up of the Euler equations based on this inviscid regularization. The coupling of a magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid irresistive magneto-hydrodynamic (MHD) system. Global regularity of the regularized MHD system is also established.
Keywords: $\alpha-$Models, Navier-Stokes-Voigt, MHD Equations, Turbulence Models, Blow-Up Criterion for Euler., Simplified Bardina Model, Euler-Voigt, Inviscid Regularization.
Mathematics Subject Classification: Primary: 35Q30, 76A10, 76B03, 76D03, 76F20, 76F55, 76F65, 76W0.
Citation: Adam Larios, E. S. Titi. On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 603-627. doi: 10.3934/dcdsb.2010.14.603
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