Depletion of nonlinearity for space-analytic space-periodic solutions to equations of the hydrodynamic type

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Abstract

We consider solenoidal space-periodic space-analytic solutions to the equations of hydrodynamics. An elementary bound shows that due to the special structure of the nonlinear terms in the equations for modified solutions, effectively they lack a half of the spatial gradient. This appears to be a novel mechanism for depletion of nonlinearity. We present a two-phase iterative procedure yielding an expanded bound for the guaranteed time of the space analyticity of the hydrodynamic solutions. Each iteration involves two phases: In phase 1, the enstrophy of the modified solution and the bound for the radius of the analyticity of the original solution simultaneously increase (the bound is proportional to the elapsed time since the beginning of phase 1). In phase 2, the enstrophy and bound simultaneously decrease. It is straightforward to generalise this construction for the equations of magnetohydrodynamics.

About the authors

V. A. Zheligovsky

Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences

Author for correspondence.
Email: vlad@mitp.ru
Russian Federation, Moscow

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