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Local Smoothness of Weak Solutions to the Magnetohydrodynamics Equations via Blowup Methods

Basil NicolaenkoAlex MahalovTimofey Shilkin

Séminaire Équations aux dérivées partielles

We demonstrate that there exist no self-similar solutions of the incompressible magnetohydrodynamics (MHD) equations in the space L 3 ( R 3 ) . This is a consequence of proving the local smoothness of weak solutions via blowup methods for weak solutions which are locally L 3 . We present the extension of the Escauriaza-Seregin-Sverak method to MHD systems.

Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics

Anatoli BabinAlex MahalovBasil Nicolaenko — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D "primitive" Navier-Stokes equations for strong stratification (large stratification parameter ). This uniform existence is proven for periodic or stress-free boundary conditions for domain aspect ratios, including the case of three wave resonances which yield nonlinear " 2 1 2 dimensional" limit equations...

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