Global existence of strong solutions to nonhomogeneous MHD system in thin three-dimensional domains

We establish the global existence of strong solutions to the nonhomogeneous incompressible magnetohydrodynamics (MHD) equations in a thin three-dimensional domain <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi mathvariant="normal">Ω</a:mi> <a:mo>=</a:mo> <a:msup> <a:mrow class="MJX-TeXAtom-ORD"> <a:mi mathvariant="double-struck">R</a:mi> </a:mrow> <a:mn>2</a:mn> </a:msup> <a:mo>×</a:mo> <a:mo stretchy="false">(</a:mo> <a:mn>0</a:mn> <a:mo>,</a:mo> <a:mi>ϵ</a:mi> <a:mo stretchy="false">)</a:mo> </a:math>, with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>ϵ</g:mi> <g:mo>∈</g:mo> <g:mo stretchy="false">(</g:mo> <g:mn>0</g:mn> <g:mo>,</g:mo> <g:mn>1</g:mn> <g:mo stretchy="false">]</g:mo> </g:math>, subject to Dirichlet boundary conditions on the top and bottom boundaries. Global well-posedness may hold for large initial data, provided the vertical thickness <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mi>ϵ</j:mi> </j:math> is sufficiently small. Moreover, when <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mi>ϵ</k:mi> <k:mo stretchy="false">→</k:mo> <k:msup> <k:mn>0</k:mn> <k:mo>+</k:mo> </k:msup> </k:math>, both the velocity and magnetic field tend to vanish away from the initial time. The analysis is based on a priori <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> estimates of the solutions, with particular attention to the dependence on the vertical parameter <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mi>ϵ</n:mi> </n:math>.