<abstract><p>In this paper, we are concerned with the Cauchy problem of inhomogeneous incompressible magnetic Bénard equations with vacuum as far-field density in $ \Bbb R^2 $. We prove that if the initial density and magnetic field decay not too slowly at infinity, the system admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even has compact support. Moreover, we extend the result of [16, 17] to the global one.</p></abstract>
<abstract><p>We are concerned with the initial value problem of non-isothermal incompressible nematic liquid crystal flows in $ \Bbb R^3 $. Through some time-weighted a priori estimates, we prove the global existence of a strong solution provided that $ \Big(\|\sqrt{\rho_0}u_0\|_{L^2}^2+\|\nabla d_0\|_{L^2}^2\Big)\Big(\|\nabla u_0\|_{L^2}^2+\|\nabla^2d_0\|_{L^2}^2\Big) $ is reasonably small, which extends the corresponding Li's (Methods Appl. Anal. 2015 <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>) and Ding-Huang-Xia's (Filomat 2013 <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>) results to the whole space $ \Bbb R^3 $ and non-isothermal case. Furthermore, we also derive the algebraic decay estimates of the solution.</p></abstract>
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