Global solvability of 3D non-isothermal incompressible nematic liquid crystal flows

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>