Global well‐posedness and large‐time behavior for non‐isothermal nematic liquid crystal flows with temperature‐dependent viscosity

Abstract We investigate an initial‐boundary value problem of non‐isothermal nematic liquid crystal flows with temperature‐dependent viscosity in three‐dimensional bounded domains. Based on the time‐weighted energy estimates, we derive the existence of a unique global strong solution under some smallness condition. In addition, the exponential decay‐in‐time rates of the corresponding strong solution are also obtained. We improve the previous results [D. Bian and Y. Xiao, Global solution to the nematic liquid crystal flows with heat effect, J. Differential Equations, 263 (2017), 5298–5329; Global well‐posedness of non‐isothermal inhomogeneous nematic liquid crystal flows, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1243–1272] in the sense that the large oscillations are allowed and the assumptions on the viscosity coefficient are weakened.