2013/10/31 by Eduard Feireisl, Feireisl, Eduard, Elisabetta Rocca +5 · 3 citations
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1310.8474
openalex publication_date 2013/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove the existence of global in time weak solutions for an\nevolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid\ncrystal (LC) flows in three dimensions of space. In our model, the\nincompressible Navier-Stokes system for the macroscopic velocity vu is\ncoupled to a nonlinear convective parabolic equation describing the evolution\nof the Q-tensor QQ, namely a tensor-valued variable representing the\nnormalized second order moments of the probability distribution function of the\nLC molecules. The effects of the (absolute) temperature vt are prescribed in\nthe form of an energy balance identity complemented with a global entropy\nproduction inequality. Compared to previous contributions, we can consider here\nthe physically realistic singular configuration potential f introduced by\nBall and Majumdar. This potential gives rise to severe mathematical\ndifficulties since it introduces, in the Q-tensor equation, a term which is at\nthe same time singular in QQ and degenerate in vt. To treat it a careful\nanalysis of the properties of f, particularly of its blow-up rate, is carried\nout.\n