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Energy dissipation for weak solutions of incompressible liquid crystal flows

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  • In this paper, we are concerned with the simplified Ericksen-Leslie system (1)--(3), modeling the flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0, d_0)\in {\bf H}\times H^1(\Omega, \mathbb{S}^2)$, with $d_0(\Omega)\subset\mathbb{S}^2_+$. We define a dissipation term $D(u,d)$ that stems from an eventual lack of smoothness in the solutions, and then obtain a local equation of energy for weak solutions of liquid crystals in dimensions three. As a consequence, we consider the 2D case and obtain $D(u,d)=0$.
    Mathematics Subject Classification: 35D30, 76A15, 35Q35.

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