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Zero inertia limit of incompressible Qian-Sheng model. (English) Zbl 1489.35212

Summary: The Qian-Sheng model is a system describing the hydrodynamics of nematic liquid crystals in the Q-tensor framework. When the inertial effect is included, it is a hyperbolic-type system involving a second-order material derivative coupling with forced incompressible Navier-Stokes equations. If formally letting the inertial constant \(\varepsilon\) go to zero, the resulting system is the corresponding parabolic model. We provide the result on the rigorous justification of this limit in \(\mathbb{R}^3\) with small initial data, which validates mathematically the parabolic Qian-Sheng model. To achieve this, an initial layer is introduced to not only overcome the disparity of the initial conditions between the hyperbolic and parabolic models, but also make the convergence rate optimal. Moreover, a novel \(\varepsilon \)-dependent energy norm is carefully designed, which is non-negative only when \(\varepsilon\) is small enough, and handles the difficulty brought by the second-order material derivative.

MSC:

35Q35 PDEs in connection with fluid mechanics
76A15 Liquid crystals
35L05 Wave equation
35L30 Initial value problems for higher-order hyperbolic equations
35L81 Singular hyperbolic equations
58J37 Perturbations of PDEs on manifolds; asymptotics
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