paper

On well-posedness of Ericksen-Leslie's paraboloc-hyperbolic liquid crystal model

arXiv:1612.04616 · doi:10.1137/18M1167310

Abstract

We establish the following well-posedness results on Ericksen-Leslie's parabolic-hyperbolic liquid crystal model: 1, if the dissipation coefficients β= μ_4 - 4 μ_6 > 0, and the size of the initial energy E^{in} is small enough, then the life span of the solution is at least -O(\ln E^{in}); 2, for the special case that the coefficients μ_1 = μ_2 = μ_3 = μ_5 = μ_6 = 0, for which the model is the Navier-Stokes equations coupled with the wave map from \mathbb{R}^n to \mathbb{S}^2, the same existence result holds but without the smallness restriction on the size of the initial data; 3, with further constraints on the coefficients, namely α= μ_4 - 4 μ_6 - \tfrac{ (|λ_1| - 7 λ_2)^2 }η - \tfrac{ 2 ( 7 |λ_1| - 2λ_2 )^2 }{ |λ_1| } > 0 and μ_2 < μ_3, the global classical solution with small initial data can be established. A relation between the Lagrangian multiplier and the geometric constraint |d|=1 plays a key role in the proof.

34 pages. arXiv admin note: text overlap with arXiv:1105.2180 by other authors

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