Gevrey analyticity of solutions to the 3D nematic liquid crystal flows in critical Besov space

Abstract

We show that the solution to the Cauchy problem of the 3D nematic liquid crystal flows, with initial data belonging to a critical Besov space, belongs to a Gevrey class. More precisely, it is proved that for any (u0,d0−d¯0)∈Ḃp,13p−1(R3)×Ḃq,13q(R3) with some suitable conditions imposed on p,q∈(1,∞), there exists T∗>0 depending only on initial data, such that the nematic liquid crystal flows admit a unique solution (u,d) on R3×(0,T∗), and satisfies ‖etΛ1u(t)‖L˜T∗∞(Ḃp,13p−1)∩L˜T∗1(Ḃp,13p+1)+‖etΛ1(d(t)−d¯0)‖L˜T∗∞(Ḃq,13q)∩L˜T∗1(Ḃq,13q+2)<∞. Here, d¯0∈S2 is a constant unit vector, and Λ1 is the Fourier multiplier whose symbol is given by |ξ|1=|ξ1|+|ξ2|+|ξ3|. Moreover, if the initial data is sufficiently small, then T∗=∞. As a consequence of the results, decay estimates of higher-order derivatives of solutions in Besov spaces are deduced.