Abstract
The present paper is devoted to the compressible nematic liquid crystal flow in the whole space $ \mathbb{R}^N\,(N≥ 2)$. Here we concentrate on the incompressible limit in the $ L^p$ type critical Besov spaces setting. We first establish the existence of global solutions in the framework of $ L^p$ type critical spaces provided that the initial data are close to some equilibrium states. Based on the global existence, we then consider the incompressible limit problem in the ill prepared data case. We justify the low Mach number convergence to the incompressible flow of liquid crystals in proper function spaces. In addition, the accurate converge rates are obtained.
Highlights
Introduction and main resultsWe address the convergence of solutions to the compressible flow of nematic liquid crystals when the Mach number goes to 0
In [2], the authors of this paper considered the incompressible limit in L2 type critical Besov spaces with ill prepared data
We introduce the following Besov-Chemin-Lerner space LρT (Bps,r): LρT (Bps,r) = f ∈ (0, +∞) × Y (RN ) : f LρT (Bps,r) < +∞, where f =def
Summary
Qu converges weakly to 0 when goes to 0, and, if (Pu0, d0) (m0, n0) (Pu , d ) converges in the sense of distributions to the solution of (2). For some sufficiently small η, there exists a positive constant Γ such that (b, u, d − d) X1p,1 ≤ Γη This uniform estimates will enable us to extend the local solution (b, u, d − d) obtained by using a Friedrichs method as in [1] to be a global one. To estimate the high frequencies of Qu, we employ the approach of [21, 22], and introduce the following velocity field v d=ef Qu + (−∆)−1∇b. To handle Q(I(b)Au), we decompose it into Q(I(b)Au) = TI(b)∆Qu + QR(I(b), Au) + Q(TAuI(b)) + [Q, TI(b)]Au. using product estimate, we see that for large j0, 2−j0 bu h bu for the last term of (34), arguing as in deriving (53), we get ξ (56).
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