On instability of Rayleigh–Taylor problem for incompressible liquid crystals under L1$L^{1}$-norm

*Correspondence: linxueyun90@fzu.edu.cn 1College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350116, China Abstract We investigate the nonlinear Rayleigh–Taylor (RT) instability of a nonhomogeneous incompressible nematic liquid crystal in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution. Thus we construct solutions of the linearized problem that grow in time in the Sobolev space H4, then we show that the RT equilibrium state is linearly unstable. With the help of the established unstable solutions of the linearized problem and error estimates between the linear and nonlinear solutions, we establish the nonlinear instability of the density, the horizontal and vertical velocities under L1-norm.

⊂ R 3 : (1.1) Here the unknown function ρ is the density of the nematic liquid crystals, u the velocity, and p the pressure, d represents the macroscopic average of the nematic liquid crystal orientation field. Also μ > 0 is the coefficient of viscosity, g > 0 is the gravitational constant, e 3 = (0, 0, 1) T is the vertical unit vector, and -ρge 3 is the gravitational force.
In this paper we study the Rayleigh-Taylor (RT) instability of system (1.1). To this purpose, we consider a density profileρ :=ρ(x 3 ) ∈ C 5 (¯ ), which satisfies inf x∈ {ρ} > 0, (1.2) and an RT condition The condition (1.3) means that there is a region in which the RT density profile has a larger density with increasing x 3 (height), thus leading to RT instability.

Main results
Before stating our main result, we shall introduce some mathematical notations of Sobolev spaces: a b means that a≤cb for some positive constant c, such that, for any given δ ∈ (0, δ 0 ), there is a unique classical solution ( , u, σ ) ∈ C 0 (Ī T , to the LCRT problem with the initial data 0 , u 0 , q 0 := δ ˜ 0 ,ũ 0 ,q 0 + δ 2 0, u r , q r , but the solution satisfies for some escape time T δ := 1 ln 2ε m 0 δ ∈ I T . In addition, the initial data 0 , u 0 , q 0 , and σ 0 satisfy the compatibility conditions: The proof of Theorem 1 is based on a bootstrap instability method, which has its origin in [18,19]. We mention that many authors have established various versions of the bootstrap methods for mathematical proofs of various flow instabilities, see [20][21][22], for example. We complete the proof of Theorem 1 in four steps. Firstly, we introduce unstable solutions to the linearized LCRT problem, in view of the linearized LCRT problem, we can obtain a growing mode ansatz of solutions, i.e., for some > 0, ( , u, q, σ ) := e t (-ρ ũ 3 / ,ũ,q, 0), see Proposition 1. Secondly, by using the standard energy method, we establish a Gronwall-type energy inequality of the local-in-time solution of the LCRT problem, see Proposition 2. Thirdly, we use initial data of solutions of the linearized LCRT problem to construct initial data for solutions of the LCRT problem, so that the modified initial data ( δ 0 , u δ 0 , q δ 0 ) := δ(˜ 0 ,ũ 0 ,q 0 ) + δ 2 (0, u r , q r ) belongs to H 4 × H 4 σ × H 3 and satisfies necessary compatibility condition, see Proposition 4. Finally, we introduce the error estimates between the solutions of the linearized and nonlinear LCRT problems, and then prove the nonlinear solution is unstable under L 1 -norm. Now, we will introduce some well-known mathematical results, which will be used in the proof of Theorem 1.

Linear instability
Proposition 1 Under the assumptions of Theorem 1, the LCRT equilibrium state R C is linearly unstable, that is, there is an unstable solution in the form with the constant growth rate defined by

Moreover,ũ satisfies
Proof Please refer to the proof of [24, Theorem 1.1].

Gronwall-type energy inequality of nonlinear solutions
We derive that any small solution of the LCRT problem enjoys a Gronwall-type energy inequality. We will derive such an inequality by the a priori estimate method for simplicity. Let ( , u, σ ) be a solution of LCRT problem such that Moreover, the solution enjoys fine regularity, which makes valid the procedure of formal deduction. In addition, we rewrite (1.4) with the boundary-value condition in (1.5) as a nonhomogeneous form: Lemma 2 Under the assumption (3.1) with sufficiently small δ, it holds that Proof Multiplying (3.2) 2 by u in L 2 and using integration by parts, we get By (3.1) and product estimate, it holds that Proof Let α: = (α 1 , α 2 , α 3 ) be a multiindex of order |α| := α 1 + α 2 + α 3 ≤ 4, and ∂ α := 3 . Using integration by parts, we can get u · ∇∂ α ∂ α dx = 0, thus, applying ∂ α to (3.2) 1 , and then multiplying the resulting identity by ∂ α in L 2 , we get By (3.1) and product estimate, it holds that Multiplying (3.11) 1 with i = 2 by u tt in L 2 , and using the integration by parts and (3.2) 1 , we can get that By (3.1) and product estimate, it holds that Moreover, by using (3.1), integration by parts, and product estimate, we can obtain Putting the above two estimates into (3.12), and then using Friedrichs's and Young's inequalities, we get (3.10). Similarly, we can easily derive (3.9) from (3.11) with i = 1.

Lemma 5
Under the assumption (3.1) with sufficiently small δ, it holds that multiplying (3.17) by σ in L 2 with i = 0, then using integration by parts, we get By (3.1), integration by parts and product estimate, it holds that Putting the above estimate into (3.18), and then using Young's inequality, we get (3.14). Similarly, we can easily derive (3.15) and (3.16) from (3.17) with i = 1 and i = 2, respectively.
Remark 2 For any classical solution ( , u, σ ) constructed by Proposition 3, and for any given t 0 ∈ (0, T max ), we take ( , u, σ )| t=t 0 as a new initial datum. Then the new initial data can define a unique local-in-time classical solution (˜ ,ũ,σ ,q) constructed by Proposition 3, moreover, the initial data ofq is equal to q| t=t 0 by unique solvability of (3.69).

Error estimates and existence of escape times
where T max denotes the maximal time of existence of the solution ( , u) ∈ C([0, T max ), H 4 × H 4 σ ). Obviously, T * T * * > 0 and We denote T min := min{T δ , T * , T * * }. By the definition of T * * , we can deduce from the estimate (3.60) that, for all t < T min , Applying Gronwall's inequality to the above estimate, we arrive at, for some constant C 4 , E ≤ C 4 δ 2 e 2 t for all t < T min . (5.7) In addition, we have the following error estimate between the nonlinear solution ( , u) and the linear solution ( a , u a ).