Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals

Yanbo Hu

doi:10.1515/anona-2022-0268

Open Access Published by De Gruyter November 19, 2022

Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals

Yanbo Hu

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2022-0268

Abstract

This article focuses on the singularity formation of smooth solutions for a one-dimensional nonlinear degenerate hyperbolic-parabolic coupled system originating from the Poiseuille flow of nematic liquid crystals. Without assuming that the wave speed of the hyperbolic equation is a positive function, we show that its smooth solution will break down in finite time even for an arbitrarily small initial energy. Based on an estimate of the solution for the heat equation, we use the method of characteristics to control the wave speed and its derivative so that the wave speed does not degenerate and its derivative does not change sign in a period of time.

Keywords: hyperbolic-parabolic coupled system; singularity; characteristic method

MSC 2010: 35L72; 35K05; 35B44

1 Introduction

We are interested in the singularity formation for a nonlinear hyperbolic-parabolic coupled system

(1.1) θ t t + 2 θ t − c ( θ ) ( c ( θ ) θ x ) x + u x = 0 , u t − ( u x + θ t ) x = 0 ,

where ( u , θ ) are unknown functions, and the wave speed c of the hyperbolic equation is a given smooth function of θ . The first hyperbolic equation and the second parabolic equation in (1.1) describe, respectively, the “crystal” property and “liquid” property in nematic liquid crystals.

System (1.1) is derived from the full Ericksen-Leslie equations for Poiseuille flow of nematic liquid crystals by Chen et al. in a recent article [8]. The full Ericksen-Leslie equations read that [10,20, 21,22]

(1.2) ν n ¨ = λ n − ∂ W ∂ n − g + ∇ ⋅ ∂ W ∂ ∇ n , ρ u ˙ + ∇ P = ∇ ⋅ σ − ∇ ⋅ ∂ W ∂ ∇ n ⊗ ∇ n , ∇ ⋅ u = 0 , ∣ n ∣ = 1 ,

where f ˙ is the material derivative, that is, f ˙ = f t + u ⋅ ∇ f . In system (1.2), n is the director field, u is the velocity field, ρ is the density, P is the pressure, W is the Oseen-Frank potential energy density, σ is the viscous stress tensor, g is the kinematic transport tensor, ν is the inertial coefficient of n , and λ is the Lagrangian multiplier. For one-dimensional Poiseuille flows, Chen et al. [8] considered solutions of (1.2) of the special form [7]

(1.3) n ( x , t ) = ( sin θ ( x , t ) , 0 , cos θ ( x , t ) ) T , u ( x , t ) = ( 0 , 0 , u ( x , t ) ) T ,

and reduced system (1.2) to

(1.4) ν θ t t + γ 1 θ t = c ( θ ) ( c ( θ ) θ x ) x − h ( θ ) u x , ρ u t = a + ( g ( θ ) u x + h ( θ ) θ t ) x ,

where γ 1 and a are constants, and

c 2 ( θ ) = k 1 cos 2 θ + k 3 sin 2 θ , h ( θ ) = α 3 cos 2 θ − α 2 sin 2 θ , g ( θ ) = α 1 sin 2 θ cos 2 θ + α 5 − α 2 2 sin 2 θ + α 3 + α 6 2 cos 2 θ + α 4 2 ,

where α i are Leslie coefficients of the material arising from the viscous stress tensor σ , and k i are elastic coefficients of the material arising from the Oseen-Frank potential energy density W . System (1.1) was obtained from (1.4) by choosing appropriate parameters such that h ( θ ) = g ( θ ) ≡ 1 and ν = ρ = 1 , a = 0 , γ 1 = 2 . For a detailed derivation of (1.4) and more information on it, the reader is referred to [8]. Some relevant studies for the full Ericksen-Leslie equations can be found among others in [16,17].

When neglecting the fluid effect, system (1.4) reduces to the well-known variational wave equation

(1.5) ν θ t t + γ 1 θ t − c ( θ ) ( c ( θ ) θ x ) x = 0 ,

which has been widely studied since it was introduced by Hunter and Saxton [15]. Under the assumption that the wave speed c is a strictly positive function, the cusp-type singularity formations in finite time were established in [9,11]. The results of well-posedness of dissipative and conservative weak solutions to (1.5) and its related models were presented in [2,3, 4,5,6,12,25,26, 27,28]. Physically, the elastic coefficients may be negative in some cases, see e.g. [1,23], so the wave speed c ( ⋅ ) may degenerate at some time. Moreover, equation (1.5) with c ( θ ) = θ and γ 1 = 0 corresponded to the second sound equation [18,19]. By allowing either k 1 or k 3 to be zero, Saxton [24] examined the boundary blowup properties of smooth solutions to (1.5) with γ 1 = 0 . Hu and Wang [13,14] discussed the local existence of its smooth solutions near the degenerate line.

In this article, we are concerned with the singularity formation of smooth solutions for the hyperbolic-parabolic coupled system (1.1) without assuming that the wave speed c is a positive function. More precisely, we assume that the wave speed c satisfies

(1.6) c ( ⋅ ) ∈ C 2 , ∣ c ′ ( ⋅ ) ∣ ≤ c ¯

for some positive constant c ¯ . In the previous article, Chen et al. [8] established the singularity formation result to system (1.1) under condition (1.6) with the assumption that

0 < C L ≤ c ( ⋅ ) ≤ C U

for two constants C L , C U . Moreover, they also obtained the global existence of Hölder continuous weak solutions to its Cauchy problem for initial data of finite energy. In the current article, the main difficulty is to control the wave speed c so that it does not degenerate for a period of time.

The main result of this article can be stated as follows.

Theorem 1

Assume that c ( θ 0 ) = A 0 > 0 , c ′ ( θ 0 ) = A 1 > 0 for some constant θ 0 . Let ϕ ( z ) be a smooth function satisfying

(1.7) ϕ ( z ) ∈ C c 1 ( ( − 1 , 1 ) ) , ϕ ′ ( 0 ) ≤ − max 4 A 0 , 96 A 1 .

Suppose that ( θ ( x , t ) , u ( x , t ) ) ∈ C 1 is a solution of (1.1) for ( x , t ) ∈ R + × [ 0 , T ) with initial data

(1.8) u ( x , 0 ) = 0 , θ ( x , 0 ) = θ 0 + ε ϕ x − x 0 ε , θ t ( x , 0 ) = [ − c ( θ ( x , 0 ) ) + ε ] θ x ( x , 0 ) ,

where ε > 0 and x 0 are two numbers. Then there exists a small positive constant ε 0 (the number ε 0 is given in (2.39)) such that T < T ∗ with some T ∗ < 1 for any ε < ε 0 . Moreover,

(1.9) θ t ( x , t ) → + ∞ , θ x ( x , t ) → − ∞ , u x ( x , t ) → − ∞ ,

as ( x , t ) → ( x ∗ , T ∗ − ) at some x ∗ .

2 The proof of Theorem 1

We shall show Theorem 1 in this section. The proof is based on an estimate for the quantity J ( x , t ) ≔ u x + θ t established by Chen et al. in [8]. Employing their result, we estimate the wave speed c ( θ ) by the condition c ( θ 0 ) > 0 and the energy inequality. Moreover, the sign of c ′ ( θ ) is also determined in a period of time by the condition c ′ ( θ 0 ) > 0 and the smallness of ε > 0 .

The proof is divided into seven steps. In Step 1, we present the result about the quantity J obtained in [8] and derive the energy inequality for the Riemann variables. Step 2 is devoted to establishing the estimate of the energy. The estimate for the wave speed c ( θ ( x , t ) ) is acquired in Step 3 by applying the estimate of the energy. In Step 4, we establish the estimate of the quantity J and then deduce the estimate of Riemann variables in the characteristic triangle region. Step 5 is focused on controlling the sign of c ′ ( θ ) . In Step 6, we prove that smooth solutions breakdown before t = 1 . Finally, we complete the proof of Theorem 1 in Step 7.

Step 1. Following Chen et al. [8], we introduce

(2.1) v ( x , t ) ≔ ∫ − ∞ x u ( z , t ) d z , and then u = v x , v t = u x + θ t = J .

In terms of variables ( θ , v ) , system (1.1) can be transformed into

(2.2) θ t t − c ( θ ) ( c ( θ ) θ x ) x = − θ t − v t , v t − v x x = θ t .

Making use of the theory of fundamental solutions for one-dimensional heat equation, Chen et al. established the estimates of ( v t , v x ) for smooth solutions.

Lemma 1

[8] Let ( θ , v ) be a smooth solution of system (2.2). Then for any t > 0 , the function v satisfies

(2.3) ‖ J ‖ L ∞ ( R × ( 0 , t ) ) = ‖ v t ‖ L ∞ ( R × ( 0 , t ) ) ≤ C ( ‖ u x ( x , 0 ) ‖ L 2 ( R ) + ‖ θ t ( x , 0 ) ‖ L 2 ( R ) ) + C t 1 4 ‖ θ t ‖ L ∞ ( ( 0 , t ) , L 2 ( R ) ) + ‖ θ x ‖ L ∞ ( ( 0 , t ) , L 2 ( R ) ) 2 + t 1 4 ‖ v x ‖ L ∞ ( R × ( 0 , t ) )

and

(2.4) ‖ u ‖ L ∞ ( R × ( 0 , t ) ) = ‖ v x ‖ L ∞ ( R × ( 0 , t ) ) ≤ C ‖ u ( x , 0 ) ‖ L 2 ( R ) + t 1 4 ‖ θ t ‖ L ∞ ( ( 0 , t ) , L 2 ( R ) ) ,

for some positive constant C.

For the equation θ in (2.2), we introduce the Riemann variables as follows:

(2.5) R ≔ θ t + c ( θ ) θ x , S ≔ θ t − c ( θ ) θ x ,

so that

(2.6) θ t = R + S 2 , θ x = R − S 2 c ( θ ) .

Then one has

(2.7) R t − c R x = c ′ 4 c ( R 2 − S 2 ) − ( R + S ) − u x , S t + c S x = c ′ 4 c ( S 2 − R 2 ) − ( R + S ) − u x , θ t + c θ x = R ,

(2.8) R t − c R x = c ′ 4 c ( R 2 − S 2 ) − 1 2 ( R + S ) − J , S t + c S x = c ′ 4 c ( S 2 − R 2 ) − 1 2 ( R + S ) − J , θ t + c θ x = R .

We multiply the first equation in (2.7) by 2 R and the second one by 2 S and add the resulting equations to acquire

(2.9) ( R 2 + S 2 ) t + [ c ( θ ) ( S 2 − R 2 ) ] x = − 2 ( R + S ) 2 − 2 ( R + S ) u x .

Now multiplying the second equation in (1.1) by 2 u gives

(2.10) ( u 2 ) t = 2 u u x x + 2 u θ t x = ( 2 u u x ) x − 2 ( u x ) 2 + ( 2 u θ t ) x − 2 u x θ t = ( 2 u u x ) x − 2 ( u x ) 2 + ( u ( R + S ) ) x − ( R + S ) u x .

We add (2.9) and (2.10) to find that

(2.11) ( R 2 + S 2 + u 2 ) t + [ c ( θ ) ( S 2 − R 2 ) − 2 u u x − u ( R + S ) ] x = − 2 ( R + S ) 2 − 3 ( R + S ) u x − 2 ( u x ) 2 = − 2 R + S + 3 4 u x 2 − 7 8 ( u x ) 2 ,

which means that

(2.12) ( R 2 + S 2 + u 2 ) t + [ c ( θ ) ( S 2 − R 2 ) − 2 u u x − u ( R + S ) ] x ≤ 0 .

Step 2. Corresponding to (1.8), we have by (2.1)

(2.13) u ( x , 0 ) = 0 , θ x ( x , 0 ) = ϕ ′ x − x 0 ε , R ( x , 0 ) = ε θ x ( x , 0 ) , S ( x , 0 ) = − [ 2 c ( θ ( x , 0 ) ) − ε ] θ x ( x , 0 ) .

According to the assumptions c ( ⋅ ) ∈ C 2 and ϕ ∈ C c 1 , we know by c ( θ 0 ) = A 0 > 0 that there exists a small number ε 1 > 0 satisfying

(2.14) ε 1 ≤ A 0 2 ,

such that for any ε ≤ ε 1 it holds

(2.15) c ( θ ( x , 0 ) ) = c θ 0 + ε ϕ x − x 0 ε ≥ c ( θ 0 ) 2 = A 0 2 , c ( θ ( x , 0 ) ) = c θ 0 + ε ϕ x − x 0 ε ≤ 3 c ( θ 0 ) 2 = 3 A 0 2 ,

which implies that

(2.16) A 0 2 ≤ [ 2 c ( θ ( x , 0 ) ) − ε ] ≤ 3 A 0 ,

for any ε ≤ ε 1 .

Denote

(2.17) E ( t ) ≔ ∫ − ∞ ∞ [ R 2 ( x , t ) + S 2 ( x , t ) ] d x , ℰ ( t ) ≔ ∫ − ∞ ∞ [ R 2 ( x , t ) + S 2 ( x , t ) + u 2 ( x , t ) ] d x .

Next integrating the energy inequality (2.12) and applying the initial data (2.13) derives the estimate of the energy E ( t )

(2.18) E ( t ) ≤ ℰ ( t ) ≤ ℰ ( 0 ) = ∫ − ∞ ∞ [ R 2 ( x , 0 ) + S 2 ( x , 0 ) + u 2 ( x , 0 ) ] d x = ∫ − ∞ ∞ [ ε 2 + ( 2 c ( θ ( x , 0 ) ) − ε ) 2 ] ϕ ′ x − x 0 ε 2 d x ≤ ε [ A 0 2 + ( 3 A 0 ) 2 ] ∫ − 1 1 ( ϕ ′ ( z ) ) 2 d z = K ε ,

where

K = 10 A 0 2 ∫ − 1 1 ( ϕ ′ ( z ) ) 2 d z .

Step 3. We now use the energy estimate (2.18) to control the wave speed c ( θ ( x , t ) ) . We claim that there exists a positive number ε 2 ≤ ε 1 such that for any ε < ε 2 it holds

(2.19) A 0 2 ≤ c ( θ ( x , t ) ) ≤ 3 A 0 2 .

It is first noted by the mean value theorem that

∣ c ( θ ( x , 0 ) ) − c ( θ 0 ) ∣ ≤ c ¯ ∣ θ ( x , 0 ) − θ 0 ∣ = c ¯ ⋅ ε ⋅ ϕ x − x 0 ε .

Then one has by (2.18) for t < 1

(2.20) ∫ − ∞ ∞ [ c ( θ ( x , t ) ) − c ( θ 0 ) ] 2 d x = ∫ − ∞ ∞ c ( θ ( x , 0 ) ) − c ( θ 0 ) + ∫ 0 t c ′ ( θ ) θ t ( x , τ ) d τ 2 d x ≤ ∫ − ∞ ∞ 2 ∣ c ( θ ( x , 0 ) ) − c ( θ 0 ) ∣ 2 d x + ∫ − ∞ ∞ 2 c ¯ 2 ∫ 0 t ∣ θ t ∣ d τ 2 d x ≤ 2 c ¯ 2 ∫ − 1 1 ∣ ϕ ( z ) ∣ 2 d z ⋅ ε 3 + 2 c ¯ 2 ∫ − ∞ ∞ ∫ 0 1 ∣ θ t ∣ 2 d τ d x ≤ 2 c ¯ 2 ∫ − 1 1 ∣ ϕ ( z ) ∣ 2 d z ⋅ ε 3 + 2 c ¯ 2 ∫ 0 1 ∫ − ∞ ∞ ( R 2 + S 2 ) ( x , τ ) d x d τ ≤ 2 c ¯ 2 ∫ − 1 1 ∣ ϕ ( z ) ∣ 2 d z ⋅ ε 3 + 2 c ¯ 2 ⋅ K ε ≤ K 1 ε ,

where

K 1 = 2 c ¯ 2 A 0 2 ∫ − 1 1 ∣ ϕ ( z ) ∣ 2 d z + 2 c ¯ 2 K .

Set

(2.21) ε 2 = min ε 1 , A 0 4 8 [ 2 c ¯ A 0 ( K + K 1 ) + 1 ] .

We next show that inequality (2.19) holds for any ε < ε 2 . For any fixed time t < 1 , since ( c ( θ ( x , t ) ) − c ( θ 0 ) ) → 0 as x → − ∞ , then there exists a positive number X such that for all x ≤ − X it holds

∣ c ( θ ( x , t ) ) − c ( θ 0 ) ∣ < A 0 4 .

We employ the contradiction argument to verify

(2.22) ∣ c ( θ ( x , t ) ) − c ( θ 0 ) ∣ < A 0 2 ,

for any x ∈ [ − X , ∞ ) , which leads to (2.19). Assume without the loss of generality that there exists a number x ˆ > − X such that ∣ c ( θ ( x , t ) ) − c ( θ 0 ) ∣ < A 0 / 2 for any x < x ˆ and ∣ c ( θ ( x ˆ , t ) ) − c ( θ 0 ) ∣ = A 0 / 2 . Then it suggests by (2.6), (2.18), and (2.20) that

(2.23) A 0 2 4 ≤ c 2 ( θ ( x ˆ , t ) ) [ c ( θ ( x ˆ , t ) ) − c ( θ 0 ) ] 2 = ∫ − ∞ x ˆ ∂ x { c 2 ( θ ( x , t ) ) [ c ( θ ( x , t ) ) − c ( θ 0 ) ] 2 } d x ≤ ∫ − ∞ x ˆ 2 ∣ c ′ ( θ ) ∣ ⋅ ∣ c θ x ∣ ⋅ [ c ( θ ) − c ( θ 0 ) ] 2 + 2 ∣ c ( θ ) ∣ ⋅ ∣ c θ x ∣ ⋅ ∣ c ′ ( θ ) ∣ ⋅ ∣ c ( θ ) − c ( θ 0 ) ∣ d x ≤ ∫ − ∞ x ˆ 2 c ¯ ⋅ A 0 2 ⋅ ∣ c θ x ∣ ⋅ ∣ c ( θ ) − c ( θ 0 ) ∣ + 2 c ¯ ⋅ 3 A 0 2 ⋅ ∣ c θ x ∣ ⋅ ∣ c ( θ ) − c ( θ 0 ) ∣ d x ≤ 4 c ¯ A 0 ∫ − ∞ ∞ ∣ c θ x ∣ ⋅ ∣ c ( θ ) − c ( θ 0 ) ∣ d x ≤ 2 c ¯ A 0 ∫ − ∞ ∞ ( R 2 + S 2 ) d x + ∫ − ∞ ∞ ∣ c ( θ ) − c ( θ 0 ) ∣ 2 d x ≤ 2 c ¯ A 0 ( K + K 1 ) ε ≤ 2 c ¯ A 0 ( K + K 1 ) ⋅ A 0 4 8 [ 2 c ¯ A 0 ( K + K 1 ) + 1 ] < A 0 4 8 ,

a contradiction. The proof of (2.19) is completed.

Step 4. In view of (2.18)–(2.19) and Lemma 1, we can obtain the estimate for the quantity J with the current initial data (2.13). It follows by (2.4) that for t < 1 and ε < ε 2

(2.24) ‖ u ‖ L ∞ ( R × ( 0 , t ) ) = ‖ v x ‖ L ∞ ( R × ( 0 , t ) ) ≤ C ( 0 + ‖ θ t ‖ L ∞ ( ( 0 , t ) , L 2 ( R ) ) ) ≤ C E ( t ) ≤ C K ⋅ ε .

Inserting (2.24) into (2.3) and utilizing (2.19) yields

(2.25) ‖ J ‖ L ∞ ( R × ( 0 , t ) ) = ‖ v t ‖ L ∞ ( R × ( 0 , t ) ) ≤ C ( 0 + E ( 0 ) ) + C E ( t ) + 2 A 0 2 E ( t ) + C K ⋅ ε ≤ 2 C K ⋅ ε + 2 C A 0 2 ⋅ K ε + C 2 K ⋅ ε ≤ M ε ,

where

M = 2 C K + 2 C K A 0 2 + C 2 K .

Based on the estimate of J in (2.25), we can establish the estimate of Riemann variables in the characteristic triangle region. Let x 1 < x 2 be any two numbers satisfying x 2 − x 1 < A 0 . From points ( x 1 , 0 ) and ( x 2 , 0 ) , we draw the positive and negative characteristic curves x + ( t ) (or t + ( x ) ) and x − ( t ) (or t − ( x ) ), respectively, which are defined as

(2.26) d x + ( t ) d t = c ( θ ( x + ( t ) , t ) ) , x + ( 0 ) = x 1 , d x − ( t ) d t = − c ( θ ( x − ( t ) , t ) ) , x − ( 0 ) = x 2 .

It is asserted by (2.19) that the intersection of x = x + ( t ) and x = x − ( t ) , denoted ( x m , t m ) , satisfies t m < 1 (Figure 1). In fact, from (2.26), one obtains

x m = x 1 + ∫ 0 t m c ( θ ( x + ( t ) , t ) ) d t , x m = x 2 − ∫ 0 t m c ( θ ( x − ( t ) , t ) ) d t ,

thus

(2.27) ∫ 0 t m c ( θ ( x + ( t ) , t ) ) d t + ∫ 0 t m c ( θ ( x − ( t ) , t ) ) d t = x 2 − x 1 < A 0 .

On the other hand, we use (2.19) to obtain

(2.28) ∫ 0 t m c ( θ ( x + ( t ) , t ) ) d t + ∫ 0 t m c ( θ ( x − ( t ) , t ) ) d t ≥ ∫ 0 t m A 0 2 d t + ∫ 0 t m A 0 2 d t = A 0 t m .

Combining (2.27) and (2.28) gives t m < 1 .

$Figure 1 The characteristic triangle region D ( x 1 , x 2 ) D\left({x}_{1},{x}_{2}) .$

Figure 1

The characteristic triangle region D ( x 1 , x 2 ) .

Now we consider the characteristic triangle region bounded by x ± ( t ) and the segment [ x 1 , x 2 ] . Denote this characteristic triangle region by D ( x 1 , x 2 ) . Integrating equation (2.9) over D ( x 1 , x 2 ) and employing the divergence theorem arrives at

(2.29) ∫ x 1 x m R 2 ( x , t + ( x ) ) d x + ∫ x m x 2 S 2 ( x , t − ( x ) ) d x = 1 2 ∫ x 1 x 2 [ R 2 ( x , 0 ) + S 2 ( x , 0 ) ] d x + ∬ D ( x 1 , x 2 ) 1 2 ( R + S ) 2 + ( R + S ) J d x d t .

Here we used the relation u x = J − θ t = J − ( R + S ) / 2 . Thanks to (2.18) and (2.25), we achieve

(2.30) ∬ D ( x 1 , x 2 ) 1 2 ( R + S ) 2 + ( R + S ) J d x d t ≤ ∫ 0 t m ∫ x + ( t ) x − ( t ) 1 2 ( R + S ) 2 + ( ∣ R ∣ + ∣ S ∣ ) ⋅ ∣ J ∣ d x d t ≤ ∫ 0 1 ∫ − ∞ ∞ ( R 2 + S 2 ) d x d t + ∫ 0 1 ∫ x 1 x 2 ( ∣ R ∣ + ∣ S ∣ ) ⋅ ∣ J ∣ d x d t ≤ K ε + M ε ⋅ 2 A 0 ⋅ K ε = ( K + 2 M A 0 K ) ε .

Putting (2.30) into (2.29) leads to

(2.31) ∫ x 1 x m R 2 ( x , t + ( x ) ) d x + ∫ x m x 2 S 2 ( x , t − ( x ) ) d x ≤ 2 ( K + M A 0 K ) ε .

Step 5. Let x = x ˆ ( t ) (or t = t ˆ ( x ) ) for t < 1 be the positive characteristic curve starting from point ( x 0 , 0 ) (Figure 1). It is easy to see that

(2.32) d x ˆ ( t ) d t = c ( θ ( x ˆ ( t ) , t ) ) , x ˆ ( 0 ) = x 0 .

Along the curve x = x ˆ ( t ) , we know by the equation for θ in (2.8) that

(2.33) d θ ( x ˆ ( t ) , t ) d t = R ( x ˆ ( t ) , t ) ,

which is employed to control the sign of c ′ ( θ ) on the curve x = x ˆ ( t ) in a period of time by the initial condition c ′ ( θ 0 ) = A 1 > 0 . We integrate (2.33) from 0 to t < 1 and apply (2.19), (2.31) to conclude

(2.34) ∣ θ ( x ˆ ( t ) , t ) − θ ( x 0 , 0 ) ∣ = ∫ 0 t R ( x ˆ ( τ ) , τ ) d τ ≤ ∫ 0 t ∣ R ( x ˆ ( τ ) , τ ) ∣ 2 d τ = ∫ x 0 x ∣ R ( z , t ˆ ( z ) ) ∣ 2 c ( θ ) d z ≤ 2 A 0 ⋅ ∫ − ∞ ∞ R 2 d z ≤ 2 K A 0 ⋅ ε ,

from which and the C 2 regularity assumption of c ( ⋅ ) , we see that there exists a small number ε 3 ≤ ε 2 such that

(2.35) c ′ ( θ ( x ˆ ( t ) , t ) ) ≥ c ′ ( θ ( x 0 , 0 ) ) 2 ≥ c ′ ( θ 0 ) 4 = A 1 4 > 0 ,

holds on the curve x = x ˆ ( t ) for any ε < ε 3 .

Step 6. Finally, we show that the function S will become infinite along the curve x = x ˆ ( t ) before time t = 1 . From (1.7), (2.13), and (2.16), one can obtain the information of S ( x 0 , 0 )

(2.36) S ( x 0 , 0 ) = − [ 2 c ( θ ( x 0 , 0 ) ) − ε ] ϕ ′ ( 0 ) ≥ A 0 2 ⋅ max 4 A 0 , 96 A 1 ≥ max 2 , 48 A 0 A 1 ,

which means that

(2.37) 1 S ( x 0 , 0 ) < min 1 2 , A 1 48 A 0 .

Introduce two positive numbers K ^ and ε 0 defining as follows:

(2.38) K ^ = c ¯ K A 0 2 A 0 2 + 2 K A 0 + M

and

(2.39) ε 0 = min ε 3 , 1 2 K ^ + 1 , A 1 48 A 0 K ^ + 1 .

Here ε 3 ≤ ε 2 ≤ ε 1 , and the numbers ε 1 , ε 2 , ε 3 come from (2.14), (2.21), and (2.35), respectively. We claim that S ( x ˆ ( t ) , t ) goes to infinite at some point t ∗ < 1 for any ε < ε 0 . In order to prove it, we first utilize the contradiction argument to show the following fact that if S ( x ˆ ( t ) , t ) is smooth for t ∈ [ 0 , 1 ] , then there must be S ( x ˆ ( t ) , t ) > 1 . Suppose without the loss of generality that there exists a number t ˜ ≤ 1 such that the function S ( x ˆ ( t ) , t ) ∈ C 1 ( [ 0 , t ˜ ] ) satisfies S ( x ˆ ( t ) , t ) > 1 for t ∈ [ 0 , t ˜ ) and S ( x ˜ , t ˜ ) = 1 , where x ˜ = x ˆ ( t ˜ ) . This means the time t ˜ is the first time such that S ( x ˆ ( t ) , t ) = 1 . Consider the time period [ 0 , t ˜ ] and rewrite the equation of S in (2.8) as

(2.40) d d t 1 S ( x ˆ ( t ) , t ) = − c ′ 4 c + 1 S 2 c ′ 4 c R 2 + 1 2 ( R + S ) + J ≤ 1 S 2 c ′ 4 c R 2 + 1 2 ( ∣ R ∣ + ∣ S ∣ ) + ∣ J ∣

(2.41) ≤ c ¯ 2 A 0 R 2 + 1 2 ( ∣ R ∣ + ∣ S ∣ ) + M ε .

Here we used the estimate of c ( θ ) in (2.19), the estimate of J in (2.25), and the assumption S ( x ˆ ( t ) , t ) ≥ 1 on [ 0 , t ˜ ] . Integrating (2.41) from t = 0 to t = t ˜ and applying (2.37)–(2.39) yield

(2.42) 1 = 1 S ( x ˆ ( t ˜ ) , t ˜ ) ≤ 1 S ( x 0 , 0 ) + ∫ 0 t ˜ c ¯ 2 A 0 R 2 + 1 2 ( ∣ R ∣ + ∣ S ∣ ) + M ε d t ≤ 1 2 + ∫ x 0 x ˜ c ¯ 2 A 0 ⋅ R 2 c ( θ ) d x + t ˜ ⋅ ∫ x 0 x ˜ R 2 + S 2 c ( θ ) d x + M ε ⋅ t ˜ ≤ 1 2 + c ¯ K ε 0 A 0 2 + 2 K A 0 + M ε ≤ 1 2 + K ^ ⋅ ε 0 ≤ 1 2 + K ^ ⋅ 1 2 K ^ + 1 < 1 ,

a contradiction. Hence, it is concluded that S ( x ˆ ( t ) , t ) > 1 in t ∈ [ 0 , 1 ] before the blowup time. Furthermore, we integrate (2.40) from t = 0 to t < 1 and use (2.19), (2.25), (2.35), (2.37)–(2.39) again to obtain

(2.43) 1 S ( x ˆ ( t ) , t ) = 1 S ( x 0 , 0 ) + ∫ 0 t − c ′ 4 c + 1 S 2 c ′ 4 c R 2 + 1 2 ( R + S ) + J d τ ≤ 1 S ( x 0 , 0 ) + ∫ 0 t − A 1 4 4 ⋅ 3 A 0 2 d τ + K ^ ε = 1 S ( x 0 , 0 ) − A 1 24 A 0 t + K ^ ε ≤ A 1 48 A 0 − A 1 24 A 0 t + K ^ ⋅ A 1 48 A 0 K ^ + 1 = A 1 24 A 0 1 2 + 24 A 0 K ^ 48 A 0 K ^ + 1 − t ,

which implies that, for ε < ε 0 , there exists a number T ∗ < 1 such that S ( x ˆ ( t ) , t ) → + ∞ as t → T ∗ − .

Step 7. To determine (1.9), it suffices to show by (2.1) and (2.5) that the function R ( x ˆ ( t ) , t ) is uniformly bounded over [ 0 , T ∗ ) . For any t ∗ < T ∗ , we draw the negative characteristic curve x = x ¯ − ( t ) (or t = t ¯ − ( x ) ) from point ( x ∗ , t ∗ ) down to the x axis at ( x ¯ , 0 ) , where x ∗ = x ˆ ( t ∗ ) . Integrating the equation of R in (2.8) along the curve x = x ¯ − ( t ) from t = 0 to t = t ∗ arrives at

(2.44) R ( x ˆ ( t ∗ ) , t ∗ ) = R ( x ¯ , 0 ) + ∫ 0 t ∗ c ′ 4 c ( R 2 − S 2 ) − 1 2 ( R + S ) − J ( x ¯ − ( t ) , t ) d t

from which and (2.13), (2.18), (2.19), (2.25), one has

(2.45) ∣ R ( x ∗ , t ∗ ) ∣ ≤ ∣ R ( x ¯ , 0 ) ∣ + ∫ 0 t ∗ c ′ 4 c ( R 2 + S 2 ) + 1 2 ( ∣ R ∣ + ∣ S ∣ ) + ∣ J ∣ d t ≤ ε max z ∈ [ − 1 , 1 ] ∣ ϕ ′ ( z ) ∣ + ∫ x ∗ x ¯ c ¯ 4 ⋅ A 0 2 4 ( R 2 + S 2 ) d x + t ∗ ⋅ ∫ x ∗ x ¯ R 2 + S 2 A 0 2 d x + M ε ⋅ t ∗ ≤ ε 0 max z ∈ [ − 1 , 1 ] ∣ ϕ ′ ( z ) ∣ + c ¯ K ε 0 A 0 2 + 2 K A 0 + M ε 0 ,

which means by the arbitrariness of point ( x ∗ , t ∗ ) on curve x = x ˆ ( t ) that the function R ( x ˆ ( t ) , t ) is uniformly bounded over [ 0 , T ∗ ) . The proof of Theorem 1 is completed.

Acknowledgments

The author would like to thank the editors and referees for very helpful comments and suggestions to improve the quality of the article.

Funding information: This work was partially supported by the Natural Science Foundation of Zhejiang province of China (LY21A010017) and National Natural Science Foundation of China (12071106, 12171130).
Conflict of interest: The author states no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2022-03-14

Accepted: 2022-07-14

Published Online: 2022-11-19

This work is licensed under the Creative Commons Attribution 4.0 International License.

Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals

Abstract

1 Introduction

Theorem 1

2 The proof of Theorem 1

Lemma 1

Acknowledgments

References

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