# Wave-Breaking Criterion for the Generalized Weakly Dissipative Periodic Two-Component Hunter-Saxton System

Authors: Jianmei Zhang; Lixin Tian
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper studies the wave-breaking criterion for the generalized weakly dissipative two-component Hunter-Saxton system in the periodic setting. We get local well-posedness for the generalized weakly dissipative two-component Hunter-Saxton system. We study a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles.


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## Body

## 1. Introduction

In recent years, the Hunter-Saxton equation [1]
(1)utxx+2uxuxx+uuxxx=0
models the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal. In Hunter and Saxton [1], x is the space variable in a reference frame moving with the linearized wave velocity, t is a slow-time variable, and u(t,x) is a measure of the average orientation of the medium locally around x at time t. In order to be more precise, the orientation of the molecules is described by the field of unit vectors (cosu(t,x), sinu(t,x)) [2]. The Hunter-Saxton equation also arises in a different physical context as the high-frequency limit [3, 4] of the Camassa-Holm equation for shallow water waves [5, 6] and a reexpression of the geodesic flow on the diffeomorphism group of the circle [7] with a bi-Hamiltonian structure [1, 8] which is completely integrable [4, 9]. Hunter and Saxton [1] explored the initial value problem for the Hunter and Saxton equation on the line (nonperiodic case) and on the unit circle S=R/Z by using the method of characteristics, while Yin [2] studied it by using the Kato semigroup method. In addition, the two classes of admissible weak solutions, dissipative and conservative solutions, and their stability were investigated in [10–12]. Lenells [13] confirmed that the Hunter-Saxton equation also describes the geodesic flows on the quotient space of the infinite-dimensional group Ds(S) modulo the subgroup of rotations  Rot(S).The Camassa-Holm equation admits many integrable multicomponent generalizations. So many authors studied the two-component Camassa-Holm system [14, 15]. Inspired by this, recently, the researchers have made a study of the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting as follows:
(2)utxx+2σuxuxx+σuuxxx-ρρx+Aux=0,t>0,x∈R,ρt+(ρu)x=0,t>0,x∈R,u(t,x+1)=u(t,x),ρ(t,x+1)=ρ(t,x),ssssssssssssssssssst≥0,x∈R,u(0,x)=u0(x),ρ(0,x)=ρ0(x),x∈R.The authors of [16] have explored the particular choice of the parameter σ=1. The authors of [17] have further studied the wave breaking and global existence for the system for the parameter σ∈R to determine a wave-breaking criterion for strong solutions by using the localization analysis in the transport equation theory.In general, avoiding energy dissipation mechanisms in a real world is not so easy. Wu and Yin [18, 19] have investigated the blow-up phenomena and the blow-up rate of the strong solutions of the weakly dissipative CH equation and DP equation. Inspired by the results mentioned above, we are going to discuss the initial value problem associated with the generalized weakly dissipative periodic two-component Hunter-Saxton system
(3)utxx+2σuxuxx+σuuxxx-ρρx+Aux+λ(u-uxx)=0,t>0,x∈R,ρt+(ρu)x=0,t>0,x∈R,u(t,x+1)=u(t,x),ρ(t,x+1)=ρ(t,x),GGfGGt≥0,x∈R,u(0,x)=u0(x),ρ(0,x)=ρ0(x),x∈R,
where σ∈R is the new free parameter and  A≥0, λ<0.Our major results of this paper are Theorems11 and 12 (wave-breaking criterion). The remainder of the paper is organized as follows. Section 2 establishes the local well-posedness for (3) with the initial data in Hs×Hs-1, s≥2. Section 3 deals with the wave breaking of this new system. Theorem 11, using transport equation theory, states a wave-breaking criterion which says that the wave breaking only depends on the slope of u, not the slope of ρ. Theorem 12 improves the blow-up criterion with a more precise condition.Notation 1.
Throughout this paper,S=R/Z will denote the unit circle. By Hs, s≥0, we will represent the Sobolev spaces of equivalence classes of functions defined on the unit circle S which have square-integrable distributional derivatives up to order s. The Hs-norm will be designated by ∥·∥Hs, and the norm of a vector v∈Hs×Hs-1 will be written as ∥v∥Hs×Hs-1. Also, the Lebesgue spaces of order p∈[1,∞] will be denoted by Lp(S), and the norm of their elements will be denoted by ∥f∥Lp(S). Finally, if p=2, we agree on the convention ∥·∥L2(S)=∥·∥.


## 2. Preliminaries

In this part, we will establish the local well-posedness for the Cauchy problem of system (3) by using Kato’s theory. To pursue our goal, we give the results we wanted in brief.We now provide the framework in which we will reformulate (3). To do this, we observe that we can write the first equation of (3) in the following integrated form:
(4)utx+σ2ux2+σuuxx-12ρ2+Au+λ∂x-1u-λux=g(t),
where ∂x-1f(x)=∫0x‍f(y)dy and g(t) is determined by the periodicity of u to be
(5)g(t)=-∫S(σ2ux2+12ρ2-Au)dx.Integrating both sides of (4) with respect to variable  x, we get
(6)ut+σuux=∂x-1(σ2ux2+12ρ2-Au+g+λux-λ∂x-1u)+h(t),
where h(t):[0,∞)→R is an arbitrary continuous function. Therefore, (3) can be written in the “transport” form as follows:
(7)ut+σuux=∂x-1(σ2ux2+12ρ2-Au-λ∂x-1u+λux+g)+h(t),t>0,x∈R,ρt+uρx=-uxρ,t>0,x∈R,u(t,x+1)=u(t,x),ρ(t,x+1)=ρ(t,x),ffffddfft≥0,x∈R,u(0,x)=u0(x),ρ(0,x)=ρ0(x),x∈R,
where h(t):[0,∞)→R is an arbitrary continuous function.Next, we apply Kato’s theory to establish the local well-posedness for the system (3). Consider the abstract quasi-linear evolution equation
(8)dvdt+A(v)v=f(v),t≥0,v(0)=v0.Proposition 1 (see [20]).
Given the evolution equation (8), assume that the Kato conditions hold. For a fixed v0∈Y, there is a maximal T>0 depending only on ∥v0∥Y and a unique solution v to the abstract quasi-linear evolution equation (8) such that
(9)v=v(·,v0)∈C([0,T);Y)∩C1([0,T);X).
Moreover, the mapv0→v(·,v0) is continuous from Y to
(10)C([0,T);Y)∩C1([0,T);X).
One may follow the similar argument as in [17] to obtain the following local well-posedness for (3).Theorem 2.
Given anyX0=(u0ρ0)∈Hs×Hs-1, s≥2,   there exist a maximal  T=T(σ,A;∥X0∥Hs×Hs-1)>0 and a unique solution X=(uρ) to (3) such that
(11)X=X(·,X0)∈C([0,T);Hs(S)×Hs-1(S))∩C1([0,T);Hs-1(S)×Hs-2(S)).
Moreover, the solution depends continuously on the initial data, that is, the mappingX0→X(·,X0):Hs×Hs-1→C([0,T);Hs(S)×Hs-1(S))∩C1([0,T);andHs-1(S)×Hs-2(S))  is continuous, and the maximal existence time T can be chosen independently of the Sobolev order s.
Now, discuss the initial value problem for the Lagrangian flow map as follows:(12)∂φ∂t=u(t,φ(t,x)),t∈[0,T),φ(0,x)=x,x∈R,
where u is the first component of the solution X to (3). Using classical results from ordinary differential equations, one can acquire the following result on φ which is of vital importance in the proof of the blow-up scenarios.Lemma 3 (see [17]).
Letu∈C([0,T);Hs)∩C1([0,T);Hs-1), s≥2. Then, initial value problem (12) admits a unique solution  φ∈C1([0,T)×R,R). Moreover, {φ(t,·)}t∈[0,T) is increasing diffeomorphism of R with
(13)φx(t,x)=e∫0T‍ux(τ,φ(τ,x))dτ>0,(t,x)∈[0,T)×R.Remark 4.
Sinceφ(t,·):R→R is a diffeomorphism of the linear for every t∈[0,T), the L∞-norm of any function v(t,·)∈L∞, t∈[0,T) is preserved under the family of diffeomorphisms φ(t,·) with t∈[0,T), that is,
(14)∥v(t,·)∥L∞(S)=∥v(t,φ(t,·))∥L∞(S),t∈[0,T).
Similarly, we have(15)infx∈Sv(t,x)=infx∈Sv(t,φ(t,x)),t∈[0,T),supx∈Sv(t,x)=supx∈Sv(t,φ(t,x)),t∈[0,T).Lemma 5.
LetX0=(u0ρ0)∈Hs×Hs-1, s≥2, and let T be the maximal existence time of the solution X=(uρ) to (3) with initial data X0. Then, for all t∈[0,T), we have the following results:
(16)∫Sρ(t,x)dx=∫Sρ0(t)dx,(17)∫Sux2(t,x)+ρ2(t,x)dx≤∫Su0,x2(x)+ρ02(x)dx≜E0.Proof.
On the one hand, integrating the second equation in (3) by parts and using the periodicity of u and ρ, we acquire
(18)ddt∫Sρdx=-∫S(uρ)xdx=0.
On the other hand, multiplying (4) by ux and integrating by parts, considering the periodicity of u, we obtain
(19)ddt∫Sux2dx=-2∫Suρρxdx+2λ∫Su2dx+2λ∫Sux2dx.
Multiplying the second equation in (3) by ρ and integrating by parts, we have
(20)ddt∫Sρ2dx=2∫Suρρxdx.
Adding the above two equations, we get(21)ddt∫Sux2+ρ2dx=2λ∥u∥H22,λ<0.
We acquire(22)∫Sux2(t,x)+ρ2(t,x)dx≤∫Su0,x2(x)+ρ02(x)dx≜E0.
This completes the proof of Lemma5.Lemma 6.
LetX0=(u0ρ0)∈Hs×Hs-1, s≥2, and let T be the maximal existence time of the solution X=(uρ) to (3) with initial data X0. Then, for all t∈[0,T), we have the following results:
(23)∫Su2(t,x)dx≤eC2t(∫Su02(x)dx+1),
where C1=max(|σ|-λ,1)E0+supt∈[0,∞)|h(t)|>0, C2=C1+4A-2λ.Proof.
By computing directly, we have(24)|∂x-1(σ2ux2+12ρ2-Au+g+λux-λ∂x-1u)+h(t)|≤∫0x‍|σ2ux2+12ρ2-Au+g+λux-λ∂x-1u|dx+|h(t)|≤12max(|σ|-λ,1)E0+|g(t)|+|h(t)|+(|A|-λ)∫01‍|u|dx-λ≤max(|σ|-λ,1)E0+|h(t)|+(2|A|-λ)∫S‍|u|dx-λ:=C1+(2|A|-λ)∫S‍|u|dx-λ,
where C1=max(|σ|-λ,1)E0+supt∈[0,∞)|h(t)|>0 and
(25)|g(t)|=|-∫S(σ2ux2+12ρ2-Au)dx|≤12max(|σ|,1)E0+A∫S|u|dx.
Multiplying (6) by u and integrating with respect to x, using the periodicity of u and (24), we obtain
(26)12ddt∫Su2(t,x)dx=∫Suutdx=-σ∫Suxu2dx+∫Su[∂x-1(σ2ux2+12ρ2-Au+λuxddddddddddd-λ∂x-1u+g12ρ2σ2ux2)+h(t)∂x-1]dx=∫Su[∂x-1(σ2ux2+12ρ2-Au+λuxfffffffff-λ∂x-1u+gσ2ux212ρ2)+h(t)∂x-1(σ2ux2+12ρ2-Au+λux]dx≤(C1+(2|A|-λ)∫S|u|dx-λ)∫S|u|dx≤C1∫S|u|dx+(2|A|-λ)(∫S|u|dx)2-λ∫Su2dx≤(C12+(2|A|-2λ))∫Su2dx+C1-λ2=C22∫Su2dx+C1-λ2,
where C2=C1+4A-2λ; note that C2>C1.
By Gronwall's inequality, we get(27)∫Su2(t,x)dx≤eC2t(∫Su02(x)dx+C1-λC2)-C1-λC2≤eC2t(∫Su02(x)dx+1).
This completes the proof of Lemma6.Lemma 7.
Assume thatu0∈Hs(S), s≥2, u0≠0, and that the corresponding solution u(t,x) of (3) has a zero point for any time t≥0. Then, for all t∈[0,T) we have
(28)∫Su2(t,x)dx≤∫Sux2(t,x)dx≤E0.Proof.
By assumption, there isx0∈[0,1] such that u(t,x0)=0 for each t∈[0,T).
Then, forx∈S, by holder equality, we have
(29)u2(t,x)=(∫x0x‍uxdx)2≤(x-x0)∫x0x‍ux2dx,x∈[x0,x0+12].
This impliessupx∈Su2(t,x)≤(1/2)∫Sux2dx(30)∫Su2(t,x)dx≤supx∈Su2(t,x)≤12∫Sux2dx≤∫Sux2+ρ2dx≤E0.


## 3. Wave-Breaking Criteria

In this section, by using transport equation theory, we obtain the wave-breaking criteria for solutions to (3). We first recall the following propositions.Proposition 8 (1D Moser-type estimates).
The following estimates hold:(a)
Fors≥0,
(31)∥fg∥Hs(R)≤C(∥f∥L∞(R)∥g∥Hs(R)+∥f∥Hs(R)∥g∥L∞(R)).(b)
Fors>0,
(32)∥f∂xg∥Hs(R)≤C(∥f∥L∞(R)∥∂xg∥Hs(R)+∥f∥Hs+1(R)∥g∥L∞(R)).(c)
Fors1≤1/2, s2>1/2,  s1+s2>0,
(33)∥fg∥Hs1(R)≤C∥f∥Hs1(R)∥g∥Hs2(R), where C′s are constants that are independent of f and g.Proposition 9 (see [21]).
Suppose thats>-d/2. Let v be a vector field such that ∇v belongs to L1([0,T];Hs-1) if s>1+d/2 or to L1([0,T];Hd/2∩L∞), otherwise. Suppose also that f0∈Hs, F∈L1([0,T];Hs) and that f∈L∞([0,T];Hs)∩C([0,T];S′)  solves the d-dimensional linear transport equations
(34)(T){∂tf+v·∇f=F,f|t=0=f0.
Thenf∈C([0,T];Hs). More precisely, there exists a constant C depending only on s, p, and d such that the following statements hold:(1)
Ifs≠1+d/2,
(35)∥f∥Hs≤∥f0∥Hs+∫0t∥F(τ)∥Hsdτ+C∫0tV′(τ)∥f(τ)∥Hsdτ,or
(36)∥f∥Hs≤eCV(t)(∥f0∥Hs+∫0te-CV(τ)∥F(τ)∥Hsdτ),
with V(t)=∫0t‍∥∇v(τ)∥Hd/2∩L∞dτ if s<1+d/2 and  V(t)=∫0t‍∥∇v(τ)∥Hs-1dτ else.(2)
Iff=v, then for alls>0, estimates (35) and (36) hold with(37)V(t)=∫0t‍∥∂xu(τ)∥L∞dτ.Proposition 10 (see [21]).
Let0<s<1. Suppose that f0∈Hs, g∈L1([0,T];Hs)v, ∂xv∈L1([0,T];L∞) and that f∈L∞([0,T];Hs)∩C([0,T];S′) solves the 1-dimensional linear transport equation
(38)(T){∂tf+v·∇f=g,f|t=0=f0.
Thenf∈C([0,T];Hs). More precisely, there exists a constant C depending only on s such that the following statements hold:
(39)∥f∥Hs≤∥f0∥Hs+C∫0t∥g(τ)∥Hsdτ+C∫0tV′(τ)∥f(τ)∥Hsdτ
or
(40)∥f∥Hs≤eCV(t)(∥f0∥Hs+C∫0t∥g(τ)∥Hsdτ),
with V(t)=∫0t‍(∥v(τ)∥L∞+∥∂xv(τ)∥L∞)dτ.The above proposition was proved in [8] using Littlewood-Paley analysis for the transport equation and Moser-type estimates. Using this result and performing the same argument, as in [17], we can obtain the following blow-up criterion.Theorem 11.
LetX0=(u0ρ0)∈Hs×Hs-1 with  s≥2, and X=(uρ) be the corresponding solution to (3). Assume that T>0 is the maximal time of existence. Then
(41)T<∞⟹∫0T∥∂xu(τ)∥L∞dτ=∞.Our next result describes the necessary and sufficient condition for the blow-up of solutions to (3).Theorem 12.
Suppose thatσ∈R∖{0}. Let X0=(u0ρ0)∈Hs×Hs-1, with s≥2, and let T be the maximal existence time of the solution X=(uρ) to (3) with initial data X0. Then, the solution blows up in finite time if and only if
(42)liminft→T-{infx∈Sσux(t,x)}=-∞.The approach one takes here is the method of characteristics. Applying the following lemma, we may carry out the estimates along the characteristicsφ(t,x) which captures supx∈Sux(t,x) and infx∈Sux(t,x).Lemma 13 (see [22]).
LetT>0 and let v∈C1([0,T];H2(R)). Then, for every t∈[0,T), there exists at least one point ξ(t)∈R with
(43)m(t):=infx∈Svx(t,x)=vx(t,ξ(t)),
and the function m(t) is almost everywhere differentiable on (0,T) with
(44)dm(t)dt=vtx(t,ξ(t)),a.e.on(0,T).Lemma 14.
LetX0=(u0ρ0)∈Hs×Hs-1 with s≥2, and let T be the maximal existence time of the solution X=(uρ) to (3) with initial data X0. Then one has the following:(1)
σ≠0(45)supx∈Sux(t,x)≤∥u0,x∥L∞(S)+∥ρ0∥L∞(S)2+K12(T)σ+λ2σ2+λσdddddddddddddddddd(σ>0)(46)infx∈Sux(t,x)≥-∥u0,x∥L∞(S)-λ2σ2-K22(T)σ+λσ(σ<0).(2)
σ=0(47)supx∈Sux(t,x)≤supx∈Su0,x(x)+12(supx∈Sρ02(x)+K12(T))eλt-1λ(48)infx∈Sux(t,x)≥infx∈Su0,x(x)+12(infx∈Sρ02(x)-K22(T))eλt-1λ.
The constants above are defined as follows:(49)K1(T)=2A-λ+A2E0+3A-2λ2[eC2T(∥u0∥L2(S)2+1)],(50)K2(T)=2A-λ+A+22E0+3A-2λ2[eC2T(∥u0∥L2(S)2+1)].Proof of Lemma14.
By Theorem2 and a simple density argument, we show that the desired results are valid when  s≥3, so we take s=3 in the proof.
Letσ>0. Using Lemma 13 and the fact that
(51)supx∈S[vx(t,x)]=-infx∈S[-vx(t,x)].
We can considerM(t) and γ(t) as follows:
(52)M(t):=ux(t,ξ(t))=supx∈S[ux(t,x)],t∈[0,T).
Hence,(53)uxx(t,ξ(t))=0,a.e.ont∈[0,T).
Take the trajectoryφ(t,x) defined in (12). Then we know that φ(t,·):R→R is a diffeomorphism for every t∈[0,T). Therefore, there exists x0(t)∈R such that
(54)φ(t,x0(t))=ξ(t),t∈[0,T).
Now, let(55)γ(t)=ρ(t,φ(t,x0)),t∈[0,T).
Therefore, along the trajectoryφ(t,x0), (4) and the second equation of (3) become
(56)M′(t)=-σ2M2(t)+λM(t)+12γ2(t)+f(t,φ(t,x0))γ′(t)=-γM,a.e.t∈[0,T),
where the notation denotes the derivative with respect to t and f represents the function
(57)f=-Au-λ∂x-1u+g(t)=-Au-λ∂x-1u-∫S(σ2ux2+12ρ2-Au)dx.
We first compute the upper and lower bounds forf for later use in getting the blow-up result as follows:
(58)f=-Au-λ∫0x‍udx-σ2∫Sux2dx-12∫Sρ2dx+A∫Sudx≤-Au-λ∫0x‍udx+A∫Sudx≤A2(1+u2)+A-λ2(1+∫Su2dx).
Sinceu2≤(1/2)∫S(u2+ux2)dx, (17), we obtain the upper bound for f(59)f≤A2(1+12∫S(u2+ux2)dx)+A-λ2(1+∫Su2dx)≤A-λ2+A4∫S(ρ2+ux2)dx+3A-2λ4∫Su2dx≤A-λ2+A4E0+3A-2λ4[eC2t(∫Su02(x)dx+1)]≤A-λ2+A4E0+3A-2λ4×[eC2T(∥u0∥L2(S)2+1)]:=12K12(T).
Now we turn to the lower bound of f. Using previous arguments, we get
(60)-f=Au+λ∫0x‍udx+σ2∫Sux2dx+12∫Sρ2dx-A∫Sudx≤A1+u22+max(|σ|,1)2∫S(ρ2+ux2)dx+A-λ2(1+∫Su2dx)≤A-λ2+A+2max(|σ|,1)4∫S(ρ2+ux2)dx+3A-2λ4∫Su2dx≤A-λ2+A+2max(|σ|,1)4E0+3A-2λ4[eC2T(∥u0∥L2(S)2+1)].
Whenσ<0, we have a finer estimate
(61)-f≤A-λ2+A+24∫S(ρ2+ux2)dx+3A-2λ4∫Su2dx≤A-λ2+A+24E0+3A-2λ4×[eC2T(∥u0∥L2(S)2+1)]:=12K22(T).
Combining (59) and (60), we obtain
(62)|f|≤A-λ2+A+2max(|σ|,1)4E0+3A-2λ4[eC2T(∥u0∥L2(S)2+1)]:=12K32(T).
Sinces≥3, we have u∈C01(S). Therefore,
(63)supx∈Sux(t,x)≥0,infx∈Sux(t,x)≤0,t∈[0,T).
Hence,M(t)>0 for t∈[0,T). From the second equation of (55), we obtain
(64)γ(t)=γ(0)e-∫0t‍M(τ)dτ.
Hence,(65)|ρ(t,φ(t,x0))|=|γ(t)|≤|γ(0)|≤∥ρ0∥L∞(S).
For any givenx∈S, define
(66)p1(t)=M(t)-∥u0,x∥L∞(S)-∥ρ0∥L∞(S)2+K12(T)σ+λ2σ2-λσ(σ>0).
Observing thatp1(t) is a C1-differentiable function on [0,T) and satisfies
(67)p1(0)=M(0)-∥u0,x∥L∞(S)-∥ρ0∥L∞(S)2+K12(T)σ+λ2σ2-λσ≤M(0)-∥u0,x∥L∞(S).
We now claim thatp1(t)≤0t∈[0,T).
Assume the contrary that there ist0∈[0,T) such that p1(t0)>0.
Lett1=max{t<t0:p1(t)=0}. Then p1(t1)=0 and p1′(t1)≥0, or equivalently,
(68)M(t1)=∥u0,x∥L∞(S)+∥ρ0∥L∞(S)2+K12(T)σ+λ2σ2+λσ
and M′(t1)≥0 a.e. t∈[0,T). On the other hand, we have
(69)M′(t1)=-σ2M2(t1)+λM(t1)+12γ2(t1)+f(t1,φ(t1,x0))a.e.t∈[0,T)≤-σ2(M(t1)-λσ)2+12∥ρ0∥L∞(S)2+λ22σ+K12(T)2<0,
which is a contradiction. Therefore, p1(t)≤0 for all t∈[0,T). Since x is arbitrarily chosen, we obtain (45).
To derive (46) in the case of σ<0, we consider M~(t) and γ~(t) as in Lemma 13:
(70)M~(t):=ux(t,ζ(t))=infx∈S[ux(t,x)]t∈[0,T).
Hence,(71)uxx(t,ζ(t))=0a.e.t∈[0,T).
Using previous arguments, we take the characteristicφ(t,x) defined in (13) and choose x1(t)∈R such that
(72)φ(t,x1(t))=ζ(t).
Let(73)γ~(t)=ρ(t,φ(t,x)),t∈[0,T).
Hence, along the trajectoryM~′(t)=λM~(t) +(1/2)γ2(t) +f(t,φ(t,x0)) ≥λM~(0)+(1/2)γ2(0) +(1/2)K22(T), (4) and the second equation of (3) become
(74)M~′(t)=-σ2M~2(t)+λM~(t)+12γ2(t)+f(t,φ(t,x0))γ~′(t)=-γ~M~,a.e.t∈[0,T).
Define(75)p2(t)=M~(t)+∥u0,x∥L∞(S)+λ2σ2-K22(T)σ-λσ(σ<0).
For any givenx∈S, Note that p2(t) is also C1-differentiable function on [0,T) and satisfies
(76)p2(0)=M~(0)+∥u0,x∥L∞(S)+λ2σ2-K22(T)σ-λσ≥M~(0)+∥u0,x∥L∞(S)≥0.
We now claim thatp2(t)≥0, for any t∈[0,T).
Suppose not, then there ist~∈[0,T) such that p2(t0~)<0. Define
(77)t2=max{t<t0~:p2(t)=0}.
Then,p2(t2)=0 and p2′(t2)<0, or equivalently,
(78)M~(t2)=-∥u0,x∥L∞(S)-λ2σ2-K22(T)σ+λσ
and M~′(t2)≤0 a.e. t∈[0,T). However, we have
(79)M~′(t2)=-σ2M~2(t2)+λM~(t2)+12γ2(t2)+f(t2,φ(t2,x0))a.e.t∈[0,T)≥-σ2(-∥u0,x∥L∞(S)-λ2σ2-K22(T)σ)2+λ22σ-12K22(T)>0.
Therefore,p2(t)≥0 for any t∈[0,T). Since x is chosen arbitrarily, we obtain (46).
Letσ=0. Using previous arguments, (56) becomes
(80)M′(t)=-σ2M2(t)+λM(t)+12γ2(t)+f(t,φ(t,x0))γ′(t)=-γM,a.e.t∈[0,T),
where the notation denotes the derivative with respect to t and f represents the function
(81)f=-Au-λ∂x-1u+g(t)=-Au-λ∂x-1u-∫S(12ρ2-Au)dx.
We first compute the upper and lower bounds forf for later use in getting the blow-up result:
(82)f=-Au-λ∫0x‍udx-12∫Sρ2dx+A∫Sudx≤-Au-λ∫0x‍udx+A∫Sudx≤A2(1+u2)+A-λ2(1+∫Su2dx)≤A2(1+12∫S(u2+ux2)dx)+A-λ2(1+∫Su2dx)≤A-λ2+A4∫S(ρ2+ux2)dx+3A-2λ4∫Su2dx≤A-λ2+A4E0+3A-2λ4[eC2t(∫Su02(x)dx+1)]≤A-λ2+A4E0+3A-2λ4[eC2T(∥u0∥L2(S)2+1)].
Now, we turn to the lower bound off:
(83)-f=Au+λ∫0x‍udx+12∫Sρ2dx-A∫Sudx≤A1+u22+12∫Sρ2dx+A-λ2(1+∫Su2dx)≤A-λ2+A+24∫S(ρ2+ux2)dx+3A-2λ4∫Su2dx≤A-λ2+A+24E0+3A-2λ4[eC2T(∥u0∥L2(S)2+1)].
Combining (82) and (83), we obtain
(84)|f|≤A-λ2+A+24E0+3A-2λ4[eC2T(∥u0∥L2(S)2+1)].
We knowM(t)>0 for t∈[0,T). From the second equation of (81), we obtain that
(85)γ(t)=γ(0)e-∫0t‍M(τ)dτ.
Hence,(86)|ρ(t,φ(t,x0))|=|γ(t)|≤|γ(0)|.
Therefore, we have(87)M′(t)=λM(t)+12γ2(t)+f(t,φ(t,x0))≤λM(0)+12γ2(0)+12K12(T),(88)M′(t)-λM(t)≤12(supx∈Sρ02(x)+K12(T))a.e.t∈[0,T).
Integrating (88) on [0,t], we prove (47) as follows:
(89)M(t)≤supx∈Su0,x(x)+12(supx∈Sρ02(x)+K12(T))eλt-1λ.
To obtain a lower bound forinfx∈Sux(t,x), we use the same argument.
Sinceσ=0, (80) becomes
(90)M~′(t)=λM~(t)+12γ2(t)+f(t,φ(t,x1)),γ~′(t)=-γ~M~,a.e.t∈[0,T).
Because ofM~(t)<0, we get from the second equation of (90) that
(91)γ~(t)=γ~(0)e-∫0t‍M(τ)dτ.
This means that(92)|ρ(t,φ(t,x1))|=|γ(t)|≥|γ(0)|.
Then,(93)M~′(t)=λM~(t)+12γ2(t)+f(t,φ(t,x0))≥λM~(0)+12γ2(0)+12K22(T),(94)M~′(t)-λM~(t)≥12(infx∈Sρ02(x)-K22(T))a.e.t∈[0,T).
Integrating (94) on [0,t], we prove (48). This completes the proof of Lemma 14.Lemma 15.
Suppose thatσ∈R∖{0}. Suppose X0=(u0ρ0)∈Hs×Hs-1 with s≥2, and let T be the maximal existence time of the solution X=(uρ) to (3) with initial data X0. Then we have
(95)ρ(t,φ(t,x))φx(t,x)=ρ0(x),(t,x)∈[0,T]×S.
Moreover, if there existsM>0 such that
(96)inf(t,x)∈[0,T]×Sσux(t,x)≥-M,(t,x)∈[0,T]×S.
Then(97)∥ρ(t,·)∥L∞(S)=∥ρ(t,φ(t,·))∥L∞(S)≤eMT/σ∥ρ0∥L∞(S)(σ>0)(98)∥ρ(t,·)∥L∞(S)=∥ρ(t,φ(t,·))∥L∞(S)≤eNT/σ∥ρ0∥L∞(S)(σ<0),
where N=∥u0,x∥L∞(S)+λ2/σ2-K22(T)/σ-λ/σ and K2(T) is given in (50).Proof.
Differentiating the left hand side of (95) with respect to t, in view of the relations (12) and (3), we obtain
(99)ddt{ρ(t,φ(t,x))φx(t,x)}=[ρt(t,φ)+ρx(t,φ)φt(t,x)]×φx(t,x)+ρ(t,φ)φxt(t,x)=[ρt(t,φ)+ρx(t,φ)φt(t,x)]φx(t,x)+ρ(t,φ)ux(t,φ)φx(t,x)=[ρt(t,φ)+ρx(t,φ)φt(t,x)+ρ(t,φ)ux(t,φ)]×φx(t,x)=0.
This completes the proof of (95). In view of the assumption (96) and σ>0, we obtain u(t,x)≥-(M/σ)(t,x)∈[0,T]×S.
By Lemma3 and (95), we have
(100)∥ρ(t,·)∥L∞=∥ρ(t,φ(t,·))∥L∞=∥e-∫0t‍ux(τ,·)dτρ0(·)∥L∞≤eMT/σ∥ρ0(·)∥L∞.
To obtain (98), we use a similar argument as before. Using (13) and the lower bound for ux(t,x) in (46), it follows that
(101)∥ρ(t,·)∥L∞=∥ρ(t,φ(t,·))∥L∞=∥e-∫0t‍ux(τ,·)dτρ0(·)∥L∞≤eNT/σ∥ρ0(·)∥L∞,
which proves (98). This completes the proof of Lemma 15.Proof of Theorem12.
Suppose thatT<∞ and that (42) is not valid. Then, there is some positive number M>0 such that
(102)σux(t,x)≥-M,(t,x)∈[0,T]×S.
It now follows from Lemma14 that |ux(t,x)|≤C, where C=C(A,M,σ,E0,λ,∥u0∥,T). Therefore, Theorem 11 implies that the maximal existence time T=∞, which contradicts with the assumption that T<∞.
Conversely, the Sobolev embedding theoremHs(S)→L∞(S) with s>1/2 implies that if (70) holds, the corresponding solution blows up in finite time, which completes the proof of Theorem 12.



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*Source: 809824-2013-08-28.xml*