# TheH1(R) Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

Authors: Zhaowei Sheng; Shaoyong Lai; Yuan Ma; Xuanjun Luo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the spaceC([0,∞) × R)∩L∞([0,∞); H1(R)) provided that its initial value u0(x) belongs to the space H1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.


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## Body

## 1. Introduction

In [1], the author investigated the following weakly dissipative Camassa-Holm model
(1.1)ut-uxxt+3uux+λ(u-uxx)=2uxuxx+uuxxx,
where λ≥0. When λ=0, (1.1) becomes the classical Camassa-Holm equation [2]. The authors in [1] obtained the local well-posedness of the solution for the model by using the Kato theorem. A necessary and sufficient condition of the blow-up of the solution and some criteria guaranteeing the blow-up of the solution are derived. The blow-up rate of the solution is discussed. It is also shown in [1] that the equation has global strong solutions, and these strong solutions decay to zero as time goes to infinite provided the potentials associated to their initial data are of one sign. However, the existence of global weak solutions in the space H1(R) is not discussed in paper [1]. This will constitute the objective of this work.More relevant for the present paper, here we state several works on the global weak solution for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for global weak solutions to the Camassa-Holm equation have been proved by Constantin and Escher [3], Constantin and Molinet [4], and Danchin [5, 6]. Xin and Zhang [7] proved that the global existence of the weak solution for the Camassa-Holm equation in the energy space H1(R) without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under certain conditions on the solution [8]. Coclite et al. [9] investigated the global weak solutions for a generalized hyperelastic-rod wave equation or a generalized Camassa-Holm equation. The existence of a strongly continuous semigroup of global weak solutions for the generalized hyperelastic-rod equation with any initial value in the space H1(R) was established in [9]. Under the sign condition imposed on the initial value, Yin et al. [10] proved the existence and uniqueness results of global weak solution for a nonlinear shallow water equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases. For other dynamic properties about various generalized Camassa-Holm models and other partial differential equations, the reader is referred to [11–16].The aim of this work is to study the existence of global weak solutions for (1.1) in the space C([0,∞)×R)⋂‍L∞([0,∞);H1(R)) under the assumption u0(x)∈H1(R). The limits of viscous approximations for the equation are used to establish the existence of the global weak solution. Here, we should mention that up to now, there have been no global existence results for weak solutions to the generalized Camassa-Holm equation (1.1).The rest of this paper is as follows. The main result is given in Section2. In Section 3, we state the viscous problem and give a corresponding well-posedness result. An upper bound, higher integrability estimate, and basic compactness properties for the viscous approximations are also established in Section 3. Strong compactness of the derivative of the viscous approximations is obtained in Section 4, where the main result for the existence of (1.1) is proved.


## 2. Main Result

Consider the Cauchy problem for (1.1)
(2.1)ut-utxx+(3u22)x+λ(u-uxx)=2uxuxx+uuxxx,u(0,x)=u0(x),
which is equivalent to
(2.2)ut+uux+∂P∂x=0,∂P(t,x)∂x=Λ-2∂x[u2+12ux2]+λu,u(0,x)=u0(x),
where operator Λ2=1-(∂2/∂x2). For a fixed 1≤p0<∞, it has
(2.3)Λ-2g(x)=12∫R‍e-|x-y|g(y)dyforg(x)∈Lp0(R),1<p0<∞.In fact, problem (2.1) satisfies the following conservation law:
(2.4)∫R‍(u2+ux2)dx+2λ∫0t‍∫R‍(u2+ux2)dxdt=∫R‍(u02+u0x2)dx.Now, we introduce the definition of a weak solution to the Cauchy problem (2.1) or (2.2).Definition 2.1.
A continuous functionu:[0,∞)×R→R is said to be a global weak solution to the Cauchy problem (2.2) if(i)
u∈C([0,∞)×R)⋂‍L∞([0,∞);H1(R));(ii)
∥u(t,.)∥H1(R)≤∥u0∥H1(R);(iii)
u=u(t,x) satisfies (2.2) in the sense of distributions and takes on the initial value pointwise.
The main result of this paper is stated as follows.Theorem 2.2.
Assumeu0(x)∈H1(R). Then, the Cauchy problem (2.1) or (2.2) has a global weak solution u(t,x) in the sense of Definition 2.1. Furthermore, this weak solution satisfies the following properties.(a)
There exists a positive constantc0 depending on ∥u0∥H1(R) and λ such that the following one-sided L∞ norm estimate on the first order spatial derivative holds
(2.5)∂u(t,x)∂x≤4t+c0,for(t,x)∈[0,∞)×R.(b)
Let0<γ<1, T>0, and a,b∈R,a<b. Then, there exists a positive constant c1 depending only on ∥u0∥H1(R), γ,T,a,b and λ such that the space higher integrability estimate holds
(2.6)∫0t‍∫ab‍|∂u(t,x)∂x|2+γdx≤c1.


## 3. Viscous Approximations

Defining(3.1)ϕ(x)={e1/(x2-1),|x|<1,0,|x|≥1,
and setting the mollifier ϕε(x)=ε-(1/4)ϕ(ε-(1/4)x) with 0<ε<1/4 and uε,0=ϕε⋆u0, we know that uε,0∈C∞ for any u0∈Hs,s>0 (see [10]). In fact, suitably choosing the mollifier, we have
(3.2)∥uε,0∥H1(R)≤∥u0∥H1(R),uε,0⟶u0inH1(R).The existence of a weak solution to the Cauchy problem (2.2) will be established by proving compactness of a sequence of smooth functions {uε}ε>0 solving the following viscous problem:
(3.3)∂uε∂t+uε∂uε∂x+∂Pε∂x=ε∂2uε∂x2,∂Pε(t,x)∂x=Λ-2∂x[uε2+12(∂uε∂x)2]+λuε,uε(0,x)=uε,0(x).The beginning point of our analysis is the following well-posedness result for problem (3.3).Lemma 3.1.
Provided thatu0∈H1(R), then for any σ≥3, there exists a unique solution uε∈C([0,∞);Hσ(R)) to the Cauchy problem (3.3). Moreover, for any t>0, it holds that
(3.4)∫R‍(uε2+(∂uε∂x)2)dx+2λ∫0t‍∫R‍(uε2+(∂uε∂x)2)dxdt+2ε∫0t‍∫R‍((∂uε∂x)2+(∂2uε∂x2)2)(s,x)dxds=∥uε,0∥H1(R)2,
or
(3.5)∥uε(t,.)∥H1(R)2+2ε∫0t‍∥∂uε∂x(s,.)∥H1(R)2ds+2λ∫0t‍∥uε(t,.)∥H1(R)2dt=∥uε,0∥H1(R)2.Proof.
For anyσ≥3 and u0∈H1(R), we have uε,0∈C([0,∞);Hσ(R)). From Theorem 2.1 in [9] or Theorem 2.3 in [11], we conclude that problem (3.3) has a unique solution uε∈C([0,∞);Hσ(R)) for an arbitrary σ>3.
We know that the first equation in system (3.3) is equivalent to the form
(3.6)∂uε∂t-∂3uε∂tx2+32∂uε2∂x+λ(uε-∂2uε∂x2)=2∂uε∂x∂2uε∂x2+uε∂3uε∂x3+ε(∂2uε∂x2-∂4uε∂x4),
from which we derive that
(3.7)12ddt∫R‍(uε2+(∂uε∂x)2)dx+λ∫R‍(uε2+(∂uε∂x)2)dx+ε∫R‍((∂uε∂x)2+(∂2uε∂x2)2)dx=0,
which completes the proof.From Lemma3.1, we have
(3.8)∥uε∥L∞(R)≤∥uε∥H1(R)≤∥uε,0∥H1(R)≤∥u0∥H1(R).
Differentiating the first equation of problem (3.3) with respect to the variable x and writing ∂uε/∂x=qε, we obtain
(3.9)∂qε∂t+uε∂qε∂x-ε∂2qε∂x2+12qε2+λux=uε2-Λ-2[uε2+12qε2]=Qε(t,x).Lemma 3.2.
Let0<γ<1, T>0, and a,b∈R,a<b. Then, there exists a positive constant c1 depending only on ∥u0∥H1(R), γ,T,a, b and λ, but independent of ε, such that the space higher integrability estimate holds
(3.10)∫0T‍∫ab‍|∂uε(t,x)∂x|2+γdx≤c1,
where uε=uε(t,x) is the unique solution of problem (3.3).Proof.
The proof is a variant of the proof presented by  Xin and Zhang [7] (also see Coclite et al. [9]). Let χ∈C∞(R) be a cut-off function such that 0<χ<1 and
(3.11)ϕ(x)={1,ifx∈[a,b],0,ifx∈(-∞,a-1]⋃‍[b+1,∞).
Consider the map θ(ξ):=ξ(1+|ξ|)γ, ξ∈R,0<γ<1, and observe that
(3.12)θ′(ξ)=(1+(1+γ)|ξ|)(1+|ξ|)γ-1,θ′′(ξ)=γsign(ξ)(1+|ξ|)γ-2(2+(1+γ)|ξ|)=γ(1+γ)sign(ξ)(1+|ξ|)γ-1+(1-γ)γsign(ξ)(1+|ξ|)γ-2.
We have
(3.13)|θ(ξ)|≤|ξ|+|ξ|1+γ,|θ′(ξ)|≤1+(1+γ)|ξ|,|θ′′(ξ)|≤2γ,(3.14)ξθ(ξ)-12ξ2θ′(ξ)=1-γ2ξ2(1+|ξ|)γ+γ2ξ2(1+|ξ|)γ-1≥1-γ2ξ2(1+|ξ|)γ.
Differentiating the first equation of problem (3.3) with respect to the variable x and writing u=uε and ∂uε/∂x=qε=q for simplicity, we obtain
(3.15)utx+uuxx+12ux2+λux=u2-Λ-2[u2+12ux2]=Q(t,x).
Multiplying (3.15) by χθ'(q), using the chain rule and integrating over ΠT:=[0,T]×R, we have
(3.16)∫ΠT‍χ(x)qθ(q)dtdx-12∫ΠT‍q2χ(x)θ′(q)dtdx=∫R‍χ(x)(θ(q(t,x))-θ(q(0,x)))dx-∫ΠT‍uχ′(x)θ(q)dtdx+ε∫ΠT‍∂q∂xχ′(x)θ′(q)dtdx+ε∫ΠT‍(∂q∂x)2χ(x)θ′′(q)dtdx+λ∫ΠT‍qχ(x)θ′(q)dtdx-∫ΠT‍Q(t,x)χ(x)θ′(q)dtdx.
From (3.14), we get
(3.17)∫ΠT‍χ(x)qθ(q)dtdx-12∫ΠT‍q2χ(x)θ′(q)dtdx=∫ΠT‍χ(x)(qθ(q)-12q2θ′(q))dtdx≥(1-γ)2∫ΠT‍χ(x)q2(1+|q|)γdtdx.
Using the Hölder inequality, (3.8) and (3.13), yields
(3.18)|∫R‍χ(x)θ(q)dx|≤∫R‍χ(x)(|q|1+γ+|q|)dx≤∥χ∥L2/(1-γ)(R)∥q∥L2(R)1+γ+∥χ∥L2(R)∥q∥L2(R)≤(b-a+2)(1-γ)/2∥u0∥H1(R)1+γ+(b-a+2)1/2∥u0∥H1(R),(3.19)|∫ΠT‍uχ′(x)θ(q)dtdx|≤∫ΠT‍|u||χ′(x)|(|q|1+γ+|q|)dtdx≤∥u0∥H1(R)∫0T‍(∥χ'∥L2/(1-γ)(R)∥q∥L2(R)1+γ+∥χ'∥L2(R)∥q∥L2(R))dt≤T∥u0∥H1(R)(∥χ′∥L2/(1-γ)(R)∥u0∥H1(R)1+γ+∥χ′∥L2(R)∥u0∥H1(R)).
Integration by parts gives rise to
(3.20)∫ΠT‍∂q∂xχ′(x)θ′(q)dtdx=-∫ΠT‍θ(q)χ′′(x)dtdx.
From (3.13), (3.20), and the Hölder inequality, we have
(3.21)|∫ΠT‍∂q∂xχ′(x)θ′(q)dtdx|≤ε∫ΠT‍|θ(q)||χ′′(x)|dtdx≤ε∫ΠT‍|χ′′(x)|(|q|1+γ+|q|)dtdx≤εT(∥χ′′∥L2/(1-γ)(R)∥u0∥H1(R)1+γ+∥χ′′∥L2(R)∥u0∥H1(R)).
Using (3.13) and Lemma 3.1, we have
(3.22)ε|∫ΠT‍(∂q∂x)2χ(x)θ′′(q)dtdx|≤2γε∫ΠT‍(∂q∂x)2dtdx≤γ∥u0∥H1(R)2.
From (3.13), it has
(3.23)|λ∫ΠT‍qχ(x)θ′(q)dtdx|≤|λ|∫ΠT‍|χ(x)|((1+γ)q2+|q|)dtdx≤|λ|((1+γ)∥χ∥L∞∥u0∥H1(R)2+∥χ(x)∥L2(R)∥u0∥H1(R)).
Applying (3.8), the Hölder inequality, Lemma 3.1 and ∫R‍e-|x-y|dy=2, we have
(3.24)|Q(t,x)|=|u2-Λ-2[u2+12q2]|≤∥u∥L∞(R)2+12∥u∥H1(R)2≤32∥u0∥H1(R)2.
From (3.24), we obtain
(3.25)|∫ΠT‍Q(t,x)χ(x)θ′(q)dtdx|≤c∫ΠT‍|χ(x)|((1+γ)|q|+1)dtdx≤32∥u0∥H1(R)2T((1+γ)∥χ(x)∥L2(R)∥u0∥H1(R)+∫R‍|χ(x)|dx).
The inequalities (3.15)–(3.23) and (3.25) derive the desired result (3.10).Lemma 3.3.
There exists a positive constantC depending only on ∥u0∥H1(R) and λ such that
(3.26)∥Qε(t,.)∥L∞(R)≤C,(3.27)∥Qε(t,.)∥L1(R)≤C,(3.28)∥Qε(t,.)∥L2(R)≤C,(3.29)∥∂Pε(t,.)∂x∥L∞(R)≤C,(3.30)∥∂Pε(t,.)∂x∥L2(R)≤C,
where uε=uε(t,x) is the unique solution of system (3.3).Proof.
For simplicity, settingu(t,x)=uε(t,x), we have
(3.31)Qε(t,x)=u2-Λ-2[u2+12ux2],∂Pε(t,x)∂x=Λ-2∂x[u2+12(∂u∂x)2]+λu.
The inequality (3.26) is proved in Lemma 3.2 (see (3.24)). Now, we prove (3.27). Using
(3.32)∫R‍|Λ-2[u2+12ux2]|dx=12∫R‍|∫R‍e-|x-y|(u2+12ux2)dy|dx≤∥u∥H1(R)2≤∥u0∥H1(R)2,
and (3.8) result in (3.27).
Applying the Tonelli theorem, (3.26) and (3.27), we get
(3.33)∥Qε(t,.)∥L2(R)2≤∥Qε(t,.)∥L∞∥Qε(t,.)∥L1(R)≤C.
Since
(3.34)|Λ-2∂x[u2+12(∂u∂x)2]|=|12e-x∫-∞x‍ey∂y[u2+12(∂u∂y)2]dy+12ex∫x∞‍e-y∂y[u2+12(∂u∂y)2]dy|=|-12e-x∫-∞x‍ey[u2+12(∂u∂y)2]dy+12e-x∫-∞x‍ey[u2+12(∂u∂y)2]dy|≤∫R‍e-|x-y|[u2+12(∂u∂y)2]dy≤∥u∥H1(R)2≤∥u0∥H1(R)2,∫R‍|Λ-2∂x[u2+12(∂u∂y)2]|dx=∫R‍|-12e-x∫-∞x‍ey[u2+12(∂u∂y)2]dy+12e-x∫-∞x‍ey[u2+12(∂u∂y)2]dy|dx≤∫R‍∫R‍e-|x-y|[u2+12(∂u∂y)2]dydx≤2∥u∥H1(R)2≤2∥u0∥H1(R)2,
we have
(3.35)∫R‍|Λ-2∂x[u2+12(∂u∂x)2]|2dx≤∥u0∥H1(R)2∫R‍|Λ-2∂x[u2+12(∂u∂x)2]|dx≤2∥u0∥H1(R)4,
which, together with Lemma 3.1, we get (3.29) and (3.30). The proof of Lemma 3.3 is completed.Lemma 3.4.
Assumeuε=uε(t,x) is the unique solution of problem (3.3). There exists a positive constant C depending only on ∥u0∥H1(R) and λ such that the following one-sided L∞ norm estimate on the first order spatial derivative holds
(3.36)∂uε(t,x)∂x≤4t+C,for(t,x)∈[0,∞)×R.Proof.
From (3.9) and Lemma 3.3, we know that there exists a positive constant C depending only on ∥u0∥H1(R) and λ such that ∥Qε(t,x)∥L∞(R)≤C. Therefore,
(3.37)∂qε∂t+uε∂qε∂x+12qε2+λqε=Qε(t,x)≤C.
Letf=f(t) be the solution of
(3.38)dfdt+12f2+λf=C,t>0,f(0)=∥∂uε,0∂x∥L∞(R).
Letting supx∈Rqε(t,x)=f(t), due to the comparison principle for parabolic equations, we get
(3.39)qε(t,x)≤f(t).
Using-λf≤ρ2f2+(1/4ρ2)λ2, we derive that
(3.40)dfdt=C-12f2-λf≤C-12f2+ρ2f2+14ρ2λ2≤C-14f2+λ2,
where (1/4ρ2)λ2=λ2 and ρ=1/2. Setting M0=C+λ2, we obtain
(3.41)dfdt+14f2≤M0.
Letting F(t)=(4/t)+2M0, we have (dF(t)/dt)+(1/4)F2(t)-M0=(4M0/t)>0. Due to the comparison principle for ordinary differential equations, we get f(t)≤F(t) for all t>0. Therefore, by this and (3.39), the estimate (3.36) is proved.Lemma 3.5.
There exists a sequence{εj}j∈N tending to zero and a function u∈L∞([0,∞);H1(R))⋂‍H1([0,T]×R), such that
(3.42)uεj⇀uinH1([0,T]×R),foreachT≥0,(3.43)uεj→uinLloc∞([0,∞)×R),
where uε=uε(t,x) is the unique solution of (3.3).Proof.
For fixedT>0, using Lemmas 3.1 and 3.3, and
(3.44)∂uε∂t+uε∂uε∂x+∂Pε∂x=ε∂2uε∂x2,
we obtain
(3.45)∥∂uε∂t∥L2([0,T]×R)≤c(1+ε∥u0∥H1(R)),
where c depends on T. Hence, {uε} is uniformly bounded in L∞([0,∞);H1(R))⋂‍H1([0,T]×R) and (3.42) follows.
Observe that, for each0≤s,t≤T,
(3.46)∥uε(t,.)-uε(s,.)∥L2(R)2=∫R‍(∂uε∂t(τ,x)dτ)2dx≤|t-s|∫R‍∫0T‍(∂uε∂t(τ,x))2dτdx.
Moreover,{uε} is uniformly bounded in L∞([0,T];H1(R)) and H1(R)⊂Lloc∞⊂Lloc2(R). Then, (3.43) is valid.Lemma 3.6.
There exists a sequence{εj}j∈N tending to zero and a function Q∈L∞([0,∞);W1,∞(R)) such that for each 1<p<∞(3.47)Qεj→QstronglyinLlocp([0,∞)×R).Proof.
Using Lemma3.3, we have the existence of pointwise convergence subsequence Qεj which is uniformly bounded in L∞([0,∞)×R). This implies (3.47).Throughout this paper we use overbars to denote weak limits (the space in which these weak limits are taken isLlocr([0,∞)×R) with 1<r<(3/2)).Lemma 3.7.
There exists a sequence{εj}j∈N tending to zero and two functions q∈Llocp([0,∞)×R), q2¯∈Llocr([0,∞)×R) such that
(3.48)qεj⇀qinLlocp([0,∞)×R),qεj⇀⋆qinLloc∞([0,∞);L2(R)),(3.49)qεj2⇀q2¯inLlocr([0,∞)×R),
for each 1<p<3 and 1<r<(3/2). Moreover,
(3.50)q2(t,x)≤q2¯(t,x),foralmostevery(t,x)∈[0,∞)×R(3.51)∂u∂x=qinthesenseofdistributionson[0,∞)×R.Proof.
(3.48) and (3.49) are direct consequence of Lemmas 3.1 and 3.2. Inequality (3.50) is valid because of the weak convergence in (3.49). Finally, (3.51) is a consequence of the definition of qε, Lemma 3.5 and (3.48).In the following, for notational convenience, we replace the sequence{uεj}j∈N, {qεj}j∈N and {Qεj}j∈N by {uε}ε>0, {qε}ε>0 and {Qε}ε>0, separately.Using (3.48), we conclude that for any convex function η∈C1(R) with η' bounded, Lipschitz continuous on R and any 1<p<3 we get
(3.52)η(qε)⇀η(q)¯inLlocp([0,∞)×R),η(qε)⇀⋆η(q)¯inLloc∞([0,∞);L2(R)).Multiplying (3.9) by η'(qε) yields
(3.53)∂∂tη(qε)+∂∂x(uεη(qε))-ε∂2∂x2η(qε)+εη′′(qε)(∂qε∂x)2=qεη(qε)-12η′(qε)qε2-λqεη′(qε)+Qε(t,x)η′(qε).Lemma 3.8.
For any convexη∈C1(R) with η' bounded, Lipschitz continuous on R, it holds that
(3.54)∂η(q)∂t¯+∂∂x(uη(q)¯)≤qη(q)¯-12η′(q)q2¯-λqη′(q)¯+Q(t,x)η′(q)¯
in the sense of distributions on [0,∞)×R. Here, qη(q)¯ and η'(q)q2¯ denote the weak limits of qεη(qε) and qε2η'(qε) in Llocr([0,∞)×R), 1<r<(3/2), respectively.Proof.
In (3.53), by the convexity of η, (3.8), Lemmas 3.5, 3.6, and 3.7, sending ε→0 gives rise to the desire result.Remark 3.9.
From (3.48) and (3.49), we know that
(3.55)q=q++q-=q+¯+q-¯,q2=(q+)2+(q-)2,q2¯=(q+)2¯+(q-)2¯,
almost everywhere in [0,∞)×R, where ξ+:=ξχ[0,+∞)(ξ),ξ-:=ξχ(-∞,0](ξ) for ξ∈R. From Lemma 3.4 and (3.48), we have
(3.56)qε(t,x),q(t,x)≤4t+C,fort>0,x∈R,
where C is a constant depending only on ∥u0∥H1(R) and λ.Lemma 3.10.
In the sense of distributions on[0,∞)×R, it holds that
(3.57)∂q∂t+∂∂x(uq)=12q2¯-λq+Q(t,x).Proof.
Using (3.9), Lemmas 3.5 and 3.6, (3.48), (3.49), and (3.51), the conclusion (3.57) holds by sending ε→0 in (3.9).The next lemma contains a generalized formulation of (3.57).Lemma 3.11.
For anyη∈C1(R) with η∈L∞(R), it has
(3.58)∂η(q)∂t+∂∂x(uη(q))=qη(q)+(12q2¯-q2)η′(q)-λqη′(q)+Q(t,x)η′(q),
in the sense of distributions on [0,∞)×R.Proof.
Let{ωδ}δ be a family of mollifiers defined on R. Denote qδ(t,x):=(q(t,.)⋆ωδ)(x). The ⋆ is the convolution with respect to the x variable. Multiplying (3.57) by η'(qδ), it has
(3.59)∂η(qδ)∂t=η′(qδ)∂qδ∂t=η′(qδ)[12q2¯⋆ωδ-λqδ+Q(t,x)⋆ωδ-q2⋆ωδ-u∂q∂x⋆ωδ],∂∂x(uη(qδ))=qη(qδ)+uη′(qδ)(∂qδ∂x).
Using the boundedness of η,η' and letting δ→0 in the above two equations, we obtain (3.58).


## 4. Strong Convergence ofqε and Existence for (1.1)

Following the ideas in [7] or [9], in this section, we improve the weak convergence of qε in (3.48) to strong convergence, and then we have an existence result for problem (3.3). Generally speaking, we will derive a “transport equation” for the evolution of the defect measure (q2¯-q2)≥0. Namely, we will prove that the measure is zero initially then it will continue to be zero at all later times t>0.Lemma 4.1.
Assumeu0∈H1(R). It holds that
(4.1)limt→0∫R‍q2(t,x)dx=limt→0∫R‍q2¯(t,x)dx=∫R‍(∂u0∂x)2dx.Lemma 4.2.
Ifu0∈H1(R), for each M>0, it has
(4.2)limt→0∫R‍(ηM±(q)¯(t,x)-ηM±(q(t,x)))dx=0,
where
(4.3)ηM(ξ):={12ξ2,if|ξ|≤M,M|ξ|-12M2,if|ξ|>M,
and ηM+(ξ):=ηM(ξ)χ[0,+∞)(ξ), ηM-(ξ):=ηM(ξ)χ(-∞,0](ξ), ξ∈R.
The proof of Lemmas4.1 and 4.2 is similar to those of Lemmas 6.1 and 6.2 in [9]. Here, we omit their proofs.Lemma 4.3 (Coclite et al. [9]).
LetM>0. Then, for each ξ∈R, it holds that
(4.4)ηM(ξ)=12ξ2-12(M-|ξ|)2χ(-∞,-M)⋂‍(M,∞)(ξ),η′M(ξ)ξ=ξ+(M-|ξ|)sign(ξ)χ(-∞,-M)⋂‍(M,∞)(ξ),ηM+(ξ)=12(ξ+)2-12(M-ξ)2χ(M,∞)(ξ),(ηM+)'(ξ)=ξ++(M-ξ)χ(M,∞)(ξ),ηM-(ξ)=12(ξ-)2-12(M+ξ)2χ(-∞,-M)(ξ),(ηM-)′(ξ)=ξ--(M+ξ)χ(-∞,-M)(ξ).Lemma 4.4.
Assumeu0∈H1(R). Then, for almost all t>0(4.5)12∫R‍((q+)2¯-q+2)(t,x)dx≤∫0t‍∫R‍Q(s,x)[q+¯(s,x)-q+(s,x)]dsdx.Proof.
For an arbitraryT>0(0<t<T), we let M be sufficiently large (see Lemma 3.4). Using (3.54) minus (3.58), and the entropy ηM+ (see Lemma 4.2) results in
(4.6)∂∂t(ηM+(q)¯-ηM+(q))+∂∂x(u[ηM+(q)¯-ηM+(q)])≤(qηM+(q)¯-qηM+(q))-12(q2(ηM+)′(q)¯-q2(ηM+)′(q))-12(q2¯-q2)(ηM+)′(q)-λ(q(ηM+)′(q)¯-q(ηM+)′(q))+Q(t,x)((ηM+)′(q)¯-(ηM+)′(q)).
By the increasing property of ηM+ and the convexity of q(ηM+)'(q), from (3.50), we have
(4.7)-12(q2¯-q2)(ηM+)′(q)≤0,-λ(q(ηM+)′(q)¯-q(ηM+)′(q))≤0.
It follows from Lemma 4.3 that
(4.8)qηM+(q)-12q2(ηM+)′(q)=-M2q(M-q)χ(M,∞)(q),qηM+(q)¯-12q2(ηM+)′(q)¯=-M2q(M-q)χ(M,∞)(q)¯.
In view of Remark 3.9. Let ΩM=(4/α(M-C),∞)×R. Applying (3.56) gives rise to
(4.9)qηM+(q)-12q2(ηM+)′(q)=qηM+(q)¯-12q2(ηM+)′(q)¯=0,inΩM.
In ΩM, it has
(4.10)ηM+=12(q+)2,(ηM+)′(q)=q+,ηM+(q)¯=12(q+)2¯,(ηM+)′(q)¯=q+¯.
From (4.6)–(4.10), we know that the following inequality holds in ΩM(4.11)∂∂t(ηM+(q)¯-ηM+(q))+∂∂x(u[ηM+(q)¯-ηM+(q)])≤Q(t,x)((ηM+)′(q)¯-(ηM+)′(q)).
Integrating the resultant inequality over (4/(M-C),t)×R yields
(4.12)12∫R‍((q+)2¯-q+2(t,x))dx≤limt→0∫R‍[ηM+(q)¯(t,x)-ηM+(q)(t,x)]dx+∫4/(α(M-C))t‍∫R‍Q(s,x)[q+¯(s,x)-q+(s,x)]dsdx,
for almost all t>4/(M-C). Sending M→∞ and using Lemma 4.2, we complete the proof.Lemma 4.5.
For anyt>0 and M>0, it holds that
(4.13)∫R‍(ηM-(q)¯-ηM-(q))(t,x)dx≤M22∫0t‍∫R‍u(M+q)χ(-∞,-M)(q)¯dsdx-M22∫0t‍∫R‍u(M+q)χ(-∞,-M)(q)dsdx+M∫0t‍∫R‍u[ηM-(q)¯-ηM-(q)]dsdx+M2∫0t‍∫R‍u(q+2¯-q+2)dsdx+∫0t‍∫R‍Q(t,x)((ηM-)′(q)¯-(ηM-)′(q))dsdx.Proof.
LetM>0. Subtracting (3.58) from (3.54) and using entropy ηM-, we deduce
(4.14)∂∂t(ηM-(q)¯-ηM-(q))+∂∂x(u[ηM-(q)¯-ηM-(q)])≤(qηM-(q)¯-qηM-(q))-12(q2(ηM-)′(q)¯-q2(ηM-)′(q))-12(q2¯-q2)(ηM-)′(q)-λ(q(ηM-)′(q)¯-q(ηM-)′(q))+Q(t,x)((ηM-)′(q)¯-(ηM-)′(q)).
Since -M≤(ηM-)'≤0, we get
(4.15)-12(q2¯-q2)(ηM-)′(q)≤M2(q2¯-q2).
By the convexity of q(ηM-)'(q), it has
(4.16)-λ(q(ηM-)'(q)¯-q(ηM-)'(q))≤0.
Using Remark 3.9 and Lemma 4.3 yields
(4.17)qηM-(q)-12q2(ηM-)′(q)=-M2q(M+q)χ(-∞,-M)(q),(4.18)qηM-(q)¯-12q2(ηM-)′(q)¯=-M2q(M+q)χ(-∞,-M)(q)¯.
Inserting the inequalities from (4.15) to (4.18) into (4.14) gives
(4.19)∂∂t(ηM-(q)¯-ηM-(q))+∂∂x(u[ηM-(q)¯-ηM-(q)])≤-M2q(M+q)χ(-∞,-M)(q)¯+M2q(M+q)χ(-∞,-M)(q)+M2(q2¯-q2)+Q(t,x)((ηM-)′(q)¯-(ηM-)′(q)).
Integrating the above inequality over (0,t)×R, we obtain
(4.20)∫R‍(ηM-(q)¯-ηM-(q))(t,x)dx≤-M2∫0t‍∫R‍q(M+q)χ(-∞,-M)(q)¯dsdx+M2∫0t‍∫R‍q(M+q)χ(-∞,-M)(q)dsdx+M2∫0t‍∫R‍(q2¯-q2)dsdx+∫0t‍∫R‍Q(t,x)((ηM-)′(q)¯-(ηM-)′(q))dsdx.
It follows from Lemma 4.3 that
(4.21)ηM-(q)¯-ηM-(q)=12((q-)2¯-(q-)2)+12(M+q)2χ(-∞,-M)(q)-12(M+q)2χ(-∞,-M)(q)¯.
Using Remark 3.9 and (4.20), we have
(4.22)∫R‍(ηM-(q)¯-ηM-(q))(t,x)dx≤-M2∫0t‍∫R‍q(M+q)χ(-∞,-M)(q)¯dsdx+M2∫0t‍∫R‍q(M+q)χ(-∞,-M)(q)dsdx+M∫0t‍∫R‍[ηM-(q)¯-ηM-(q)]dsdx+M2∫0t‍∫R‍(M+q)2χ(-∞,-M)(q)¯dsdx-M2∫0t‍∫R‍(M+q)2uχ(-∞,-M)(q)dsdx+M2∫0t‍∫R‍(q+2¯-q+2)dsdx+∫0t‍∫R‍Q(t,x)((ηM-)′(q)¯-(ηM-)′(q))dsdx.
Applying the identity M(M+q)2-Mq(M+q)=M2(M+q), we obtain (4.13).Lemma 4.6.
It holds that(4.23)q2¯=q2almosteverywherein[0,∞)×(-∞,∞).Proof.
Applying Lemmas4.4 and 4.5 gives rise to
(4.24)∫R‍(12[(q+)2¯-(q+)2]+[ηM-¯-ηM-])(t,x)dx≤M22(∫0t‍∫R‍(M+q)χ(-∞,-M)(q)¯dsdx-M22∫0t‍∫R‍(M+q)χ(-∞,-M)(q)dsdx)+M∫0t‍∫R‍[ηM-¯-ηM-]dsdx+M2∫0t‍∫R‍[(q+)2¯-(q+)2]dsdx+∫0t‍∫R‍Q(s,x)([q+¯-q+]+[(ηM-)′(q)¯-(ηM-)′(q)])dsdx.
From Lemma 3.6, we know that there exists a constant L>0, depending only on ∥u0∥H1(R), such that
(4.25)∥Q(t,x)∥L∞([0,∞)×R)≤L.
By Remark 3.9 and Lemma 4.3, it has
(4.26)q++(ηM-)′(q)=q-(M+q)χ(-∞,-M),q+¯+(ηM-)′(q)¯=q-(M+q)χ(-∞,-M)(q)¯.
Thus, by the convexity of the map ξ→ξ++(ηM-)'(ξ), we get
(4.27)0≤[q+¯-q+]+[(ηM-)′(q)¯-(ηM-)′(q)]=(M+q)χ(-∞,-M)-(M+q)χ(-∞,-M)(q)¯.
Using (4.25) derives
(4.28)Q(s,x)([q+¯-q+]+[(ηM-)′(q)¯-(ηM-)′(q)])≤-L((M+q)χ(-∞,-M)(q)¯-(M+q)χ(-∞,-M)).
Since ξ→(M+ξ)χ(-∞,-M) is concave and choosing M large enough, we have
(4.29)M22((M+q)χ(-∞,-M)(q)¯-(M+q)χ(-∞,-M))+Q(s,x)([q+¯-q+]+[(ηM-)′(q)¯-(ηM-)′(q)])≤(M22-L)((M+q)χ(-∞,-M)(q)¯-(M+q)χ(-∞,-M))≤0.
Then, from (4.24) and (4.29), it has
(4.30)0≤∫R‍(12[(q+)2¯-(q+)2]+[ηM-¯-ηM-])(t,x)dx≤cM∫0t‍∫R‍(12[(q+)2¯-(q+)2]+[ηM-¯-ηM-])(t,x)dsdx.
By making of the Gronwall inequality and Lemmas 4.1 and 4.2, for each t>0, we conclude that
(4.31)0≤∫R‍(12[(q+)2¯-(q+)2]+[ηM-¯-ηM-])(t,x)dx=0.
By the Fatou lemma, Remark 3.9, and (3.50), sending M→∞, we obtain
(4.32)0≤∫R‍(q2¯-q2)(t,x)dx=0,fort>0,
which completes the proof.Proof of the main result.
Using (3.2), (3.4), and Lemma 3.5, we know that the Conditions (i) and (ii) in Definition 2.1 are satisfied. We have to verify (iii). Due to Lemma 4.6, we have
(4.33)qε→qinLloc2([0,∞)×R).
From Lemma 3.5, (3.47), and (4.33), we know that u is a distributional solution to problem (2.2). In addition, inequalities (2.5) and (2.6) are deduced from Lemmas 3.2 and 3.4. The proof of the main result is completed.



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*Source: 693010-2012-10-22.xml*