Global weak solutions for a modified two-component Camassa–Holm equation

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Global weak solutions for a modified two-component Camassa–Holm equation

Chunxia Guan, Zhaoyang Yin

Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China Received 23 September 2010; accepted 26 April 2011

Available online 7 May 2011

Abstract

We obtain the existence of global-in-time weak solutions for the Cauchy problem of a modified two-component Camassa–Holm equation. The global weak solution is obtained as a limit of viscous approximation. The key elements in our analysis are the Helly theorem and some a priori one-sided supernorm and space–time higher integrability estimates on the first-order derivatives of approximation solutions.

©2011 Elsevier Masson SAS. All rights reserved.

Résumé

Nous obtenons l’existence globale en temps de solutions faibles pour le problème de Cauchy d’une équation modifiée Camassa–

Holm à deux composantes. La solution faible globale est obtenue comme une limite de par approximation visqueuse. Les éléments clé dans notre analyse sont le théorème de Helly et certaines estimations a priori de supernorme d’un seul côté et d’intégrabilité dans l’espace-temps des dérivées premières des solutions approchées.

©2011 Elsevier Masson SAS. All rights reserved.

MSC:35G25; 35L05

Keywords:A modified two-component Camassa–Holm equation; Well-posedness; Blow-up scenario; Strong solution; Global weak solution

1. Introduction

In this paper we consider the Cauchy problem of the following modified two-component Camassa–Holm equation:

⎧⎪

⎪⎨

⎪⎪

⎩

m_t+um_x+2mux= −ρρ_x, t >0, x∈R, ρ_t+(ρu)_x=0, t >0, x∈R, m(0, x)=m₀(x), x∈R, ρ(0, x)=ρ₀(x), x∈R,

(1.1)

wherem=u−uxxandρ=(1−∂_x²)(ρ−ρ0).

* Corresponding author.

E-mail addresses:guanchunxia123@163.com (C. Guan), mcsyzy@mail.sysu.edu.cn (Z. Yin).

0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2011.04.003

The Camassa–Holm equation has been recently extended to a two-component integrable system (CH2) by combing its integrability property with compressibility, or free-surface elevation dynamics in its shallow-water interpreta- tion [7,25]. Eq. (1.1) was recently introduced by Holm et al. in [40]. The modified two-component Camassa–Holm equation (MCH2) is written in terms of velocityuand locally averaged densityρ (or depth, in the shallow-water in- terpretation) andρ0is taken to be constant. MCH2 is defined as geodesic motion on the semidirect product Lie group with respect to a certain metric and is given as a set of Euler–Poincaré equations on the dual of the corresponding Lie algebra [39].

Forρ≡0, Eq. (1.1) becomes the Camassa–Holm equation, modeling the unidirectional propagation of shallow water waves over a flat bottom. Hereu(t, x)stands for the fluid velocity at timetin the spatialxdirection [5,14]. The Camassa–Holm equation is also a model for the propagation axially symmetric waves in hyperelastic rods [27]. It has a bi-Hamiltonian structure [9,32] and is completely integrable [5,12]. Its traveling waves (periodic as well as solitary) are peaked [6], capturing thus the shape of solitary wave solutions to the governing equations for water waves [13,19].

The orbital stability of the peaked solutions is proved in [24]. The explicit interaction of the peaked solutions is given in [2].

The Cauchy problem for the Camassa–Holm equation has been studied extensively [16,28,47,52]. It has been shown that this equation is locally well posed [15,28,47,52] for initial data u0∈H^s(R), s > ³₂. More interesting, it has global strong solutions [11,15] and also finite time blow-up solutions [11,15–17,28,47]. On the other hand, it has global weak solutions in H¹(R)[3,4,8,23,36–38,51]. Recently, it was claimed in [46] that the Camassa–Holm equation might be relevant to the modeling of tsunamis (see also the discussion in [20]).

The advantage of the Camassa–Holm equation in comparison with the KdV equation lies in the fact that while both KdV and Camassa–Holm equations are completely integrable Hamiltonian systems [10,22,30]; while the inverse scattering approach was obtained in [18] (see also [33]), in addition to KdV the Camassa–Holm equation has peaked solitons and models wave breaking [6,16] (by wave breaking we understand that the wave remains bounded while its slop becomes unbounded in finite time [50]).

Recently, two types of 2-component Camassa–Holm equations have been studied in [7,25,31,34]. These works have established the local well-posedness [25,31], derived precise blow-up scenarios [31], and proved the existence of strong solutions which blow up in finite time [25,31,34]. More recently, the Cauchy problem of Eq. (1.1) has been studied in [35].

However, the existence of global weak solutions to Eq. (1.1) has not been discussed yet. Our aim of this paper is to prove the existence of global week solutions to Eq. (1.1) provided the initial data satisfying some certain conditions.

We hope that our result sheds some light on important physical phenomena of Eq. (1.1) such as wave breaking. Up to now, we have no uniqueness result for the obtained global weak solutions to Eq. (1.1). This problem will be discussed later on. Note that no global existence results for strong solutions to Eq. (1.1) are available so far. Thus, we have to use the viscous approximation method to prove the existence of global weak solution to Eq. (1.1).

We now provide the framework in which we shall reformulate problem (1.1). Withm=u−u_xx,ρ=γ−γ_xxand γ=ρ−ρ₀, we can rewrite Eq. (1.1) as follows:

⎧⎪

⎪⎨

⎪⎪

⎩

m_t+m_xu+2mux= −ργ_x, t >0, x∈R, ρ_t+(uρ)_x=0, t >0, x∈R, m(0, x)=u0(x)−u0,xx(x), x∈R, ρ(0, x)=γ0−γ0,xx, x∈R.

(1.2)

Note that ifp(x):=¹₂e^−|x|,x∈R, then(1−∂_x²)⁻¹f=p∗f for allf ∈L²(R),p∗m=uandp∗ρ=γ. Here we denote by∗the convolution. Using this identity, we can rewrite Eq. (1.2) as follows:

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

u_t+uu_x= −∂_xp∗

u²+1 2u²_x+1

2γ²−1 2γ_x²

, t >0, x∈R, γt+uγx= −p∗

(uxγx)x+uxγ , t >0, x∈R,

u(0, x)=u₀(x), x∈R,

γ (0, x)=γ₀(x), x∈R.

(1.3)

The main result of this paper is to give the existence of a globe-in-time weak solutionz=u

γ to the Cauchy problem (1.1) with the initial dataz0=u0

γ₀ ∈H¹(R)×H¹(R). Before giving the precise statement of the main result, we first introduce the definition of a weak solution to the Cauchy problem (1.1).

Definition 1.1.z=u

γ is said to be an admissible weak solution to the Cauchy problem (1.1) if z(t, x)∈L^∞

(0,∞);H¹(R)×H¹(R)

satisfies Eq. (1.3) andz(t,·)→z₀ast→0⁺in the sense of distributions, and for anyt >0, z(t,·)

H¹(R)×H¹(R)z₀H¹(R)×H¹(R).

Notation.In our paper, the space of all Radon measures onRis denoted byM(R)andM⁺(R)denotes the set of all positive Radon measures.

The main result of this paper can be stated as follows:

Theorem 1.1.Letz₀=u₀

γ0 ∈H¹(R)×H¹(R). Ifρ₀=γ₀−γ_0,xx∈M⁺(R), then Eq.(1.1)has an admissible weak solution in the sense of Definition1.1.

Our paper is organized as follows. In Section 2, we give the approximate solutionsz=u

γ to Eq. (1.1) and derive the basic energy estimate onz. In Section 3, the crucial uniform a priori one-side supernorm estimate and local space–time higher integrability estimate foru are derived. Moreover, using Helly theorem we get that∂_xγis convergent almost everywhere onR₊×R. Finally, the strong convergence of∂xuinL²_loc(R₊×R)is carried out and we conclude the proof of the main result in Section 4.

2. Viscous approximate solutions

In the section, we construct the approximate solution sequencez=z(t, x)=_u(t,x)

γ(t,x) as solutions to the viscous problem of Eq. (1.1), i.e.,

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

u_t+uu_x+∂_xp∗

u²+1 2u²_x+1

2γ²−1 2γ_x²

=u_xx, t >0, x∈R, γ_t+uγ_x+∂_xp∗(u_xγ_x)+p∗(u_xγ )=0, t >0, x∈R,

u(0, x)=u_,0(x), x∈R,

γ (0, x)=γ,0(x), x∈R,

(2.1)

or the equivalent form:

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

u_t+uu_x= −∂_x

1−∂_x^{2 −1}

u²+1 2u²_x+1

2γ²−1 2γ_x²

+u_xx, t >0, x∈R, γt+uγx= −

1−∂_x^{2 −1}

(uxγx)x+uxγ , t >0, x∈R,

u(0, x)=u,0(x), x∈R,

γ (0, x)=γ,0(x), x∈R,

(2.2)

hereu_,0(x)=(φ∗u₀)(x),γ_,0(x)=(φ∗γ₀)(x)∈H^s(R),s >2, and φ(x):=

R

φ(ξ ) dξ ₋1

1 φ

x

, x∈R, >0,

whereφ∈C_c^∞(R)is defined by φ(x)=

e^1/(x2−1), |x|<1,

0, |x|1.

Then, we have

u_,0H¹(R)u₀H¹(R), γ_,0H¹(R)γ₀H¹(R), and

u_,0→u₀, γ_,0→γ₀, inH¹(R).

Theorem 2.1.Assumez0=u0

γ₀ ∈H¹(R)×H¹(R)andρ0=γ0−γ0,xx∈M⁺(R). Then there exists a unique solu- tionz=u

γ ∈C(R₊;H^s(R)×H^s(R))∩C¹(R₊;L²(R)×L²(R)),s >2, to the Cauchy problem(2.1). Moreover, for eacht0,

R

u²+γ²+ ∂u

∂x 2

+ ∂γ

∂x 2

(t, x) dx+2 t 0

R

∂u

∂x 2

+ ∂²u

∂x² 2

(s, x) dx ds

= z_,0²_H1(R)×H¹(R)z₀²_H1(R)×H¹(R), (2.3) or

z(t,·)²

H¹(R)×H¹(R)+2 t

0

∂u

∂x (s,·) ²

H¹(R)

ds= z_,0²_H1(R)×H¹(R)z₀²_H1(R)×H¹(R).

Before proving Theorem 2.1, we recall some useful lemmas.

Lemma 2.1.(See [44].) Letf∈H^s(R),s >³₂. Then Λ^−r

Λ^r+t+1, M_f Λ^−t

L²(R)cfH^s(R), |r|,|t|s−1,

whereMf is the operator of multiplication byf andcis a positive constant depending only onr, t.

Lemma 2.2.(See [43].) Letr, tbe real numbers such that−r < tr. Then f gH^t(R)cfH^rgH^t(R), ifr >1

2, f g

H^t+r−12(R)cfH^r(R)gH^t(R), ifr <1 2, wherecis a positive constant depending onr, t.

Lemma 2.3.(See [26].) IfX1andX2are Banach spaces andAi∈G(Xi,1, β),i=1,2, then the operator A=

A₁ 0 0 A2

∈G(X1×X2,1, β) withD(A)=D(A1)×D(A2).

The strategy of the proof of Theorem 2.1 is rather routine. We use some lemmas to prove this theorem. For the convenience of presentation, we will omit the subscripts inz, u, γin the following proofs.

Lemma 2.4.Givenz₀=u0

γ₀ ∈H^s(R)×H^s(R),s >2, there exist a maximalT =T (z₀H^s(R)×H^s(R)) >0, and a unique solutionz=u

γ to Eq.(2.1)such that z=z(·, z0)∈C

[0, T );H^s(R)×H^s(R) . Proof. LetA(z)=_{u∂x−∂xx 0}

0 u∂x =_{A1(z)+A0 0}

0 A₁(z) and f (z)=

−∂_x(1−∂_x²)⁻¹(u²+¹₂u²_x+¹₂γ²−¹₂γ_x²)

−(1−∂_x²)⁻¹((u_xγ_x)_x+u_xγ )

.

SetY=H^s(R)×H^s(R),X=L²(R)×L²(R),Λ=(1−∂_x²)¹² andQ=_Λs 0

0 Λ^s . Obviously,Qis an isomorphism of H^s(R)×H^s(R)ontoL²(R)×L²(R)andΛ^sis an isomorphism ofH^s(R)ontoL²(R). In order to prove Lemma 2.4, we use Kato’s semigroup approach [43].

We know thatA1(z)∈G(L²(R),1, β0(z))in [52]. It is obvious thatA0is a densely defined closed linear operator inXand is self-adjoint. For everyu∈D(A0)=H²(R),

(A₀u, u)_L2(R)= −(∂_xxu, u)_L2(R)=∂_xu²_L2(R).

By Corollary 4.4 in Chapter 1 of [49] we have that−A0is the infinitesimal generator of aC0semigroup of contractions onL²(R). ByA1(z)∈G(L²(R),1, β0(z)), therefore−A1(z)−βIis dissipative for allββ0(z). Moreover, for every y∈D(A1),

−

A₁(z)+βI y

L²(R)uL^∞∂_xyL²(R)+βyL²(R). Using integration by parts, for everyy∈H²(R), we have

∂_xy²_L2(R)y_L²_(R)∂_xxy_L²_(R) and

−

A1(z)+βI y

L²(R)

2∂xxyL²(R)+CyL²(R)=1

2 −A0yL²(R)+CyL²(R).

By Corollary 3.3.3 in [49], −A₀−A₁(z)−βI is the infinitesimal generator of aC₀ semigroup of contractions of L²(R)for everyββ₀(z). Therefore,A₁(z)+A₀∈G(L²(R),1, β(z)). Then, in view of Lemma 2.3 andA₁(z)∈ G(L²(R),1, β(z)), we haveA(z)=_{A1(z)+A0 0}

0 A₁(z) ∈G(L²(R)×L²(R),1, β(z)).

Now we prove thatA(y)∈L(H^s(R)×H^s(R), L²(R)×L²(R))fory∈H^s(R)×H^s(R)and we have that A(y)−A(z) w

L²(R)×L²(R)μ₁y−zL²(R)×L²(R)wH^s(R)×H^s(R), fory, z, w∈H^s(R)×H^s(R).

Indeed, by the definition ofA(z), we deduce that for allz, w∈H^s(R)×H^s(R), A(z)w

L²(R)×L²(R)

zH^s(R)×H^s(R)+ wH^s(R)×H^s(R). For ally, z, w∈H^s(R)×H^s(R),

A(y)−A(z) w

L²(R)×L²(R)

∂_xw₁L^∞(R)+ ∂_xw₂L^∞(R) y−zL²(R)×L²(R)

y−zL²(R)×L²(R)wH^s(R)×H^s(R). Note that

B(z)y=QA(z)Q⁻¹y−A(z)y=

[Λ^s, u∂_x]Λ^−sy₁ 0 0 [Λ^s, u∂_x]Λ^−sy₂

. Letw∈L²(R)×L²(R)andy, z∈H^s(R)×H^s(R). Then we have

B(y)w−B(z)w

L²(R)×L²(R)Λ^s, (y₁−u)∂_x

Λ^−sw₁

L²(R)+Λ^s, (y₁−u)∂_x

Λ^−sw₂

L²(R)

Λ^s, y₁−u Λ^1−s

L(L²(R))Λ⁻¹∂_xw₁

L²(R)+Λ⁻¹∂_xw₂

L²(R)

cy1−uH^s(R)wL²(R)×L²(R)cy−zH^s(R)×H^s(R)wL²(R)×L²(R), where we applied Lemma 2.1 withr=0,t=s−1 in view ofs >2. Takingy=0 in the above inequality, we obtain thatB(z)∈L(L²(R)×L²(R)).

Finally, we have that (see [35]) f (y)−f (z)

H^s(R)×H^s(R)μ3y−zH^s(R)×H^s(R), ∀y, z∈H^s(R)×H^s(R).

Thus, we only need to verify that

f (y)−f (z)

L²(R)×L²(R)

∂_x

1−∂_x^{2 −1}

y₁²−u² +1 2

y_1,x² −u²_x +1 2

y₂²−γ² −1 2

y_2,x² −γ_x²

L²(R)

+

1−∂_x^{2 −1}

(y1,xy2,x−uxγx)x+y1,xy2−uxγ

L²(R)

y₁²−u² +1 2

y_1,x² −u²_x +1 2

y₂²−γ² −1 2

y²_2,x−γ_x²

H⁻¹(R)

+(y_1,xy_2,x−u_xγ_x)_x+y_1,xy₂−u_xγ

H⁻²(R)

y1−uL²(R)y1+uL^∞(R)+c

2y1,x−uxH⁻¹(R)y1,x+uxH^s−1(R)

+1

2y₂−γL²(R)y₂+γL^∞(R)+c

2y_2,x−γ_xH−1(R)y_2,x+γ_xH^s−1(R)

+ y_1,xy_2,x−y_1,xγ_xH⁻¹(R)+ y_1,xγ_x−u_xγ_xH⁻¹(R)

+ y_1,xy₂−y_1,xγ_H−1(R)+ y_1,xγ−u_xγ_H−1(R)

y1−uL²(R)y1+uL∞(R)+c

2y1,x−uxH⁻¹(R)y1,x+uxH^s−1(R)

+1

2y₂−γ_L²_(R)y₂+γL^∞(R)+c

2y_2,x−γ_xH−1(R)y_2,x+γ_xH^s−1(R)

+cy_1,xH^s−1(R)y_2,x−γ_xH⁻¹(R)+cy_1,x−u_xH⁻¹(R)γ_xH^s−1(R)

+ y_1,xH^s−1(R)y₂−γ_H−1(R)+cy_1,x−u_xH−1(R)γ_H^s−1(R)

=(3+4c)

yH^s(R)×H^s(R)+ zH^s(R)×H^s(R) y−z_L²_(R)×L²_(R), where we applied Lemma 2.2 withr=s−1,t= −1. This proves Lemma 2.4. 2

Lemma 2.5. Letz0∈H^s(R)×H^s(R), s >2 and let T >0 be the maximal existence time of the corresponding solutionz=u

γ to Eq.(2.1). Then(2.3)holds for everyt∈(0, T ).

Proof. Applying Lemma 2.4 and a simple density argument, we only need to show that the above lemma holds for somes >2. Here we assumes=3 to prove the lemma. Differentiating Eq. (2.1) with respect tox and using the identity∂_x²p∗f=p∗f −f, we have

u_{t x}+uu_xx+u²_x=u_xxx+f −p∗f, γ_{t x}+u_xγ_x+uγ_xx= −∂_xp∗

(u_xγ_x)_x+u_xγ , (2.4)

wheref =u²+¹₂u²_x+¹₂γ²−¹₂γ_x². Denoteg=(uxγx)x+uxγandE(t )=

(u²+u²_x+γ²+γ_x²) dx. Using Eq. (2.1) and Eq. (2.4), and integrating by parts, we obtain

d

dtE(t )=2

R

(uu_t+u_xu_xt+γ γ_t+γ_xγ_xt) dx

=2

R

u(−uu_x−∂_xp∗f )+u_x

−uu_xx−u²_x+f−p∗f

+γ (−uγ_x−p∗g)+γ_x(−u_xγ_x−uγ_xx−∂_xp∗g)

dx+2

R

(uu_xx+u_xu_xxx) dx

=2

R

−u∂_xp∗f−1

2u³_x+u_xf−u_xp∗f +1 2u_xγ²

−γp∗g−1

2u_xγ_x²−γ_x∂_xp∗g

dx−2

R

u²_x+u²_xx (s, x) dx

= −2

R

u²_x+u²_xx (s, x) dx.

Integrating the above equality over(0, t ), we get (2.3). 2

Ifz∈H^s(R)×H^s(R),s >2, then by the above lemma we have the following useful inequality u(t,·)²

L^∞(R)+γ (t,·)²

L^∞(R)1

2u²_H1(R)+1

2γ²_H1(R)

1

2

u₀²_H1(R)+ γ₀²_H1(R) =1

2z₀²_H1(R)×H¹(R) (2.5)

for allt∈ [0, T ).

Now we present the blow-up scenario for the strong solutions to Eq. (2.1).

Lemma 2.6.Letz₀=u0

γ₀ ∈H^s(R)×H^s(R),s >2be given and assume thatT is the maximal existence time of the corresponding solutionz=u

γ of Eq.(2.1)with the initial dataz0. Then theH^s(R)×H^s(R)-norm ofz(t,·)blows up if and only if

lim sup

t→T

u_x(t,·)

L^∞(R)+γx(t,·)

L∞(R)

= ∞.

Proof. Letz=u

γ be the solution to Eq. (2.1) with the initial dataz0∈H^s(R)×H^s(R),s >2, and let T be the maximal existence time of the corresponding solutionz, which is guaranteed by Lemma 2.4. Throughout this proof, c >0 stands for a generic constant depending only ons.

Applying the operatorΛ^sto the first equation in Eq. (2.2), multiplying byΛ^su, and integrating overR, we obtain d

dtu²_H^s_(R)= −2(uux, u)s−2

u, f (u, γ ) _s+2(uxx, u)s

where

f (u, γ )=∂_x

1−∂_x^{2 −1}

u²+1 2u²_x+1

2γ²−1 2γ_x²

. From the proof of Theorem 3.1 in [35], we know that

(uu_x, u)_scu_xL^∞(R)u²H^s(R)

and f (u, γ ), u _sc

uxL^∞(R)+ γxL^∞(R)+ z0H¹(R)×H¹(R) γ²H^s(R)+ u²H^s(R) . Note that 2(uxx, u)s= −2uxH^s(R)0. Combining the above three inequalities, we get

d

dtu²_Hs(R)c

u_xL^∞(R)+ γ_x|L^∞(R)+ z_0H¹_(R)×H¹_(R) γ²_Hs(R)+ u²_Hs(R) . (2.6) From the proof of Theorem 3.1 in [35] again, we have

d

dtγ²H^s(R)c

uxL^∞(R)+ γxL^∞(R)+ z0H¹(R)×H¹(R) γ²H^s(R)+ u²H^s(R) . (2.7) By (2.6) and (2.7), we obtain

d dt

u²_H^s_(R)+ γ²_H^s_(R) c

u_xL^∞(R)+ γ_xL^∞(R)+ z₀H¹(R)×H¹(R) γ²_H^s_(R)+ u²_H^s_(R) .

If there existsM >0 such that lim sup_t→Tu_x(t,·)L^∞+ γ_x(t,·)L∞ M, then an application of Gronwall’s in- equality and the assumption of the theorem yield

u²_H^s_(R)+ γ²_H^s_(R)exp(cM1t )

u0²_H^s_(R)+ γ0²_H^s_(R) , whereM₁=M+ z₀H¹(R)×H¹(R).

On the other hand, by Sobolev’s imbedding theorem, we see that if lim sup

t→T

u_x(t,·)

L^∞(R)+γ_x(t,·)

L∞(R)

= ∞,

then the solution will blow up in finite time. This completes the proof of the lemma. 2 Consider now the following initial value problem

q_t=u(t, q), t∈ [0, T ),

q(0, x)=x, x∈R, (2.8)

whereudenotes the first component of the solutionzto Eq. (1.1). The introduction ofqis suggested by working in the Lagrangian framework, cf. [1] and [21] (see also [45]). Applying classical results in the theory of ordinary differential equations, one can obtain the following result onq which is crucial in the proof of blow-up scenarios.

Lemma 2.7.(See [11].) Letu∈C([0, T );H^s(R))∩C¹([0, T );H^s−1(R)),s2. Then Eq.(2.8)has a unique solution q∈C¹([0, T )×R;R). Moreover, the mapq(t,·)is an increasing diffeomorphism ofRwith

qx(t, x)=exp t

0

ux

s, q(s, x) ds

>0, ∀(t, x)∈ [0, T )×R.

Lemma 2.8.Letz₀∈H^s(R)×H^s(R),s >2andT >0be the maximal existence time of the corresponding solution z=u

γ to Eq.(2.1)guaranteed by Lemma2.4. Ifρ₀=γ₀−γ_0,xx does not change sign onR, then we have for all t∈ [0, T ),

γ_x(t,·)

L^∞(R)γ (t,·)

L^∞(R)

√2

2 z₀H¹(R)×H¹(R). (2.9)

Moreover, ifρ₀∈L¹(R)we have γ (t,·)

L¹(R)=ρ(t,·)

L¹(R)= ρ₀L¹(R), ∀t∈ [0, T ). (2.10)

Proof. Note thatρ=γ−γ_xx. Then, form the second equation of (2.4) we have

ρ_t+(uρ)_x=0. (2.11)

By (2.8) and (2.11), we get d

dt ρ

t, q(t, x) q_x(t, x) =(ρ_t+ρ_xq_t)q_x+ρq_xt=(ρ_t+ρ_xq_t+ρu_x)q_x=0.

Thus, we obtain ρ

t, q(t, x) qx(t, x)=ρ0(x). (2.12)

In view of Lemma 2.7, if ρ0does not change sign onR, then for anyt∈ [0, T ),ρ(t, x)andρ0have the same sign by (2.12). Sinceγ=p∗ρwherep=¹₂e^−|x|, we deduce

|γ_x| =1 2 e^x

∞ x

e^−yρ(y) dy−e^−x x

−∞

e^yρ(y) dy

1

2 e^x

∞ x

e^−yρ(y) dy+e^−x x

−∞

e^yρ(y) dy = |γ|. Then, by (2.5) we obtain (2.9).

In view of (2.12) and Lemma 2.7, ifρ₀does not change sign onRandρ₀∈L¹(R), by the relationγ =p∗ρwe have thatγ (t,·)does not change sign onR. Then, by Eq. (2.1) and (2.11) we obtain (2.10). This completes the proof of the lemma. 2