# Several Dynamical Properties for a Nonlinear Shallow Water Equation

Authors: Ls Yong; Haibo Yan
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

A nonlinear third order dispersive shallow water equation including the Degasperis-Procesi model is investigated. The existence of weak solutions for the equation is proved in the spaceL1(R)∩BV(R) under certain assumptions. The Oleinik type estimate and L2N(R)  (N is a natural number) estimate for the solution are obtained.


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## Body

## 1. Introduction

Constantin and Lannes [1] derived the shallow water wave equation
(1)ut+ux+32ρuux+μ(αuxxx+βuxxt)=ρμ(γuxuxx+δuuxxx),
where the constants α, β, γ, δ, ρ, and μ satisfy certain conditions. As stated in [1], using suitable mathematical transformations, one can turn (1) into the form
(2)ut-uxxt+2kux+muux=auxuxx+buuxxx,
where a, b, k, and m are constants. Clearly, (2) contains both the Camassa-Holm and Degasperis-Procesi models.The aim of this paper is to investigate the existence of weak solutions for the special case of (2). Namely, we study the shallow water equation
(3)∂tu-∂txx3u+mu∂xu=3∂xu∂xx2u+u∂xxx3u,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii(t,x)∈R+×R,
where m>0 is a constant. Letting y=u-∂xx2u, v=(m-∂xx2)-1u and using (3), we derive the conservation law
(4)∫R‍yvdx=∫R‍1+ξ2m+ξ2|u^(ξ)|2dξ=∫R‍1+ξ2m+ξ2|u^0(ξ)|2dξ~∥u0∥L2(R),
where u0=u(0,x). In fact, the conservation law (4) takes an important role in our further investigations of (3).Form=4, (3) reduces to the Degasperis-Procesi equation [2]
(5)∂tu-∂txx3u+4u∂xu=3∂xu∂xx2u+u∂xxx3u,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii(t,x)∈R+×R,
which has been studied by many scholars (see [3–5]). Lundmark and Szmigielski [6] developed an inverse scattering approach for computing n-peakon solutions to (5). The traveling wave solutions of (5) were investigated in Vakhnenko and Parkes [5]. Holm and Staley [7] studied stability of solitons and peakons numerically. Lin and Liu [8] proved the stability of peakons for the Degasperis-Procesi equation (5) under certain assumptions. The precise blowup scenario result, a blowup result, and the global existence of strong solutions and global weak solutions to (5) can be found in [9]. Matsuno [10] studied multisoliton solutions and their peakon limits. Analogous to the case of the Camassa-Holm equation, Henry [11] and Mustafa [12] showed that smooth solutions to (5) have infinite speed of propagation. For other methods to handle the problems relating to various dynamic properties of the Degasperis-Procesi equation and other shallow water equations, the reader is referred to [13–15] and the references therein.Coclite and Karlsen [16] established the existence, uniqueness, and L1(R) stability of entropy weak solutions belonging to the class L1(R)∩BV(R) for (5). They obtained existence of at least one weak solution satisfying a restricted set of entropy inequalities in the space L2(R)∩L4(R) and extended these results to a class of generalized Degasperis-Procesi equations in [16].Motivated by the desire to extend parts of the results presented in Coclite and Karlsen [16], we consider (3) with its Cauchy problem in the form
(6)∂tu-∂txx3u=-∂x(m2u2)+3∂xu∂xx2u+u∂xxx3u=-(m2u2)x+12∂xxx3u2,u(0,x)=u0(x),
which is equivalent to
(7)ut+uux=-m-12Λ-2(u2)x,u(0,x)=u0(x),
where m>0 is a constant and Λ=(1-∂x2)1/2.The objective of this paper is to study (3). We investigate the existence of weak solutions in the space L1(R)∩BV(R) under certain conditions. Several dynamical properties such as Oleinik type estimate and L2N(R) (N is a natural number) are obtained. As (3) includes the Degasperis-Procesi equation (5), parts of results presented in [16] are extended. Here we should mention that the generalized Degasperis-Procesi equation discussed in [16] does not include the model (3). We state that the ideas and approaches to prove our main results come from those in [16].The rest of this paper is organized as follows. Section2 establishes the L2, BV, and L∞ estimates for the viscous approximations of problem (6). The main result is given in Section 3.


## 2. Viscous Approximations and Estimates

Firstly, we give some notations.SetR+=[0,+∞). The space of all infinitely differentiable functions ϕ(t,x) with compact support in R+×R is denoted by C0∞. We let Lp=Lp(R)(1≤p<+∞) be the space of all measurable functions h such that ∥h∥Lpp=∫R‍|h(t,x)|pdx<∞. We define L∞=L∞(R) with the standard norm ∥h∥L∞=inf⁡m(e)=0sup⁡x∈R∖e|h(t,x)|. For any real number s, we let Hs=Hs(R) denote the Sobolev space with the norm defined by
(8)∥h∥Hs=(∫R‍(1+|ξ|2)s|h^(t,ξ)|2dξ)1/2<∞,
where h^(t,ξ)=∫R‍e-ixξh(t,x)dx.ForT>0 and nonnegative number s, let C([0,T);Hs(R)) denote the Frechet space of all continuous Hs-valued functions on [0,T).Defining(9)ϕ(x)={e1/(x2-1),|x|<1,0,|x|≥1
and letting ϕɛ(x)=ɛ-1/4ϕ(ɛ-1/4x) with 0<ɛ<1/4 and u0,ɛ=ϕɛ⋆u0, we know that u0,ɛ∈C∞ for any u0∈Hs with s≥0.For simplicity, throughout this paper, we letc0 denote any positive constant, which is independent of parameter ɛ and time t.To establish the existence of solutions to the Cauchy problem (6), we will analyze the limiting behavior of a sequence of smooth functions {uɛ}ɛ>0, where each function uɛ satisfies the viscous problem
(10)∂tuɛ-∂txx3uɛ+muɛ∂xuɛ=3∂xuɛ∂xx2uɛ+uɛ∂xxx3uɛ+ɛ∂xx2uɛ-ɛ∂xxxx4uɛ,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii(t,x)∈R+×R,uɛ(0,x)=u0,ɛ(x),x∈R,
which is equivalent to the parabolic-elliptic system
(11)∂tuɛ+∂x(uɛ22)+∂xPɛ=ɛ∂xxuɛ,Pɛ-∂xx2Pɛ=m-12uɛ2,uɛ(0,x)=u0,ɛ(x).
From the second identity of (11), we get
(12)Pɛ(t,x)=m-14∫R‍e|x-y|uɛ2(t,y)dy.


### 2.1.L2 Estimates and Several Consequences

Several properties for the smooth functionu0,ɛ are given in the following Lemma.Lemma 1.
The following estimates hold for anyɛ with 0<ɛ<1/4 and s≥0:
(13)∥u0,ɛ∥L2(R)≤c0∥u0∥L2(R),∥u0,ɛ∥L1(R)≤c0∥u0∥L1(R),∥u0,ɛ∥BV(R)≤c0∥u0∥BV(R),∥u0,ɛ∥Lp(R)≤c0∥u0∥Lp(R)for1≤p≤∞,u0,ɛ⟶u0(ɛ⟶0)inLp(R)for1≤p≤∞,∥u0,ɛ∥Hq≤c0∥u0∥Hs,ifq≤s,
where c0 is a constant independent of ɛ.The proof of Lemma1 is similar to that of Lemma 5 presented in [14]. Here we omit it.Lemma 2.
Provided thatu0∈L2(R), for any fixed ɛ>0, there exists a unique global smooth solution uɛ=uɛ(t,x) to the Cauchy problem (11) belonging to C([0,∞);Hs(R)) with s≥0.Proof.
We omit the proof since it is similar to the one found in [16] or [17] by using u0,ɛ∈C∞(R).Lemma 3.
Assume thatu0∈L2(R) holds and uɛ is a solution of problem (10). Then, the following bounds hold for any t≥0:
(14)∥uɛ∥L2≤c0∥u0∥L2,(15)ɛ∥∂xuɛ∥L2≤c0∥u0∥L2,
where c0 is a positive constant independent of ɛ and t.Proof.
Letting(16)mvɛ-∂xx2vɛ=uɛ
derives
(17)vɛ=(m-∂xx2)-1uɛ.
Multiplying the first equation of problem (11) by vɛ-∂xx2vɛ and integrating over R yield
(18)∫R‍∂tuɛ(vɛ-∂xx2vɛ)dx-ɛ∫R‍∂xx2uɛ(vɛ-∂xx2vɛ)dx=-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx-∫R‍∂xPɛ(vɛ-∂xx2vɛ)dx.
For the left-hand side of this identity, using (16), we get
(19)∫R‍∂tuɛ(vɛ-∂xx2vɛ)dx-ɛ∫R‍∂xx2uɛ(vɛ-∂xx2vɛ)dx=∫R‍(m∂tvɛ-∂txx3vɛ)(vɛ-∂xx2vɛ)dx-ɛ∫R‍(m∂xx2vɛ-∂xxxx4vɛ)(vɛ-∂xx2vɛ)dx=∫R(mvɛ∂tvɛ-vɛ∂txx3vɛ-m∂tvɛ∂xx2vɛ‍iiiiiiiiiiiiiii+∂xx2vɛ∂txx3vɛ)dx-ɛ∫R‍(mvɛ∂xx2vɛ-m(∂xx2vɛ)2-vɛ∂xxxx4vɛiiiiiiiiiiiiiiiiiiiii(∂xx2vɛ)2+∂xx2vɛ∂xxxx4vɛ)dx=∫R‍(mvɛ∂tvɛ-(m+1)vɛ∂txx3vɛ+∂xx2vɛ∂txx3vɛ)dx-ɛ∫R‍(mvɛ∂xx2vɛ-(m+1)vɛ∂xxxx4vɛiiiiiiiiiiiiiiiiiiiii+∂xx2vɛ∂xxxx4vɛ)dx=∫R‍(mvɛ∂tvɛ+(m+1)∂xvɛ∂tx2vɛ+∂xx2vɛ∂txx3vɛ)dx-ɛ∫R‍(-m∂xvɛ∂xvɛ-(m+1)∂xx2vɛ∂xx2vɛiiiiiiiiiiiiiiiiiiiii-∂xxx3vɛ∂xxx3vɛ)dx=12ddt∫R‍(mvɛ2+(m+1)(∂xvɛ)2+(∂xx2vɛ)2)dx+ɛ∫R‍(m(∂xvɛ)2+(m+1)(∂xx2vɛ)2+(∂xxx3vɛ)2)dx.
For the right-hand side of (18), we conclude
(20)-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx-∫R‍∂xPɛ(vɛ-∂xx2vɛ)dx=-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx+∫R‍(Pɛ-∂xx2Pɛ)∂xvɛdx=-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx+m-12∫R‍uɛ2∂xvɛdx=-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx-(m-1)∫R‍uɛ∂xuɛvɛdx=∫R‍(-muɛ∂xuɛvɛ+uɛ∂xuɛ∂xx2vɛ)dx=∫R‍uɛ∂xuɛ(-mvɛ+∂xx2vɛ)dx=-∫R‍uɛ2∂xuɛdx=0,
where we have used (16) and integration by parts.
From (18), (19), and (20), we have
(21)m∥vɛ∥L22+(m+1)∥∂xvɛ∥L22+∥∂xx2vɛ∥L22+2ɛ∫R‍(m∥∂xvɛ∥L22+(m+1)∥∂xx2vɛ∥L22iiiiiiiiiiiiiiiiiiiiii+∥∂xxx3vɛ∥L22)dx=m∥vɛ(0,·)∥L22+(m+1)∥∂xvɛ(0,·)∥L22+∥∂xx2vɛ(0,·)∥L22.
From (17), we obtain
(22)∥vɛ(0,·)∥L2,∥∂xvɛ(0,·)∥L2,∥∂xx2vɛ(0,·)∥L2≤c0∥u0,ɛ∥L2≤c0∥u0∥L2.
It follows from (16) that
(23)∥uɛ(t,·)∥L2(R)2=∫R‍(-∂xx2vɛ+mvɛ)2dx=∫R‍(∂xx2vɛ)2dx-2m∫R‍vɛ∂xx2vɛdx+m2∫R‍vɛ2dx=∫R‍(∂xx2vɛ)2dx+2m∫R‍(∂xvɛ)2dx+m2∫R‍vɛ2dx.
Using (16), (21), and Lemma 1 derives that
(24)∥uɛ∥L22≤2∥∂xx2vɛ∥L22+2m2∥vɛ∥L22≤max⁡(2,2m)(m∥vɛ∥L22+(m+1)∥∂xvɛ∥L22HHHiiiiiiiiiiiiiiiiiiiiiiiiiiii+∥∂xx2vɛ∥L22)≤max⁡(2,2m)(m∥vɛ(0,·)∥L22+(m+1)∥∂xvɛ(0,·)∥L22HHHiiiiiiiiiiiiiiiiiiiiiiiiiiii+∥∂xx2vɛ(0,·)∥L22)≤c0∥u0,ɛ∥L22≤c0∥u0∥L22,ɛ∥∂xuɛ∥L22≤2ɛ∥∂xxx3vɛ∥L22+2m2ɛ∥∂xvɛ∥L22≤ɛmax⁡(2,2m)(m∥∂xvɛ∥L22+(m+1)∥∂xx2vɛ∥L22iiiiiiiiiiiiiiiiiiiiiiiiiiiii+∥∂xxx3vɛ∥L22)≤max⁡(2,2m)(m∥vɛ(0,·)∥L22+(m+1)∥∂xvɛ(0,·)∥L22HHHiiiiiiiiiiiiiiiiiiiiiiiiiiii+∥∂xx2vɛ(0,·)∥L22)≤c0∥u0,ɛ∥L22≤c0∥u0∥L22.
The proof of Lemma3 follows from (24).We give some bounds on the nonlocal termPɛ, in which all are consequences of the L2 bound in Lemma 3.Lemma 4.
Assume thatu0∈L2(R) holds. Then,
(25)Pɛ≥0form≥1,(26)∥Pɛ(t,·)∥L1(R),∥∂xPɛ(t,·)∥L1(R)≤c0|m-1|2∥u0∥L22,(27)∥Pɛ∥L∞(R+×R),∥∂xPɛ∥L∞(R+×R)≤c0|m-1|2∥u0∥L22,(28)∥∂xx2Pɛ(t,·)∥L1(R)≤c0|m-1|2∥u0∥L22,
where c0 is a constant independent of ɛ and t.Proof.
Using (11), we get
(29)Pɛ(t,x)=m-14∫R‍e-|x-y|(uɛ(t,y))2dy,(30)∂xPɛ(t,x)=m-14∫R‍e-|x-y|sign⁡(y-x)(uɛ(t,y))2dy.
From (29), we obtain (25). Using (14) and the Tonelli theorem, we have
(31)∫R‍|Pɛ(t,x)|dx≤|m-1|4∫R‍(∫R‍e-|x-y|dx)(uɛ(t,y))2dy≤c0|m-1|2∫R‍(uɛ(t,y))2dy≤c0|m-1|2∥u0∥L22,∫R‍|∂xPɛ(t,x)|dx≤|m-1|4∫R‍(∫R‍e-|x-y|dx)(uɛ(t,y))2dy≤c0|m-1|2∫R‍(uɛ(t,y))2dy≤c0|m-1|2∥u0∥L22,|Pɛ(t,x)|≤|m-1|4∫R‍(uɛ(t,y))2dy2222222..≤c0|m-1|4∥u0∥L22,|∂xPɛ(t,x)|≤|m-1|4∫R‍(uɛ(t,y))2dy22222222.2≤c0|m-1|4∥u0∥L22.
It follows from (31) that (26) and (27) hold. Using the second identity of problem (11), Lemma 3, and (26), we obtain (28).Lemma 5.
Ifu0∈L1(R)∩L2(R), it holds that
(32)∥uɛ(t,·)∥L1≤∥u0∥L1+c0t∥u0∥L22.Proof.
Let functionsη and q:R→R be such that q′(u)=uη′(u). Multiplying the first equation in (11) with η′(uɛ) gives rise to
(33)∂tη(uɛ)+∂xq(uɛ)+η′(uɛ)∂xPɛ=ɛη′(uɛ)∂xx2uɛ=ɛ∂xx2η(uɛ)-ɛη′′(uɛ)(∂xuɛ)2.
Choosingη(u)=|u| (modulo an approximation argument, see [16]) and then integrating the resulting equation over R yield
(34)ddt∫R‍|uɛ|dx≤∫R‍|sign⁡(uɛ)∂xPɛ|dx.
Using (26), we get
(35)ddt∥uɛ(t,·)∥L1≤c0|m-1|2∥u0∥L22,
from which we have (32).


### 2.2.BV and L∞ Estimates

In this subsection we establish several supplementary estimates for the viscous approximations, which also are consequences of theL2 bound in Lemma 3. In particular, we prove that the sequence {uɛ}ɛ>0 is bounded in BV, which yields strong compactness of this sequence. To this end, we need to assume that u0∈L2(R) and u0∈BV(R).Lemma 6.
Assume thatu0∈L2(R) and u0∈BV(R) hold. Then,
(36)∥∂xuɛ(t,·)∥L1(R)≤c0(|u0|BV+t∥u0∥L22),t≥0.Proof.
Settingqɛ∶=∂xuɛ, we know that qɛ satisfies the equation
(37)∂tqɛ+uɛ∂xqɛ+qɛ2+∂xx2Pɛ=ɛ∂xx2qɛ.
Ifη=η(u) and q:R→R satisfies q′(u)=uη′(u), using the chain rule yields
(38)∂tη(qɛ)+∂x(uɛq(uɛ))-qɛη(qɛ)+η′(qɛ)qɛ2+η′(uɛ)∂xx2Pɛ=ɛ∂xx2η(qɛ)-ɛη′′(qɛ)(∂xqɛ)2.
Choosing η(u)=|u| (modulo an approximation argument) and then integrating the resulting equation over R give rise to
(39)ddt∫R‍|∂xuɛ|dx≤∫R‍|∂xx2Pɛ|dx.
Using (28), we have
(40)ddt∫R‍|∂xuɛ|dx≤c0∥u0∥L2(R)2,
from which we obtain (36).Lemma 7.
Assume thatu0∈BV(R)∩L2(R) holds. Then,
(41)∥uɛ(t,·)∥L∞(R)≤c0(|u0|BV(R)+t∥u0∥L2(R)2),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiit≥0,(42)∥∂xx2Pɛ(t,·)∥L∞≤c0[∥u0∥L2(R)2+(|u0|BV(R)+t∥u0∥L2(R))2],
where c0 is independent of ɛ and t.Proof.
Using(43)|uɛ(t,·)|≤∫R‍|∂xuɛ(t,y)|dy=∥uɛ(t,·)∥BV(R)
and Lemma 6 derives (41). Using (27), (41), and the second equation of problem (11), we obtain inequality (42).


### 2.3.L2N Estimate for Nature Number N≥2

Next we prove that the viscous approximations are bounded inL2N(R) for any nature number N≥2. From Lemmas 6 and 7, if u0∈BV(R)∩L2(R), we have the inequality u0,ɛ∈L∞(R) from which we derive
(44)u0,ɛ∈L2(R)⋂‍L2N(R).Lemma 8.
Assume thatu0∈BV(R)∩L2(R) holds. For any 0<ɛ<1/4, it has
(45)∥uɛ(t,·)∥L2N(R)≤c0(1+∥u0∥L22)2t+∥uɛ(0,·)∥L2N(R)2Nec0(1+∥u0∥L22)2t.Proof.
Choosingη(u)=(1/2N)u2N in (33), writing
(46)∂tη(uɛ)+∂xq(uɛ)+η′(uɛ)∂xPɛ=ɛ∂xx2η(uɛ)-ɛη′′(uɛ)(∂xuɛ)2=ɛη′(uɛ)∂xx2uɛ=ɛuɛ2N-1∂xx2uɛ,
and integrating (46) over R gives rise to
(47)12Nddt∥uɛ(t,·)∥L2N2N=-∫R‍uɛ2N-1∂xPɛdx+ɛ∫R‍uɛ2N-1∂xx2uɛdx.
Integration by parts shows(48)ɛ∫R‍uɛ2N-1∂xx2uɛdx=-(2N-1)ɛ∫R‍uɛ(2N-2)(∂xuɛ)2dx≤0.
Letting p1=2N/(2N-1),q1=2N, we have 1/p1+1/q1=1. Using Hölder’s inequality, (26), and (27), we obtain
(49)|∫R‍uɛ2N-1∂xPɛdx|≤(∫R‍uɛ2Ndx)(2N-1)/2N(∫R‍(∂xPɛ)2N)1/2N≤(1+∫R‍uɛ2Ndx)(2N-1)/2N×(∥∂xPɛ∥L∞(R+×R)2N-1)1/2N∥∂xPɛ∥L1(R)1/2N≤(1+∫R‍uɛ2Ndx)(1+∥∂xPɛ∥L∞(R+×R))∥∂xPɛ∥L1(R)1/2N≤c0(1+∥u0∥L22)(1+∥u0∥L22)1/2N(1+∫R‍uɛ2Ndx)≤c0(1+∥u0∥L22)2(1+∫R‍uɛ2Ndx).
Using (47)–(49) gives rise to
(50)12Nddt∥uɛ(t,·)∥L2N2N≤c0(1+∥u0∥L22)2(1+∥uɛ(t,·)∥L2N2N),
from which we obtain (45) by Gronwall’s inequality.


### 2.4.  Oleinik Type Estimate

Lemma 9 (Oleinik type estimate).
Assume thatu0∈L2(R)∩BV(R). Then, for each t∈(0,T] with T being fixed,
(51)∂xuɛ(t,x)≤1t+KT,x∈R,
where KT=c0{∥u0∥L2(R)2+(∥u0∥BV(R)+T∥u0∥L2(R)2)2}1/2.Proof.
Settingqɛ∶=∂xuɛ, it follows from (11) and (42) that
(52)∂tqɛ+uɛ∂xqɛ+qɛ2-ɛ∂xx2qɛ=-∂xx2Pɛ≤KT2.
Considering the ordinary differential equation
(53)dfdt+f2=KT2
and using comparing theorem, we have
(54)∂xuɛ(t,x)≤1t+KT,x∈R,
which completes the proof.


## 2.1.L2 Estimates and Several Consequences

Several properties for the smooth functionu0,ɛ are given in the following Lemma.Lemma 1.
The following estimates hold for anyɛ with 0<ɛ<1/4 and s≥0:
(13)∥u0,ɛ∥L2(R)≤c0∥u0∥L2(R),∥u0,ɛ∥L1(R)≤c0∥u0∥L1(R),∥u0,ɛ∥BV(R)≤c0∥u0∥BV(R),∥u0,ɛ∥Lp(R)≤c0∥u0∥Lp(R)for1≤p≤∞,u0,ɛ⟶u0(ɛ⟶0)inLp(R)for1≤p≤∞,∥u0,ɛ∥Hq≤c0∥u0∥Hs,ifq≤s,
where c0 is a constant independent of ɛ.The proof of Lemma1 is similar to that of Lemma 5 presented in [14]. Here we omit it.Lemma 2.
Provided thatu0∈L2(R), for any fixed ɛ>0, there exists a unique global smooth solution uɛ=uɛ(t,x) to the Cauchy problem (11) belonging to C([0,∞);Hs(R)) with s≥0.Proof.
We omit the proof since it is similar to the one found in [16] or [17] by using u0,ɛ∈C∞(R).Lemma 3.
Assume thatu0∈L2(R) holds and uɛ is a solution of problem (10). Then, the following bounds hold for any t≥0:
(14)∥uɛ∥L2≤c0∥u0∥L2,(15)ɛ∥∂xuɛ∥L2≤c0∥u0∥L2,
where c0 is a positive constant independent of ɛ and t.Proof.
Letting(16)mvɛ-∂xx2vɛ=uɛ
derives
(17)vɛ=(m-∂xx2)-1uɛ.
Multiplying the first equation of problem (11) by vɛ-∂xx2vɛ and integrating over R yield
(18)∫R‍∂tuɛ(vɛ-∂xx2vɛ)dx-ɛ∫R‍∂xx2uɛ(vɛ-∂xx2vɛ)dx=-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx-∫R‍∂xPɛ(vɛ-∂xx2vɛ)dx.
For the left-hand side of this identity, using (16), we get
(19)∫R‍∂tuɛ(vɛ-∂xx2vɛ)dx-ɛ∫R‍∂xx2uɛ(vɛ-∂xx2vɛ)dx=∫R‍(m∂tvɛ-∂txx3vɛ)(vɛ-∂xx2vɛ)dx-ɛ∫R‍(m∂xx2vɛ-∂xxxx4vɛ)(vɛ-∂xx2vɛ)dx=∫R(mvɛ∂tvɛ-vɛ∂txx3vɛ-m∂tvɛ∂xx2vɛ‍iiiiiiiiiiiiiii+∂xx2vɛ∂txx3vɛ)dx-ɛ∫R‍(mvɛ∂xx2vɛ-m(∂xx2vɛ)2-vɛ∂xxxx4vɛiiiiiiiiiiiiiiiiiiiii(∂xx2vɛ)2+∂xx2vɛ∂xxxx4vɛ)dx=∫R‍(mvɛ∂tvɛ-(m+1)vɛ∂txx3vɛ+∂xx2vɛ∂txx3vɛ)dx-ɛ∫R‍(mvɛ∂xx2vɛ-(m+1)vɛ∂xxxx4vɛiiiiiiiiiiiiiiiiiiiii+∂xx2vɛ∂xxxx4vɛ)dx=∫R‍(mvɛ∂tvɛ+(m+1)∂xvɛ∂tx2vɛ+∂xx2vɛ∂txx3vɛ)dx-ɛ∫R‍(-m∂xvɛ∂xvɛ-(m+1)∂xx2vɛ∂xx2vɛiiiiiiiiiiiiiiiiiiiii-∂xxx3vɛ∂xxx3vɛ)dx=12ddt∫R‍(mvɛ2+(m+1)(∂xvɛ)2+(∂xx2vɛ)2)dx+ɛ∫R‍(m(∂xvɛ)2+(m+1)(∂xx2vɛ)2+(∂xxx3vɛ)2)dx.
For the right-hand side of (18), we conclude
(20)-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx-∫R‍∂xPɛ(vɛ-∂xx2vɛ)dx=-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx+∫R‍(Pɛ-∂xx2Pɛ)∂xvɛdx=-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx+m-12∫R‍uɛ2∂xvɛdx=-∫R‍uɛ∂xuɛ(vɛ-∂xx2vɛ)dx-(m-1)∫R‍uɛ∂xuɛvɛdx=∫R‍(-muɛ∂xuɛvɛ+uɛ∂xuɛ∂xx2vɛ)dx=∫R‍uɛ∂xuɛ(-mvɛ+∂xx2vɛ)dx=-∫R‍uɛ2∂xuɛdx=0,
where we have used (16) and integration by parts.
From (18), (19), and (20), we have
(21)m∥vɛ∥L22+(m+1)∥∂xvɛ∥L22+∥∂xx2vɛ∥L22+2ɛ∫R‍(m∥∂xvɛ∥L22+(m+1)∥∂xx2vɛ∥L22iiiiiiiiiiiiiiiiiiiiii+∥∂xxx3vɛ∥L22)dx=m∥vɛ(0,·)∥L22+(m+1)∥∂xvɛ(0,·)∥L22+∥∂xx2vɛ(0,·)∥L22.
From (17), we obtain
(22)∥vɛ(0,·)∥L2,∥∂xvɛ(0,·)∥L2,∥∂xx2vɛ(0,·)∥L2≤c0∥u0,ɛ∥L2≤c0∥u0∥L2.
It follows from (16) that
(23)∥uɛ(t,·)∥L2(R)2=∫R‍(-∂xx2vɛ+mvɛ)2dx=∫R‍(∂xx2vɛ)2dx-2m∫R‍vɛ∂xx2vɛdx+m2∫R‍vɛ2dx=∫R‍(∂xx2vɛ)2dx+2m∫R‍(∂xvɛ)2dx+m2∫R‍vɛ2dx.
Using (16), (21), and Lemma 1 derives that
(24)∥uɛ∥L22≤2∥∂xx2vɛ∥L22+2m2∥vɛ∥L22≤max⁡(2,2m)(m∥vɛ∥L22+(m+1)∥∂xvɛ∥L22HHHiiiiiiiiiiiiiiiiiiiiiiiiiiii+∥∂xx2vɛ∥L22)≤max⁡(2,2m)(m∥vɛ(0,·)∥L22+(m+1)∥∂xvɛ(0,·)∥L22HHHiiiiiiiiiiiiiiiiiiiiiiiiiiii+∥∂xx2vɛ(0,·)∥L22)≤c0∥u0,ɛ∥L22≤c0∥u0∥L22,ɛ∥∂xuɛ∥L22≤2ɛ∥∂xxx3vɛ∥L22+2m2ɛ∥∂xvɛ∥L22≤ɛmax⁡(2,2m)(m∥∂xvɛ∥L22+(m+1)∥∂xx2vɛ∥L22iiiiiiiiiiiiiiiiiiiiiiiiiiiii+∥∂xxx3vɛ∥L22)≤max⁡(2,2m)(m∥vɛ(0,·)∥L22+(m+1)∥∂xvɛ(0,·)∥L22HHHiiiiiiiiiiiiiiiiiiiiiiiiiiii+∥∂xx2vɛ(0,·)∥L22)≤c0∥u0,ɛ∥L22≤c0∥u0∥L22.
The proof of Lemma3 follows from (24).We give some bounds on the nonlocal termPɛ, in which all are consequences of the L2 bound in Lemma 3.Lemma 4.
Assume thatu0∈L2(R) holds. Then,
(25)Pɛ≥0form≥1,(26)∥Pɛ(t,·)∥L1(R),∥∂xPɛ(t,·)∥L1(R)≤c0|m-1|2∥u0∥L22,(27)∥Pɛ∥L∞(R+×R),∥∂xPɛ∥L∞(R+×R)≤c0|m-1|2∥u0∥L22,(28)∥∂xx2Pɛ(t,·)∥L1(R)≤c0|m-1|2∥u0∥L22,
where c0 is a constant independent of ɛ and t.Proof.
Using (11), we get
(29)Pɛ(t,x)=m-14∫R‍e-|x-y|(uɛ(t,y))2dy,(30)∂xPɛ(t,x)=m-14∫R‍e-|x-y|sign⁡(y-x)(uɛ(t,y))2dy.
From (29), we obtain (25). Using (14) and the Tonelli theorem, we have
(31)∫R‍|Pɛ(t,x)|dx≤|m-1|4∫R‍(∫R‍e-|x-y|dx)(uɛ(t,y))2dy≤c0|m-1|2∫R‍(uɛ(t,y))2dy≤c0|m-1|2∥u0∥L22,∫R‍|∂xPɛ(t,x)|dx≤|m-1|4∫R‍(∫R‍e-|x-y|dx)(uɛ(t,y))2dy≤c0|m-1|2∫R‍(uɛ(t,y))2dy≤c0|m-1|2∥u0∥L22,|Pɛ(t,x)|≤|m-1|4∫R‍(uɛ(t,y))2dy2222222..≤c0|m-1|4∥u0∥L22,|∂xPɛ(t,x)|≤|m-1|4∫R‍(uɛ(t,y))2dy22222222.2≤c0|m-1|4∥u0∥L22.
It follows from (31) that (26) and (27) hold. Using the second identity of problem (11), Lemma 3, and (26), we obtain (28).Lemma 5.
Ifu0∈L1(R)∩L2(R), it holds that
(32)∥uɛ(t,·)∥L1≤∥u0∥L1+c0t∥u0∥L22.Proof.
Let functionsη and q:R→R be such that q′(u)=uη′(u). Multiplying the first equation in (11) with η′(uɛ) gives rise to
(33)∂tη(uɛ)+∂xq(uɛ)+η′(uɛ)∂xPɛ=ɛη′(uɛ)∂xx2uɛ=ɛ∂xx2η(uɛ)-ɛη′′(uɛ)(∂xuɛ)2.
Choosingη(u)=|u| (modulo an approximation argument, see [16]) and then integrating the resulting equation over R yield
(34)ddt∫R‍|uɛ|dx≤∫R‍|sign⁡(uɛ)∂xPɛ|dx.
Using (26), we get
(35)ddt∥uɛ(t,·)∥L1≤c0|m-1|2∥u0∥L22,
from which we have (32).


## 2.2.BV and L∞ Estimates

In this subsection we establish several supplementary estimates for the viscous approximations, which also are consequences of theL2 bound in Lemma 3. In particular, we prove that the sequence {uɛ}ɛ>0 is bounded in BV, which yields strong compactness of this sequence. To this end, we need to assume that u0∈L2(R) and u0∈BV(R).Lemma 6.
Assume thatu0∈L2(R) and u0∈BV(R) hold. Then,
(36)∥∂xuɛ(t,·)∥L1(R)≤c0(|u0|BV+t∥u0∥L22),t≥0.Proof.
Settingqɛ∶=∂xuɛ, we know that qɛ satisfies the equation
(37)∂tqɛ+uɛ∂xqɛ+qɛ2+∂xx2Pɛ=ɛ∂xx2qɛ.
Ifη=η(u) and q:R→R satisfies q′(u)=uη′(u), using the chain rule yields
(38)∂tη(qɛ)+∂x(uɛq(uɛ))-qɛη(qɛ)+η′(qɛ)qɛ2+η′(uɛ)∂xx2Pɛ=ɛ∂xx2η(qɛ)-ɛη′′(qɛ)(∂xqɛ)2.
Choosing η(u)=|u| (modulo an approximation argument) and then integrating the resulting equation over R give rise to
(39)ddt∫R‍|∂xuɛ|dx≤∫R‍|∂xx2Pɛ|dx.
Using (28), we have
(40)ddt∫R‍|∂xuɛ|dx≤c0∥u0∥L2(R)2,
from which we obtain (36).Lemma 7.
Assume thatu0∈BV(R)∩L2(R) holds. Then,
(41)∥uɛ(t,·)∥L∞(R)≤c0(|u0|BV(R)+t∥u0∥L2(R)2),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiit≥0,(42)∥∂xx2Pɛ(t,·)∥L∞≤c0[∥u0∥L2(R)2+(|u0|BV(R)+t∥u0∥L2(R))2],
where c0 is independent of ɛ and t.Proof.
Using(43)|uɛ(t,·)|≤∫R‍|∂xuɛ(t,y)|dy=∥uɛ(t,·)∥BV(R)
and Lemma 6 derives (41). Using (27), (41), and the second equation of problem (11), we obtain inequality (42).


## 2.3.L2N Estimate for Nature Number N≥2

Next we prove that the viscous approximations are bounded inL2N(R) for any nature number N≥2. From Lemmas 6 and 7, if u0∈BV(R)∩L2(R), we have the inequality u0,ɛ∈L∞(R) from which we derive
(44)u0,ɛ∈L2(R)⋂‍L2N(R).Lemma 8.
Assume thatu0∈BV(R)∩L2(R) holds. For any 0<ɛ<1/4, it has
(45)∥uɛ(t,·)∥L2N(R)≤c0(1+∥u0∥L22)2t+∥uɛ(0,·)∥L2N(R)2Nec0(1+∥u0∥L22)2t.Proof.
Choosingη(u)=(1/2N)u2N in (33), writing
(46)∂tη(uɛ)+∂xq(uɛ)+η′(uɛ)∂xPɛ=ɛ∂xx2η(uɛ)-ɛη′′(uɛ)(∂xuɛ)2=ɛη′(uɛ)∂xx2uɛ=ɛuɛ2N-1∂xx2uɛ,
and integrating (46) over R gives rise to
(47)12Nddt∥uɛ(t,·)∥L2N2N=-∫R‍uɛ2N-1∂xPɛdx+ɛ∫R‍uɛ2N-1∂xx2uɛdx.
Integration by parts shows(48)ɛ∫R‍uɛ2N-1∂xx2uɛdx=-(2N-1)ɛ∫R‍uɛ(2N-2)(∂xuɛ)2dx≤0.
Letting p1=2N/(2N-1),q1=2N, we have 1/p1+1/q1=1. Using Hölder’s inequality, (26), and (27), we obtain
(49)|∫R‍uɛ2N-1∂xPɛdx|≤(∫R‍uɛ2Ndx)(2N-1)/2N(∫R‍(∂xPɛ)2N)1/2N≤(1+∫R‍uɛ2Ndx)(2N-1)/2N×(∥∂xPɛ∥L∞(R+×R)2N-1)1/2N∥∂xPɛ∥L1(R)1/2N≤(1+∫R‍uɛ2Ndx)(1+∥∂xPɛ∥L∞(R+×R))∥∂xPɛ∥L1(R)1/2N≤c0(1+∥u0∥L22)(1+∥u0∥L22)1/2N(1+∫R‍uɛ2Ndx)≤c0(1+∥u0∥L22)2(1+∫R‍uɛ2Ndx).
Using (47)–(49) gives rise to
(50)12Nddt∥uɛ(t,·)∥L2N2N≤c0(1+∥u0∥L22)2(1+∥uɛ(t,·)∥L2N2N),
from which we obtain (45) by Gronwall’s inequality.


## 2.4.  Oleinik Type Estimate

Lemma 9 (Oleinik type estimate).
Assume thatu0∈L2(R)∩BV(R). Then, for each t∈(0,T] with T being fixed,
(51)∂xuɛ(t,x)≤1t+KT,x∈R,
where KT=c0{∥u0∥L2(R)2+(∥u0∥BV(R)+T∥u0∥L2(R)2)2}1/2.Proof.
Settingqɛ∶=∂xuɛ, it follows from (11) and (42) that
(52)∂tqɛ+uɛ∂xqɛ+qɛ2-ɛ∂xx2qɛ=-∂xx2Pɛ≤KT2.
Considering the ordinary differential equation
(53)dfdt+f2=KT2
and using comparing theorem, we have
(54)∂xuɛ(t,x)≤1t+KT,x∈R,
which completes the proof.


## 3. Existence inL1(R)∩BV(R)

Using the estimates established in Section2, we will show the existence of weak solutions to problem (6) under the assumption u0∈L1(R)∩BV(R).We state the concepts of weak solutions.Definition 10 (weak solution).
We call a functionu:R+×R→R a weak solution of the Cauchy problem (6) provided that(i)
u∈L∞(R+;L2(R)) and(ii)
∂tu+∂x(u2/2)+∂xPu(t,x)=0 in D′([0,∞)×R); that is, for all ϕ∈C0∞([0,∞)×R), there holds the identity
(55)∫R+∫R‍(u∂tϕ+u22∂xϕ-∂xPuϕ)dxdt+∫R‍u0(x)ϕ(0,x)dx=0,where
(56)Pu(t,x)=G1*(m-12u2)(t,x)=m-14∫R‍e-|x-y|(u(t,y))2dy.Remark 11.
It follows from part (i) of Definition10 that u∈L1((0,T)×R) for any T>0 and ∂xPu∈L∞(R+×R) (see Lemma 5). Therefore, (55) makes sense.We assume that(57)∥u(t2,·)-u(t1,·)∥L1(R)≤CT|t2-t1|iiiiiiiiiiiiiiiiiiiiiiiiifor∀t1,t2∈[0,T],
where CT=c0[∥u0∥L2(R)2+(|u0|BV(R)+t∥u0∥L2(R))2]. Therefore, we have
(58)lim⁡t→0+∥u(t,·)-u0∥L1(R)=0.Our main results are summarized in the following Theorem.Theorem 12.
Provided thatu0∈L1(R)∩L2(R)∩BV(R) and the solution u(t,x) satisfies (57), then there exists a weak solution to the Cauchy problem (6). The weak solution u satisfies the following estimates for any t∈(0,T):
(59)∥u(t,·)∥L1(R)≤∥u0∥L1+c0t∥u0∥L22,∥u(t,·)∥BV,∥u(t,x)∥L∞≤c0(|u0|BV(R)+t∥u0∥L2(R)2),∥uɛ(t,·)∥L2N(R)≤c0(1+∥u0∥L22)2t+∥u(0,·)∥L2N(R)2Nec0(1+∥u0∥L22)2t,
where c0 is a positive constant independent of t and ɛ.
The following Oleinik type estimate holds for a.e.(t,x)∈(0,T]×R:
(60)∂xu(t,x)≤1t+KT,
where
(61)KT=c0{∥u0∥L2(R)2(∥u0∥L22)2iiiiii+(∥u0∥BV(R)+T∥u0∥L2(R)2)2}1/2.This theorem is an immediate consequence of Theorem13 and results are presented in Section 2.Theorem 13 (existence).
Assume thatu0∈L1(R)∩L2(R)∩BV(R) and the solution u(t,x) satisfies (57). Then, there exists at least one weak solution to problem (6).Proof.
Using the estimates obtained in Section2, we take a standard argument to see that there exists a sequence of strictly positive numbers {ɛk}k=1∞ tending to zero such that as k→∞(62)uɛk⟶ua.e.inR+×R,uɛk⟶uinLloc⁡1(R+×R).
The previous estimates in Section2 imply immediately that the limit function u satisfies (59)-(60).
Let us now prove that ask→∞(63)Pɛk⟶Pu,∂xPɛk⟶∂xPuinL1((0,T)×R)∀T>0,
which follows from the following calculations:
(64)∥Pɛk-Pu∥L1((0,T)×R)≤(|m-1|4)×∫0T∫R‍(∫R‍e-|x-y|iiiiiiiiiiiiiiiiiiiiiiiiiiii∫R‍×|(uɛk(t,y))2-(u(t,y))2|dy)dxdt≤(|m-1|2)∫0T‍(∫R‍|uɛk(t,y)-u(t,y)|iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∫R‍×|uɛk(t,y)+u(t,y)|dy)dt≤CT∫0T∫R‍|uɛk(t,y)-(u(t,y))|dydt⟶0iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiask⟶∞,∥∂xPɛk-∂xPu∥L1((0,T)1×R)≤(|m-1|4)×∫0T∫R‍(∫R‍e-|x-y|iiiiiiiiiiiiiiiiiiiiiiiiiiii∫R‍×|(uɛk(t,y))2-(u(t,y))2|dy)dxdt≤(|m-1|2)∫0T‍(∫R‍|uɛk(t,y)-u(t,y)|iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∫R‍×|uɛk(t,y)+u(t,y)|dy)dt≤CT∫0T∫R‍|uɛk(t,y)-(u(t,y))|dydt⟶0iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiask⟶∞.
From (62) to (64), we complete the proof.



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