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Abstract
The existence of global weak solutions to the shallow water model with moderate amplitude, which is firstly introduced in Constantin and Lannes’s work (2009), is investigated in the space without the sign condition on the initial value by employing the limit technique of viscous approximation. A new one-sided lower bound and the higher integrability estimate act a key role in our analysis.
Public Interest Statement
In this paper, we use the limit technique of viscous approximation to prove the existence of global weak solutions for a shallow water wave model with moderate amplitude. It is shown from the proof of main Theorem that the weak solutions are stable when a regularizing term vanishes. The method is so effective and can be applied to solve some control problems and economic model. Moreover, the shallow water wave model with moderate amplitude we investigate captures breaking wave, which is a major interest in shallow water wave. Overall, the results we obtain can be applied in many hydrodynamic problems.
1. Introduction
In this paper, we consider the following model for shallow water wave with moderate amplitude(1)
(1)
system (1) is firstly found in Constantin and Lannes (Citation2009) as a model for the evolution of the free surface u. Here the function ,
and
are parameters,
is shallowness parameter,
is amplitude parameter (see Alvarez-Samaniego & Lannes, Citation2008; Constantin & Lannes, Citation2009; Mi & Mu, Citation2013). It is shown in Constantin (Citation2011) that unlike KdV and Camassa–Holm equation, system (1) does not have a bi-Hamiltonian integrable structure. However, the equation possesses solitary wave profiles that resemble those of C–H (Constantin & Escher, Citation2007). Recently, Constantin and Lannes (Citation2009) established the local well-posedness of system (1) for any initial data
with
and also claimed that if the maximal existence time is finite, then blow-up occurs in form of wave breaking. In Duruk Mutlubas (Citation2013), the local well-posedness of system (1) is proved for initial data in
with
using Kato’s semigroup method for quasi-linear equations. Orbital stability and existence of solitary waves for system (1) was obtained in Duruk Mutlubas and Geyer (Citation2013), Geyer (Citation2012). Mi and Mu (Citation2013) investigated the local well-posedness of system (1) in Besov space using Littlewood–Paley decomposition and transport equation theory, and proposed that if initial data
is analytic its solutions are analytic. Moreover, persistence properties on strong solutions were also presented (see Mi & Mu, Citation2013).
One of the close relatives of the first equation of problem (1) is the rod wave equation (Dai, Citation1998; Dai & Huo, Citation2000)(2)
(2)
where and
stands for the radial stretch relative to a prestressed state in non-dimensional variables. Equation (2) is a model for finite-length and small-amplitude axial-radial deformation waves in the cylindrical compressible hyperelastic rods. Since Equation (2) was derived by Dai (Citation1998), Dai and Huo (Citation2000, many works have been carried out to investigate its dynamic properties. In Constantin and Strauss (Citation2000), Constantin and Strauss studied the Cauchy problem of the rod equation on the line (nonperiodic case), where the local well-posedness and blow-up solutions were discussed. Moreover, they also proved the stability of solitary waves for the equation (see Constantin & Strauss, Citation2000). Later, Yin (Citation2003,Citation2004) and Hu and Yin (Citation2010) discussed the smooth solitary waves and blow-up solutions. Zhou (Citation2006), the precise blow-up scenario and several blow-up results of strong solutions to the rod equation on the circle (periodic case) were presented. For other techniques to study the problems relating to various dynamic properties of other shallow water wave equations, the reader is referred to Coclite, Holden, and Karlsen (Citation2005), Yan, Li, and Zhang (Citation2014), Fu, Liu, and Qu (Citation2012), Guo and Wang (Citation2014), Himonas, Misiolek, Ponce, and Zhou (Citation2007), Holden and Raynaud (Citation2009), Li and Olver (Citation2000), Qu, Fu, and Liu (Citation2014), Lai (Citation2013) and the reference therein.
Xin and Zhang (Citation2000) use the limit method of viscous approximations to analyze the existence of global weak solutions for Equation (2) with (Namely, Camassa–Holm equation). Motivated by the desire to extend the works (Xin & Zhang, Citation2000), the objective of this paper was to establish the existence of global weak solutions for the system (1) in the space
under the assumption
. Following the idea in Xin and Zhang (Citation2000), the limit method of viscous approximations is employed to establish the existence of the global weak solution for system(1). In our analysis, a new one-sided lower bound (see Lemma 3.4) and the higher integrability estimate (see Lemma 3.3), which ensure that weak convergence of
is equal to strong convergence, play a crucial role in establishing the existence of global weak solutions.
The rest of this paper is as follows. The main result is presented in Section 2. In Section 3, we state the viscous problem and give a corresponding result. Strong compactness of the derivative of viscous approximations is obtained in Section 4. Section 5 completes the proof of the main result.
2. The main results
Using the Green function , we have
for all
, and
, where we denote by
the convolution. Then we can rewrite system(1) as follows
(3)
(3)
which is also equivalent to the elliptic-hyperbolic system(4)
(4)
where .
Now we give the definition of a weak solution to the Cauchy problem (3) or (4).
Definition 2.1
A continuous function is said to be a global weak solution to the Cauchy problem (4) if
(i) |
| ||||
(ii) |
| ||||
(iii) |
|
The main result of present paper is collected in following theorem.
Theorem 2.2
Assume that . Then the Cauchy problem (4) has a global weak solution u(t, x) in the sense of Definition 2.1. In addition, there is a positive constant
, independent of
, such that
(6)
(6)
3. Viscous approximations
Defining(7)
(7)
and setting the mollifier with
and
, we know that
for any
,
(see Lai & Wu, Citation2010).
In fact, suitably choosing the mollifier, we have(8)
(8)
Differentiating the first equation of problem (5) with respect to variable x and letting , we have
(9)
(9)
The starting point of our analysis is the following well-posedness result for problem (5).
Lemma 3.1
Assume For any
, there exists a unique solution
to the Cauchy problem (5). Moreover, for any
, it holds that
(10)
(10)
Proof
For any and
, we have
. From Theorem 2.1 in Coclite et al. (Citation2005), we infer that problem (5) has a unique solution
.
The first equation of (5) is rewritten as(11)
(11)
Multiplying (11) by , we derive that
(12)
(12)
which finishes the proof.
For simplicity, in this paper, let c denote any positive constant which is independent of the parameter . From Lemma 3.1, we have
(13)
(13)
Lemma 3.2
For , there exists a positive constant
, independent of
, such that
(14)
(14)
and(15)
(15)
where is the unique solution of (5) and
(16)
(16)
Proof
In the proof of this lemma, we will use the identity(17)
(17)
For simplicity, setting , we have
(18)
(18)
and(19)
(19)
Note that for
. Using (22), one has
(20)
(20)
which proves (14).
In view of Lemma 3.1 and Tonelli theorem, one has(21)
(21)
and then, we get(22)
(22)
Using the Tonelli theorem and the Hölder inequality, it holds(23)
(23)
Making use of (27) and (28), we complete the proof of (15).
From (26)–(27) and the Hölder inequality, we have(24)
(24)
Hence,(25)
(25)
By (25) and (28), one has(26)
(26)
From (30) and (31), we deduce (16).
On the other hand, from (24), we derive that(27)
(27)
Inequalities (17), (18), and (19) are direct consequences of (25), (27), (28), (30), and (31).
Finally, note that(28)
(28)
Using (25), we obtain (20).
Lemma 3.3
Let ,
, and
,
. Then there exists a positive constant
depending only on
and b, but independent of
, such that
(29)
(29)
where is the unique solution of (5).
Proof
The proof of Lemma 3.3 is similar to that of Lemma 4.1 in Xin and Zhang (Citation2000). Here, we omit its proof.
Lemma 3.4
For an arbitrary , the following estimate on the first-order spatial derivative holds
(30)
(30)
Proof
Using (9), we get(31)
(31)
Let be the solution of
(32)
(32)
Since is a supersolution of the parabolic equation (36) with initial value
, due to the comparison principle for parabolic equations, we get
Consider the function , observing that
for any
and using the comparison principle for ordinary differential equations, we have
for all
. It completes the proof.
Lemma 3.5
There exists a sequence tending to zero and a function
, for each
, such that
(33)
(33)
where is the unique solution of (5).
Proof
For fixed , using Lemmas 3.1 and 3.3, and
we obtain(34)
(34)
Hence is uniformly bounded in
and (38) follows.
Observe that, for each ,
(35)
(35)
Moreover, is uniformly bounded in
and
. Using the results in Coclite et al. (Citation2005), we know that (39) holds.
Lemma 3.6
There exists a sequence tending to zero and a function
such that for each
,
(36)
(36)
Proof
Using Lemma 3.2, we have the existence of pointwise convergence subsequence which is uniformly bounded in
. Inequalities (15) and (16) derive that (42) holds.
Throughout this paper we use overbars to denote weak limits.
Lemma 3.7
There exists a sequence tending to zero and two function
,
such that
(37)
(37)
for each and
. Moreover,
(38)
(38)
and(39)
(39)
Proof
Equations (43) and (44) are direct consequences of Lemmas 3.1 and 3.3. Inequality (45) is valid because of the weak convergence in (44). Finally, (46) is a consequence of definition of , Lemma 3.5 and (43).
In the following, for notational convenience, we replace the sequence ,
, and
by
,
, and
, separately.
Using (43), we conclude that for any convex function with
bounded, Lipschitz continuous on
and any
we get
(40)
(40)
and(41)
(41)
Multiplying Equation (9) by yields
(42)
(42)
Lemma 3.8
For any convex with
bounded, Lipschitz continuous on
, it holds that
(43)
(43)
in the sense of distributions on . Here
and
denote the weak limits of
and
in
,
, respectively.
Proof
In (49), by the convexity of , Lemmas 3.5–3.7, sending
, gives rise to the desired result.
Remark 3.9
We know that(44)
(44)
almost everywhere in , where
,
for
.
Lemma 3.10
In the sense of distributions on , it holds that
(45)
(45)
Proof
Using Lemmas 3.5–3.8, (52) holds by sending in (9).
Lemma 3.11
For any with
, it has
(46)
(46)
in the sense of distributions on .
Proof
Let be a family of mollifiers defined on
. Defined
. The notation
is the convolution with respect to the x variable. Multiplying (52) by
, it has
(47)
(47)
and(48)
(48)
Using the boundedness of ,
and letting
in the above two equations, we obtain (53).
Following the ideas in Xin and Zhang (Citation2000), in next section we hope to improve the weak convergence of in (43) to strong convergence, and then we have an existence result for problem (4). Since the measure
, we will prove that if the measure is zero initially, then it will continue to be zero at all times
.
4. Strong convergence of ![](//:0)
![](//:0)
Lemma 4.1
(see Coclite et al., Citation2005) Assume . It holds that
(49)
(49)
Lemma 4.2
(see Coclite et al., Citation2005) If , for each
, it has
(50)
(50)
where(51)
(51)
and ,
for
.
Lemma 4.3
(see Coclite et al., Citation2005) Let . Then for each
(52)
(52)
Lemma 4.4
For almost all , it holds that
(53)
(53)
Proof
For an arbitrary . Using (50) minus (53), and the entropy
results in
(54)
(54)
Since is increasing and
, from (45), we have
(55)
(55)
It follows from Lemma 4.3 that(56)
(56)
From (61)–(63), we obtain the following result(57)
(57)
Integrating the resultant inequality over yields
(58)
(58)
Using Lemma 4.2, we complete the proof.
Lemma 4.5
For almost all , it holds that
(59)
(59)
Proof
Let . Subtracting (53) from (50) and using entropy
, we deduce
(60)
(60)
Since , we get
(61)
(61)
It follows from Lemma 4.3 that(62)
(62)
Using (35), we can find sufficiently large such that
. Let
. Applying Lemma 4.2 gives rise to
(63)
(63)
In , it has
(64)
(64)
Substituting (68) and (69) into (67) gives(65)
(65)
Integrating the above inequality over , by (71), we obtain
(66)
(66)
Lemma 4.6
It holds that
almost everywhere in .
Proof
It follows from Lemma 4.3 that(67)
(67)
From (60) and (74) , we have(68)
(68)
where we used the identity .
Combining (66) with (75) gets(69)
(69)
In fact, for , there exists a constant
, depending only on
and T such that
From Lemma 4.3, it has(70)
(70)
Since the map is convex and concave, we get
(71)
(71)
Therefore,(72)
(72)
Choosing M large enough,(73)
(73)
Hence, from (76) and (80), we obtain(74)
(74)
For , we conclude from Gronwall’inequality and Lemma 4.1 and 4.2 that
(75)
(75)
By the Fatou lemma, sending , we obtain
(76)
(76)
which completes the proof.
5. Proof of main theorem
Proof of Theorem 2.2. From (8), (10), 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(77)
(77)
Using (84) and Lemmas 3.5 and 3.6, we know that u is a distributional solution to problem (1). In addition, Inequality (6) is deduced from Lemma 3.4. Then the proof of Theorem 2.2 is finished.
Acknowledgements
The author thanks the referees for their valuable comments and suggestions.
Additional information
Funding
Notes on contributors
Ying Wang
Ying Wang is a senior lecturer in the college of Science at Sichuan University of Science and Engineering, Zigong city, Sichuan province, China. She has 9 years’ teaching experience. Her research interests are blow-up theory of Partial Differential Equation and exact travelling wave solutions for Partial Differential Equation. She has published some research articles in reputed international journals.
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