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Global weak solutions for a generalized Dullin-Gottwald-Holm equation in the space
Boundary Value Problems volume 2014, Article number: 203 (2014)
The existence of global weak solutions of the Cauchy problem for a generalized Dullin-Gottwald-Holm equation is established under the assumption that the initial value merely lies in the space . The limit of the viscous approximation for the equation is used to prove the global existence in the space . The elements in our study include a one-sided super bound estimate and a space-time higher-norm estimate on the first order derivative of the solution with respect to the space variable.
MSC: 35Q35, 35Q51.
Dullin, Gottwald and Holm  investigated the following equation for a unidirectional water wave:
where is the fluid velocity, , , the constants and are squares of length scales, and is the linear wave speed for undisturbed water resting at spatial infinity (see ). The Dullin, Gottwald and Holm equation (1) was derived through an asymptotic expansion from the Hamiltonian of Euler’s equation in the shallow water regime. It possesses bi-Hamiltonian and has a Lax pair formulation , . The equation is an integrable system and contains both the Korteweg-de Vries and Camassa-Holm equations ,  as limiting cases.
Extensive research has been carried out to study various dynamic properties of the Dullin, Gottwald and Holm model (DGH). Tang and Yang  found general explicit expressions of the two wave solutions for (1) by using bifurcation phase portraits of the traveling wave system. Mustafa  studied the local existence and uniqueness of solutions for the DGH equation with continuously differentiable periodic initial data. Zhou  found the best constants for two convolution problems on the unit circle via a variational method, and then applied the best constants on a nonlinear integrable shallow water equation (the Dullin, Gottwald and Holm equation) and obtained sufficient conditions required on the initial data to guarantee a finite time singularity formation for the corresponding solutions. Zhou and Guo  investigated the persistence properties of the strong solutions and infinite propagation speed for the DGH model. The existence of global weak solutions to (1) is proved by Zhang and Yin  under certain conditions imposed on the initial value. In , Tian, Gui and Liu established the global well-posedness of strong solution with provided that the initial data satisfies certain positive conditions. The blow-up of solutions for the DGH equation was also discussed in  and it was established that, similarly to the Camassa-Holm equation, singularities can arise only in the form of wave breaking, namely, the solution remains bounded but its slope becomes unbounded in finite time –). Mustafa  used the mathematical transform , , to reduce DGH (1) to a classical Camassa-Holm equation. Namely, satisfies the Camassa-Holm equation. Mustafa  applied the approaches in Bressan and Constantin  to establish the existence of global conservative solution with constant energy of provided that , and then obtained many meaningful conclusions for . As we know, is not equal to and cannot derive . In this paper, we only assume to establish the existence of global weak solutions to a generalized Dullin-Gottwald-Holm equation in the space .
In fact, we are interested in the Cauchy problem for the nonlinear model
To link with previous works in the field of study, we review here several works on the global weak solution for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for the global weak solutions of the Camassa-Holm model have been proved by Constantin and Escher  and Danchin ,  by assuming that the initial data satisfy the sign condition. Xin and Zhang  established the global existence of the weak solution for the Camassa-Holm equation in the energy space without imposing any sign conditions on the initial value, and the uniqueness of the weak solution was then obtained under certain conditions on the solution . Coclite et al. employed the analysis presented in ,  and 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 the initial value in the space was established in . Under the sign condition for the initial value, Yin and Lai  proved the existence and uniqueness of a global weak solution for a nonlinear shallow water equation, which includes the Camassa-Holm and Degasperis-Procesi equations as special cases. The existence of global weak solutions for a weakly dissipative Camassa-Holm equation was established in Lai et al..
The aim of this work is to study the existence of global weak solutions for the generalized Dullin-Gottwald-Holm equation (2) in the space under the assumption . The key elements in our analysis are that we establish a one-sided upper bound and space-time higher-norm estimates on the first order derivatives of the solution. The limit of viscous approximations for the equation is applied to establish the existence of the global weak solution. Here we should mention that our assumption has never been used as a unique condition to prove the global existence of weak solutions for DGH equation (1) or the generalized Dullin-Gottwald-Holm equation (2) in the literature.
Here we state that the ideas to prove our main result come from those presented in  (also see ). We need to show that the derivative (see (17)), which is only weakly compact, converges strongly. Namely, the strong convergence of is necessary to be established if we want to send ε to zero in the viscous problem (11). One of key factors, which is employed to prove that weak convergence is equal to strong convergence, is the higher integrability estimate (18) in Section 3. It means that the weak limit of does not contain singular measures.
The rest of this paper is organized as follows. The main result is given in Section 2. In Section 3, we present the viscous problem and give a corresponding well-posedness result. An upper bound, a higher integrability estimate and some 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 (2) is proved.
2 Main result
Consider the Cauchy problem for (2)
which is equivalent to
where the operator . For any , we have
For simplicity, throughout this article, we assume and let c denote any positive constant which is independent of parameter ε.
A continuous function is said to be a global weak solution to the Cauchy problem (3) if
satisfies (3) in the sense of distributions and takes on the initial value pointwise.
Now we illustrate the main result of this paper as follows.
There exists a positive constant depending on and the coefficients of (2) such that the following one-sided norm estimate on the first order spatial derivative holds:(7)
Let , , and , . Then there exists a positive constant depending only on , δ, T, a, b, and the coefficients of (2) such that the following space higher integrability estimate holds:(8)
3 Viscous approximations
where c is independent of parameter ε.
The existence of a weak solution to the Cauchy problem (4) will be established by proving compactness of a sequence of smooth functions solving the following viscous problem:
Now start our analysis by establishing the following well-posedness result for problem (11).
Provided that. Then for any, there exists a unique solutionto the Cauchy problem (11). Moreover, for any, we have
where c is a constant independent of ε.
We know that the first equation in system (11) is equivalent to the form
from which we derive
which completes the proof. □
From Lemma 3.1, we have
where c is a constant independent of ε.
Differentiating the first equation of problem (11) with respect to x and writing , we obtain
Let, , and, . Then there exists a positive constantdepending only on, γ, T, a, b, and the coefficients of (2), but independent of ε, such that the space higher integrability estimate holds
whereis the unique solution of problem (11).
Considering the map , , , and observing that
Differentiating the first equation of problem (11) with respect to x and writing and for simplicity, we obtain
Multiplying (22) by , using the chain rule and integrating over , we have
From (21), we get
Integration by parts gives rise to
Using (20) and Lemma 3.1, we have
which results in
There exists a positive constant c depending only onand the coefficients of (2) such that
whereis the unique solution of system (11).
For simplicity, setting , we have
Similar to the proof of (30), we have
The Parseval inequality shows that
where is the Fourier transform of with respect to x.
from which we obtain
Assume thatis the unique solution of (11). There exists a positive constant c depending only onand the coefficients of (2) such that the following one-sidednorm estimate on the first order spatial derivative holds:
Let be a supersolution of (51) associated with the initial value and satisfy
From the comparison principle for parabolic equations, we get
Letting , we have . From the comparison principle for ordinary differential equations, we get for all . Therefore, the estimate (50) is proved. □
There exist a sequenceconverging to zero and a functionsuch that, for each, we have
whereis the unique solution of (11).
For fixed , using Lemmas 3.1 and 3.3 and
where depends on T. Hence is uniformly bounded in
and (54) follows.
Observe that, for each , ,
Moreover, is uniformly bounded in and . Then (55) is valid. □
For an arbitrary, there exist a sequenceconverging to zero and a functionsuch thatinand for each
Throughout this paper we use overbars to denote weak limits (the space in which these weak limits are taken is with ).
There exist a sequenceconverging to zero and two functions, such that
for eachand. Moreover,
Equations (59) and (60) are direct consequences of Lemmas 3.1 and 3.2. Inequality (61) is valid because of the weak convergence in (60). Finally, (62) is a consequence of the definition of , Lemma 3.5, and (59). □
In the following, for notational convenience, we replace the sequence , and by , and , respectively.
Using (59), we conclude that for any convex function with being bounded and Lipschitz continuous on R and for any , we get
Multiplying (17) by yields
For any convexwithbeing bounded and Lipschitz continuous on R, we have
in the sense of distributions on. Hereanddenote the weak limits ofandin, , respectively.
almost everywhere in , where for . From Lemma 3.4 and (59), we have
where c is a constant depending only on and the coefficients of (2).
In the sense of distributions on, we have
The next lemma contains a generalized formulation of (69).
For anywith, we have
in the sense of distributions on.
Let be a family of mollifiers defined on R. Denote where the ⋆ is the convolution with respect to x variable. Multiplying (69) by yields
Using the boundedness of η, , and letting in the above two equations, we obtain (70). □
4 Strong convergence of and proof of main result
Following the work of  or , in this section we proceed to improve the weak convergence of in (59) to strong convergence, and then we establish a global existence result for problem (4). We will derive a ‘transport equation’ for the evolution of the defect measure . Namely, we will prove that if the measure is zero initially, then it will continue to be zero at all later times .
Assume. We have
If, for each, we have
and, , .
Let. Then for each
Assume. Then for almost all
By the increasing property of , from (61), we have
It follows from Lemma 4.3 that
In view of Remark 3.1, let . Applying (68) gives rise to
In , one has
Integrating the resultant inequality over yields
for almost all . Letting and using Lemma 4.2, we complete the proof. □
For anyand, we have
Since and , we get
Using Remark 3.1 and Lemma 4.3 yields
Integrating the above inequality over , we obtain
It follows from Lemma 4.3 that
Using Remark 3.1 and (91), we have
Applying the identity , we obtain (84). □
Applying Lemmas 4.4 and 4.5 gives rise to
From Lemma 3.6, we know that there exists a constant , depending only on , such that
Using Remark 3.1 and Lemma 4.3 yields
Thus, by the convexity of the map , we get
Using (95) one derives
Since is concave and choosing M large enough, we have
Using the Gronwall inequality and Lemmas 4.1 and 4.2, for each , we have
Applying the Fatou lemma, Remark 3.1, (61) and letting , we obtain
which completes the proof. □
Proof of the main result
Using Lemmas 3.1 and 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
From Lemma 3.5, (58), and (102), we know that u is a distributional solution to problem (3). In addition, inequalities (7) and (8) are deduced from Lemmas 3.2 and 3.4. The proof of the main result is completed. □
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This work is supported by National Natural Science Foundation of China (11471263).
The authors declare that they have no competing interests.
The article is a joint work of two authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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Lai, S., Wu, M. Global weak solutions for a generalized Dullin-Gottwald-Holm equation in the space . Bound Value Probl 2014, 203 (2014). https://doi.org/10.1186/s13661-014-0203-6
- global weak solution
- the Dullin-Gottwald-Holm equation
- viscous approximation