Abstract
The existence of global weak solutions to the Cauchy problem for a weakly dissipative
CamassaHolm equation is established in the space
MSC: 35G25, 35L05.
Keywords:
global weak solution; CamassaHolm type equation; existence1 Introduction
In this work, we investigate the Cauchy problem for the nonlinear model
where
To link with previous works, we review several works on global weak solutions for
the CamassaHolm and DegasperisProcesi equations. The existence and uniqueness results
for global weak solutions of the standard CamassaHolm equation have been proved by
Constantin and Escher [6], Constantin and Molinet [7] and Danchin [8,9] under the sign condition imposing on the initial value. Xin and Zhang [10] established the global existence of a weak solution for the CamassaHolm equation
in the energy space
The objective of this work is to study the existence of global weak solutions for
the Eq. (1) in the space
The rest of this paper is as follows. The main result is given in Section 2. In Section 3, we present a viscous problem of Eq. (1) and give a corresponding wellposedness 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 the existence of Eq. (1) is proved.
2 Main result
Consider the Cauchy problem for Eq. (1)
which is equivalent to
where the operator
In fact, as proved in [13], problem (2) satisfies the following conservation law:
Now we introduce the definition of a weak solution to Cauchy problem (2) or (3).
Definition 1 A continuous function
(i)
(ii)
(iii)
The main result of this paper is stated as follows.
Theorem 1 Assume
(a) There exists a positive constant
(b) Let
3 Viscous approximations
Defining
and setting the mollifier
The existence of a weak solution to Cauchy problem (3) will be established by proving
the compactness of a sequence of smooth functions
Now we start our analysis by establishing the following wellposedness result for problem (9).
Lemma 3.1Provided that
or
Proof For any
We know that the first equation in system (9) is equivalent to the form
from which we derive that
which completes the proof. □
From Lemma 3.1 and (8), we have
Differentiating the first equation of problem (9) with respect to x and writing
Lemma 3.2Let
where
The proof is similar to that of Proposition 3.2 presented in Xin and Zhang [10] (also see Coclite et al.[20]). Here we omit it.
Lemma 3.3There exists a positive constantCdepending only on
where
Due to strong similarities with the proof of Lemma 5.1 presented in Coclite et al.[20], we do not prove Lemma 3.3 here.
Lemma 3.4Assume that
Proof From (15) and Lemma 3.3, we know that there exists a positive constant C depending only on
Let
where
Using (14) and
where
Letting
Lemma 3.5For
where
Lemma 3.6There exists a sequence
The proofs of Lemmas 3.5 and 3.6 are similar to those of Lemmas 5.2 and 5.3 in [20]. Here we omit their proofs.
Throughout this paper, we use overbars to denote weak limits (the space in which these
weak limits are taken is
Lemma 3.7There exists a sequence
for each
and
Proof (28) and (29) are a direct consequence of Lemmas 3.1 and 3.2. Inequality (30) is
valid because of the weak convergence in (29). Finally, (31) is a consequence of the
definition of
In the following, for notational convenience, we replace the sequence
Using (28), we conclude that for any convex function
Multiplying Eq. (15) by
Lemma 3.8For any convex
in the sense of distributions on
Proof In (34), by the convexity of η, (14), Lemmas 3.5, 3.6 and 3.7, taking limit for
Remark 3.9 From (28) and (29), we know that
almost everywhere in
where C is a constant depending only on
Lemma 3.10In the sense of distributions on
Proof Using (15), Lemmas 3.5 and 3.6, (28), (29) and (31), the conclusion (38) holds by
taking limit for
The next lemma contains a generalized formulation of (38).
Lemma 3.11For any
in the sense of distributions on
Proof Let
and
Using the boundedness of η,
4 Strong convergence of
q
ε
Now, we will prove the strong convergence result, i.e.,
which is one of key statements to derive that
Lemma 4.1Assume
Lemma 4.2If
where
and
Lemma 4.3Let
The proofs of Lemmas 4.1, 4.2 and 4.3 can be found in [10] or [20].
Lemma 4.4Assume
Lemma 4.5For any
We do not provide the proofs of Lemmas 4.4 and 4.5 since they are similar to those of Lemmas 6.4 and 6.5 in Coclite et al.[20].
Lemma 4.6Assume
Proof
Applying Lemmas 4.4 and 4.5 gives rise to
From Lemma 3.6, we know that there exists a constant
By Remark 3.9 and Lemma 4.3, one has
Thus, by the convexity of the map
Using (51) derives
Since
Then, from (50) and (55), we have
By using the Gronwall inequality, for each
By the Fatou lemma, Remark 3.9 and (30), letting
which completes the proof. □
Proof of the main result Using (8), (10) and Lemma 3.5, we know that the conditions (i) and (ii) in Definition 1 are satisfied. We have to verify (iii). Due to Lemma 4.2 and Lemma 4.6, we have
From Lemma 3.5, (27) and (58), we know that u is a distributional solution to problem (3). In addition, inequalities (5) and (6) are deduced from Lemmas 3.2 and 3.4. The proof of Theorem 1 is completed. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The article is a joint work of three authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgements
Thanks are given to referees whose comments and suggestions were very helpful for revising our paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).
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