Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 285040, 15 pages

http://dx.doi.org/10.1155/2011/285040

## The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation

Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

Received 17 January 2011; Revised 3 August 2011; Accepted 8 August 2011

Academic Editor: Yuri V.Β Rogovchenko

Copyright Β© 2011 Shaoyong Lai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in space with . In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space with is developed.

#### 1. Introduction

Camassa and Holm [1] used the Hamiltonian method to derive a completely integrable wave equation by retaining two terms that are usually neglected in the small amplitude, shallow water limit. Its alternative derivation as a model for water waves can be found in Constantin and Lannes [2] and Johnson [3]. Equation (1.1) also models wave current interaction [4], while Dai [5] derived it as a model in elasticity (see Constantin and Strauss [6]). Moreover, it was pointed out in Lakshmanan [7] that the Camassa-Holm equation (1.1) could be relevant to the modeling of tsunami waves (see Constantin and Johnson [8]).

In fact, a huge amount of work has been carried out to investigate the dynamic properties of (1.1). For , (1.1) has traveling wave solutions of the form , called peakons, which capture the main feature of the exact traveling wave solutions of greatest height of the governing equations (see [9β11]). For , its solitary waves are stable solitons [6, 11]. It was shown in [12β14] that the inverse spectral or scattering approach was a powerful tool to handle Camassa-Holm equation. Equation (1.1) is a completely integrable infinite-dimensional Hamiltonian system (in the sense that for a large class of initial data, the flow is equivalent to a linear flow at constant speed [15]). It should be emphasized that (1.1) gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group (see [16, 17]), and this geometric illustration leads to a proof that the Least Action Principle holds. It is worthwhile to mention that Xin and Zhang [18] proved that the global existence of the weak solution in the energy space without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under some conditions on the solution [19]. Coclite et al. [20] extended the analysis presented in [18, 19] and obtained many useful dynamic properties to other equations (also see [21β24]). Li and Olver [25] established the local well-posedness in the Sobolev space with for (1.1) and gave conditions on the initial data that lead to finite time blowup of certain solutions. It was shown in Constantin and Escher [26] that the blowup occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. After wave breaking, the solution can be continued uniquely either as a global conservative weak solution [21] or a global dissipative solution [22]. For peakons, these possibilities are explicitly illustrated in the paper [27]. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water models, the reader is referred to [10, 28β32] and the references therein.

In this paper, motivated by the work in [25, 33], we study the following generalized Camassa-Holm equation where and are natural numbers, and , , and are arbitrary constants. Obviously, (1.2) reduces to (1.1) if we set , , and . Actually, Wu and Yin [34] consider a nonlinearly dissipative Camassa-Holm equation which includes a nonlinearly dissipative term , where is a differential operator or a quasidifferential operator. Therefore, we can regard the term as a nonlinearly dissipative term for the dissipative Camassa-Holm equation (1.2).

Due to the term in (1.2), the conservation laws in previous works [10, 25] for (1.1) lose their powers to obtain some bounded estimates of the solution for (1.2). A new conservation law different from those presented in [10, 25] will be established to prove the local existence and uniqueness of the solution to (2.3) subject to initial value with . We should address that all the generalized versions of the Camassa-Holm equation in previous works (see [17, 25, 34]) do not involve the nonlinear term . Lai and Wu [33] only studied a generalized Camassa-Holm equation in the case where and is an odd number. Namely, (1.2) with and arbitrary positive integer was not investigated in [33].

The main tasks of this paper are two-fold. Firstly, by using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for (1.2) with any and arbitrary positive integer in space with . Secondly, it is shown that the existence of weak solutions in lower order Sobolev space with . The ideas of proving the second result come from those presented in Li and Olver [25].

#### 2. Main Results

Firstly, we give some notation.

The space of all infinitely differentiable functions with compact support in is denoted by . is the space of all measurable functions such that . We define with the standard norm . For any real number , denotes the Sobolev space with the norm defined by where .

For and nonnegative number , denotes the Frechet space of all continuous -valued functions on . We set .

In order to study the existence of solutions for (1.2), we consider its Cauchy problem in the form which is equivalent to

Now, we state our main results.

Theorem 2.1. *Let with . Then problem (2.2) or problem (2.3) has a unique solution where depends on .*

Theorem 2.2. *Suppose that with and . Then there exists a such that (1.2) subject to initial value has a weak solution in the sense of distribution and .*

#### 3. Local Well-Posedness

We consider the abstract quasilinear evolution equation Let and be Hilbert spaces such that is continuously and densely embedded in , and let be a topological isomorphism. Let be the space of all bounded linear operators from to . If , we denote this space by . We state the following conditions in which , and are constants depending on .(i) for with and (i.e., is quasi--accretive), uniformly on bounded sets in .(ii), where is bounded, uniformly on bounded sets in . Moreover, (iii) extends to a map from into is bounded on bounded sets in , and satisfies

*Kato Theorem (see [35])*

Assume that (i), (ii), and (iii) hold. If , there is a maximal depending only on , and a unique solution to problem (3.1) such that
Moreover, the map is a continuous map from to the space

For problem (2.3), we set , , , , and . In order to prove Theorem 2.1, we only need to check that and satisfy assumptions (i)β(iii).

Lemma 3.1. *The operator with , belongs to .*

Lemma 3.2. *Let with and . Then for all . Moreover,
*

Lemma 3.3. *For , and , it holds that for and
*

Proofs of the above Lemmas 3.1β3.3 can be found in [29] or [31].

Lemma 3.4 (see [35]). *Let and be real numbers such that . Then
*

Lemma 3.5. *Let with , then is bounded on bounded sets in and satisfies
*

*Proof. *Using the algebra property of the space with , we have
from which we obtain (3.11).

Applying Lemma 3.4, , and , we get
which completes the proof of (3.12).

*Proof of Theorem 2.1. *Using the Kato Theorem, Lemmas 3.1β3.3, and 3.5, we know that system (2.2) or problem (2.3) has a unique solution

#### 4. Existence of Weak Solutions

For , using the first equation of system (2.2) derives from which we have the conservation law

Lemma 4.1 (Kato and Ponce [36]). *If , then is an algebra. Moreover,
**
where is a constant depending only on .*

Lemma 4.2 (Kato and Ponce [36]). *Let . If and , then
*

Lemma 4.3. *Let and the function is a solution of problem (2.2) and the initial data . Then the following inequality holds
**For , there is a constant , which only depends on , , , , and , such that
**For , there is a constant , which only depends on , and , such that
*

*Proof. *Using and (4.2) derives (4.5).

Using and the Parseval equality gives rise to

For , applying to both sides of the first equation of system (2.3) and integrating with respect to by parts, we have the identity
We will estimate the terms on the right-hand side of (4.9) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2, we have
Using the above estimate to the second term yields
For the third term, using the Cauchy-Schwartz inequality and Lemma 4.1, we obtain

For the last term in (4.9), using Lemma 4.1 repeatedly results in
It follows from (4.9) to (4.13) that there exists a constant depending only on and the coefficients of (1.2) such that
Integrating both sides of the above inequality with respect to results in inequality (4.6).

To estimate the norm of , we apply the operator to both sides of the first equation of system (2.3) to obtain the equation
Applying to both sides of (4.15) for gives rise to
For the right-hand side of (4.16), we have
Since
using Lemma 4.1, and , we have
Using the Cauchy-Schwartz inequality and Lemma 4.1 yields
Applying (4.17)β(4.20) into (4.16) yields the inequality
This completes the proof of Lemma 4.3.

Defining and setting with and , we know that for any and .

It follows from Theorem 2.1 that for each the Cauchy problem has a unique solution .

Lemma 4.4. *Under the assumptions of problem (4.23), the following estimates hold for any with and **
where is a constant independent of .*

The proof of this Lemma can be found in Lai and Wu [33].

Lemma 4.5. *If with such that . Let be defined as in system (4.23). Then there exist two positive constants and , which are independent of , such that the solution of problem (4.23) satisfies for any .*

*Proof. *Using notation and differentiating both sides of the first equation of problem (4.23) or (4.15) with respect to give rise to
Letting be an integer and multiplying the above equation by and then integrating the resulting equation with respect to yield the equality
Applying the HΓΆlder's inequality yields
or
where
Since as for any , integrating both sides of the inequality (4.28) with respect to and taking the limit as result in the estimate
Using the algebra property of with yields ( means that there exists a sufficiently small such that )
in which Lemma 4.3 is used. Therefore, we get

From (4.30) and (4.32), one has

From Lemma 4.4, it follows from the contraction mapping principle that there is a such that the equation
has a unique solution . Using the Theorem presented at page 51 in [25] or Theorem 2 in Section 1.1 presented in [37] yields that there are constants and independent of such that for arbitrary , which leads to the conclusion of Lemma 4.5.

Using Lemmas 4.3 and 4.5, notation and Gronwall's inequality results in the inequalities
where , and depends on . It follows from Aubin's compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively. Moreover, for any real number , is convergent to the function strongly in the space for and converges to strongly in the space for . Thus, we can prove the existence of a weak solution to (2.2).

*Proof of Theorem 2.2. *From Lemma 4.5, we know that is bounded in the space . Thus, the sequences and are weakly convergent to and in for any , respectively. Therefore, satisfies the equation
with and . Since is a separable Banach space and is a bounded sequence in the dual space of , there exists a subsequence of , still denoted by , weakly star convergent to a function in . It derives from the weakly convergent to in that almost everywhere. Thus, we obtain .

#### Acknowledgments

The author is very grateful to the reviewers for their helpful and valuable comments, which have led to a meaningful improvement of the paper. This work is supported by the Key Project of Chinese Ministry of Education (109140).

#### References

- R. Camassa and D. D. Holm, βAn integrable shallow water equation with peaked solitons,β
*Physical Review Letters*, vol. 71, no. 11, pp. 1661β1664, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin and D. Lannes, βThe hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,β
*Archive for Rational Mechanics and Analysis*, vol. 192, no. 1, pp. 165β186, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. S. Johnson, βCamassa-Holm, Korteweg-de Vries and related models for water waves,β
*Journal of Fluid Mechanics*, vol. 455, no. 1, pp. 63β82, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. S. Johnson, βThe Camassa-Holm equation for water waves moving over a shear flow,β
*Fluid Dynamics Research*, vol. 33, no. 1-2, pp. 97β111, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H.-H. Dai, βModel equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,β
*Acta Mechanica*, vol. 127, no. 1β4, pp. 193β207, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin and W. A. Strauss, βStability of a class of solitary waves in compressible elastic rods,β
*Physics Letters A*, vol. 270, no. 3-4, pp. 140β148, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Lakshmanan, βIntegrable nonlinear wave equations and possible connections to tsunami dynamics,β in
*Tsunami and Nonlinear Waves*, A. Kundu, Ed., pp. 31β49, Springer, Berlin, Germany, 2007. View at Publisher Β· View at Google Scholar - A. Constantin and R. S. Johnson, βPropagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,β
*Fluid Dynamics Research*, vol. 40, no. 3, pp. 175β211, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin, βThe trajectories of particles in Stokes waves,β
*Inventiones Mathematicae*, vol. 166, no. 3, pp. 523β535, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin and J. Escher, βParticle trajectories in solitary water waves,β
*Bulletin of American Mathematical Society*, vol. 44, no. 3, pp. 423β431, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin and W. A. Strauss, βStability of peakons,β
*Communications on Pure and Applied Mathematics*, vol. 53, no. 5, pp. 603β610, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin and H. P. McKean, βA shallow water equation on the circle,β
*Communications on Pure and Applied Mathematics*, vol. 52, no. 8, pp. 949β982, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin, βOn the inverse spectral problem for the Camassa-Holm equation,β
*Journal of Functional Analysis*, vol. 155, no. 2, pp. 352β363, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin, V. S. Gerdjikov, and R. I. Ivanov, βInverse scattering transform for the Camassa-Holm equation,β
*Inverse Problems*, vol. 22, no. 6, pp. 2197β2207, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. P. McKean, βIntegrable systems and algebraic curves,β in
*Global Analysis*, vol. 755 of*Lecture Notes in Mathematics*, pp. 83β200, Springer, Berlin, Germany, 1979. View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin and B. Kolev, βGeodesic flow on the diffeomorphism group of the circle,β
*Commentarii Mathematici Helvetici*, vol. 78, no. 4, pp. 787β804, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. MisioΕek, βA shallow water equation as a geodesic flow on the Bott-Virasoro group,β
*Journal of Geometry and Physics*, vol. 24, no. 3, pp. 203β208, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Xin and P. Zhang, βOn the weak solutions to a shallow water equation,β
*Communications on Pure and Applied Mathematics*, vol. 53, no. 11, pp. 1411β1433, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Xin and P. Zhang, βOn the uniqueness and large time behavior of the weak solutions to a shallow water equation,β
*Communications in Partial Differential Equations*, vol. 27, no. 9-10, pp. 1815β1844, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. M. Coclite, H. Holden, and K. H. Karlsen, βGlobal weak solutions to a generalized hyperelastic-rod wave equation,β
*SIAM Journal on Mathematical Analysis*, vol. 37, no. 4, pp. 1044β1069, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Bressan and A. Constantin, βGlobal conservative solutions of the Camassa-Holm equation,β
*Archive for Rational Mechanics and Analysis*, vol. 183, no. 2, pp. 215β239, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Bressan and A. Constantin, βGlobal dissipative solutions of the Camassa-Holm equation,β
*Analysis and Applications*, vol. 5, no. 1, pp. 1β27, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Holden and X. Raynaud, βDissipative solutions for the Camassa-Holm equation,β
*Discrete and Continuous Dynamical Systems A*, vol. 24, no. 4, pp. 1047β1112, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Holden and X. Raynaud, βGlobal conservative solutions of the Camassa-Holm equationβa Lagrangian point of view,β
*Communications in Partial Differential Equations*, vol. 32, no. 10β12, pp. 1511β1549, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. A. Li and P. J. Olver, βWell-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,β
*Journal of Differential Equations*, vol. 162, no. 1, pp. 27β63, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Constantin and J. Escher, βWave breaking for nonlinear nonlocal shallow water equations,β
*Acta Mathematica*, vol. 181, no. 2, pp. 229β243, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. Beals, D. H. Sattinger, and J. Szmigielski, βMulti-peakons and a theorem of Stieltjes,β
*Inverse Problems*, vol. 15, no. 1, pp. L1βL4, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Henry, βPersistence properties for a family of nonlinear partial differential equations,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 4, pp. 1565β1573, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. RodrΓguez-Blanco, βOn the Cauchy problem for the Camassa-Holm equation,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 46, no. 3, pp. 309β327, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Yin, βOn the blow-up scenario for the generalized Camassa-Holm equation,β
*Communications in Partial Differential Equations*, vol. 29, no. 5-6, pp. 867β877, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Yin, βOn the Cauchy problem for an integrable equation with peakon solutions,β
*Illinois Journal of Mathematics*, vol. 47, no. 3, pp. 649β666, 2003. View at Google Scholar Β· View at Zentralblatt MATH - Y. Zhou, βWave breaking for a periodic shallow water equation,β
*Journal of Mathematical Analysis and Applications*, vol. 290, no. 2, pp. 591β604, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Lai and Y. Wu, βThe local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,β
*Journal of Differential Equations*, vol. 248, no. 8, pp. 2038β2063, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Wu and Z. Yin, βGlobal existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation,β
*Journal of Differential Equations*, vol. 246, no. 11, pp. 4309β4321, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Kato, βQuasi-linear equations of evolution, with applications to partial differential equations,β in
*Spectral Theory and Differential Equations*, vol. 448 of*Lecture Notes in Mathematics*, pp. 25β70, Springer, Berlin, Germany, 1975. View at Google Scholar Β· View at Zentralblatt MATH - T. Kato and G. Ponce, βCommutator estimates and the Euler and Navier-Stokes equations,β
*Communications on Pure and Applied Mathematics*, vol. 41, no. 7, pp. 891β907, 1988. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - W. Walter,
*Differential and Integral Inequalities*, vol. 55 of*Ergebnisse der Mathematik und Ihrer Grenzgebiete*, Springer, New York, NY, USA, 1970.