We discuss blowup phenomena for a modified two-component Dullin-Gottwald-Holm shallow water system. In this paper, some new blowup criteria of strong solutions involving the density and suitable integral form of the momentum are established.

1. Introduction

We consider the following two-component DGH type system:

()

where u = u(x, t), (x, t)∈(ℝ, ℝ⁺) denotes the velocity field, g is the downward constant acceleration of gravity in applications to shallow water waves, and

, where

is taken to be a constant. It is obvious that if ρ ≡ 0, then (1) reduces to the well-known Dullin-Gottwald-Holm equation [1] (DGH equation for short). There are some contributions to DGH equation concerning the well-posedness, scattering problem, blowup phenomenon, and so forth; see, for example, [2–5] and references therein. We find that (1) is expressed in terms of an averaged filtered density component

in analogy to the relation between momentum and velocity by setting

and the velocity component u. The idea is actually from the recent work [6]. Our modification breaks the structure of DGH2 system derived by following Ivanov’s approach [7] by the authors in [8]. The motivation of current research is stated as follows. From geometric point of view, (1) is the model for geodesic motion on the semidirect product Lie group of diffeomorphisms acting on densities, with respect to the H¹-norm of velocity u and the H¹-norm on filtered density. From a physical point of view, (1) admits wave breaking phenomena in finite time which attracts researchers’ interest. We also find that the H¹-norm of

is conserved with respect to time variable. This makes further different discussions on the singularities, unlike those for the DGH2 system or two-component Camassa-Holm system, possible. In the previous works [9–11] on the two-component Camassa-Holm equation and its modified version, blowup conditions were established in view of the negativity of initial velocity slope at some point; basically, the initial integral form of momentum is never involved. That is why we consider this kind of blowup condition in this paper. Precisely, we show the solutions blowup in finite time provided that the initial density and momentum satisfy certain sign conditions. To our knowledge, less results exist yet for the formation of singularities of (1) although the approaches we applied here are standard. The methods in previous works cannot be moved to this model parallelly. For convenience, let

and

; then the operator Λ can be expressed by its associated Green’s function G(x) = (1/2α)e^−|x/α| with

()

Using this identity, system (1) takes an equivalent form of a quasilinear evolution equation of hyperbolic type as follows:

()

The current paper is based on some results on the Camassa-Holm equation [12–19] and its two-component generalizations [20–27]. We investigate further formation of singularities of solutions to (3) with the case of g = 1 and α > 0, just for simplicity mathematically. This paper is organized as follows. In Section 2, we recall some preliminary results on the well-posedness and blowup scenario. In Section 3, the detailed blowup conditions are presented.

2. Preliminaries

In this section, for completeness, we recall some elementary results and skip their proofs since they are not the main concern of this work. For convenience, in what follows, we let λ = −γ/α² and 2κ = 2ω + γ/α².

We can apply Kato’s theory [28] to establish the following local well-posedness theorem for (3).

Theorem 1. Assume an initial data (u₀, v₀) ∈ H^s × H^s−1, s ≥ 5/2. Then there exists a maximal and a unique solution

()

of system (3). Moreover, the solution (u, v) depends continuously on the initial value (u₀, v₀), and the maximal time of existence T > 0 is independent of s.

The proof of Theorem 1 is similar to the one in [11]. Moreover, using the techniques in [11], one can get the criterion for finite time wave breaking to (3) as follows.

Theorem 2. Let (u₀, v₀) ∈ H^s × H^s−1 with s ≥ 5/2, and let T > 0 be the maximal time of existence of the solution (u, v) to (3) with initial data (u₀, v₀). Then the corresponding solution (u, v) blowsup in finite time if and only if

()

Lemma 3 (see [29].)Assume that a differentiable function y(t) satisfies

()

with constants C, K > 0. If the initial datum

, then the solution to (9) goes to −∞ before t tends to 1/(−Cy₀ + K/y₀).

Lemma 4 (see [19].)Suppose that Ψ(t) is twice continuously differential satisfying

()

Then Ψ blowsup in finite time. Moreover the blowup time can be estimated in terms of the initial datum as

()

We also need to introduce the standard particle trajectory method for later use. Consider now the following two initial value problems:

()

where u ∈ C¹([0, T), H^s−1) is the first component of the solution (u, v) to system (3) with initial data (u₀, v₀) ∈ H^s × H^s−1 (s ≥ 5/2), and T > 0 is the maximal time of existence. By direct computation, we have

()

Then,

()

which means that q_i(t, ·): ℝ → ℝ is a diffeomorphism of the line for every t ∈ [0, T). Consequently, the L^∞-norm of any function v(t, ·) is preserved under the family of the diffeomorphisms q_i(t, ·); that is,

()

Similarly,

()

3. Blowup Phenomenon

In this section, we show that blowup phenomenon is the only one way that singularity arises in smooth solutions. We start this section with the following useful lemma.

Lemma 5. Let X₀ = (u₀, v₀) ∈ H^s × H^s−1, s ≥ 2. T is assumed to be the maximal existence time of the solution X = (u, v) to system (3) corresponding to the initial data X₀. Then for all t ∈ [0, T), one has the following conservation law:

()

Proof. We will prove that E(t) is a conserved quantity with respect to time variable. Here we use the classical energy method. Multiplying the first equation in (3) by u(x, t) and integrating by parts, we obtain

()

Similarly, we have the following inequality for the second equation (3):

()

This implies that

()

Thus, we have

()

This completes the proof.

Using this conservation law, we obtain

()

where

()

Theorem 6. Suppose that X₀ = (u₀, v₀) ∈ H^s × H^s−1, s ≥ 5/2, ρ₀(x₀) = y₀(x₀) + κ = 0, and the initial data satisfies the following conditions:

(i)
ρ₀(x) ≥ 0 on (−∞, x₀) and ρ₀(x) ≤ 0 on (x₀, ∞),
(ii)
and ,

for some point x₀ ∈ ℝ. Then the solution to system (3) with the initial value X₀ blowsup in finite time.

Proof. Differentiating the first equation of (3) with respect to x, we obtain

()

Applying the relation

yields

()

From (23) we have

()

where we used the fact proved in [30] that

()

In order to arrive at our result, we need the following three claims.

Claim 1. y(q₁(x₀, t), t)) + κ = 0 for all t in its lifespan; q₁ is defined in (9).

It is worth noting the equivalent form of the first equation in (3) in what follows:

()

From the previous equation, we can get

()

Since q₂(x, ·) defined by (10) is a diffeomorphism of the line for any t ∈ [0, T), so there exists an x₃(t) ∈ ℝ such that

()

When t = 0, we have

()

Now we prove that ρ(t, q₁(t, x₀)) = 0. It is easy to get

()

Since

()

integrating the previous equation, we can obtain

()

thus we have

()

So we can get

()

then we have

()

Our claim is proved.

Claim 2. For any fixed t, for all x ∈ ℝ. For any fixed t, if x ≤ q₁(x₀, t), then

()

where the condition (i) is used. Similarly, for x ≥ q₁(x₀, t), we also have

()

So Claim 2 is proved. Consequently, we can obtain

()

Thus, one can get

()

Claim 3. for all t ≥ 0. Furthermore, u_x(q₁(x₀, t), t) < 0 is strictly decreasing.

Suppose that there exists a t₀ such that on [0, t₀) and . From the expression of u(x, t) in terms of y(x, t), we can rewrite u(x, t) + κ and u_x(x, t) as follows:

()

Letting

()

then

()

Integrating by parts, the first term of (42) yields

()

For the second term of (42), we have the following equation in the view of Claim 1:

()

Here we have used

()

Combining the previous equations together, and with the help of (38), (42) reads as

()

where Claim 2 and the inequality [30]

()

have been used. From the continuity property, we have

()

Similarly,

()

Thus, by continuity property,

()

Summarizing (48) and (50), we obtain

()

That is a contradiction. On the other hand, from the expression of u_x(x, t) in terms of y(x, t), we can easily get that u_x(q₁(x₀, t), t) < 0. So we complete the proof of Claim 3.

Furthermore, due to (46) and (49), we can obtain

()

Integrating (39) and then substituting it into the previous inequality, we have

()

Let

; then we can complete the proof with the help of Lemma 4.

Remark 7. We note that if the condition (i) is replaced by the following one:

(i^′) ρ₀(x) ≤ 0 on (−∞, x₀) and ρ₀(x) ≥ 0 on (x₀, ∞),

then Claim 2 also holds; that is, the theorem always holds with anyone of (i) and (i^′).

As a corollary of Theorem 6, we have the following.

Theorem 8. Suppose that X₀ = (u₀, v₀) ∈ H^s × H^s−1, s ≥ 5/2, and the initial data satisfies the following conditions:

(i)
ρ₀(x) ≥ 0 on (−∞, x₀) and ρ₀(x) ≤ 0 on (x₀, ∞) ( or ρ₀(x) ≤ 0 on (−∞, x₀) and ρ₀(x) ≥ 0 on (x₀, ∞)),
(ii)
,

for some point x₀ ∈ ℝ. Then the solution to system (3) with the initial value X₀ blowsup in finite time.

Proof. As shown in Theorem 6, condition (i) guarantees that for all x ∈ ℝ. Then,

()

where K > 0 is a constant. By setting φ(t) = u_x(q₁(x₀, t), t), we obtain

()

Applying Lemma 3, we have

()

when

()

This completes the proof.

Acknowledgments

This work was partially supported by Natural Science Foundation of China under Grant no. 11226172 and Zhejiang Provincial Natural Science Foundation of China under Grant nos. LQ12A01009 and LY12A01014.

References

1 Dullin H. R., Gottwald G. A., and Holm D. D., An integrable shallow water equation with linear and nonlinear dispersion, Physical Review Letters. (2001) 87, no. 19, 4501–4504, 2-s2.0-40749103220, 194501.
10.1103/PhysRevLett.87.194501
CAS Web of Science® Google Scholar
2 Liu Y., Global existence and blow-up solutions for a nonlinear shallow water equation, Mathematische Annalen. (2006) 335, no. 3, 717–735, https://doi.org/10.1007/s00208-006-0768-1, MR2221129, ZBL1102.35021.
10.1007/s00208-006-0768-1
Web of Science® Google Scholar
3 Tian L., Gui G., and Liu Y., On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation, Communications in Mathematical Physics. (2005) 257, no. 3, 667–701, https://doi.org/10.1007/s00220-005-1356-z, MR2164948, ZBL1080.76016.
10.1007/s00220-005-1356-z
Web of Science® Google Scholar
4 Zhou Y., Blow-up of solutions to the DGH equation, Journal of Functional Analysis. (2007) 250, no. 1, 227–248, https://doi.org/10.1016/j.jfa.2007.04.019, MR2345914, ZBL1124.35079.
10.1016/j.jfa.2007.04.019
Web of Science® Google Scholar
5 Zhou Y. and Guo Z., Blow up and propagation speed of solutions to the DGH equation, Discrete and Continuous Dynamical Systems B. (2009) 12, no. 3, 657–670, https://doi.org/10.3934/dcdsb.2009.12.657, MR2525163, ZBL1180.35473.
10.3934/dcdsb.2009.12.657
Web of Science® Google Scholar
6 Holm D. D., Ó Náraigh L., and Tronci C., Singular solutions of a modified two-component Camassa-Holm equation, Physical Review E. (2009) 79, no. 1, 13, 016601, https://doi.org/10.1103/PhysRevE.79.016601, MR2552212.
10.1103/PhysRevE.79.016601
Web of Science® Google Scholar
7 Constantin A. and Ivanov R. I., On an integrable two-component Camassa-Holm shallow water system, Physics Letters A. (2008) 372, no. 48, 7129–7132, https://doi.org/10.1016/j.physleta.2008.10.050, MR2474608, ZBL1227.76016.
10.1016/j.physleta.2008.10.050
CAS Web of Science® Google Scholar
8 Guo F., Gao H., and Liu Y., On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system, Journal of the London Mathematical Society. (2012) 86, no. 3, 810–834, https://doi.org/10.1112/jlms/jds035, MR3000831.
10.1112/jlms/jds035
Web of Science® Google Scholar
9 Chen R. M. and Liu Y., Wave breaking and global existence for a generalized two-component Camassa-Holm system, International Mathematics Research Notices. (2011) no. 6, 1381–1416, https://doi.org/10.1093/imrn/rnq118, MR2806508, ZBL1225.35197.
10.1093/imrn/rnq118
Web of Science® Google Scholar
10 Guan C. and Yin Z., Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, Journal of Differential Equations. (2010) 248, no. 8, 2003–2014, https://doi.org/10.1016/j.jde.2009.08.002, MR2595712, ZBL1190.35039.
10.1016/j.jde.2009.08.002
Web of Science® Google Scholar
11 Guan C., Karlsen K. H., and Yin Z., Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, 2010, 526, American Mathematical Society, Providence, RI, USA, 199–220, Contemporary Mathematics, https://doi.org/10.1090/conm/526/10382, MR2731993, ZBL1213.35133.
10.1090/conm/526/10382
Google Scholar
12 Camassa R. and Holm D. D., An integrable shallow water equation with peaked solitons, Physical Review Letters. (1993) 71, no. 11, 1661–1664, https://doi.org/10.1103/PhysRevLett.71.1661, MR1234453, ZBL0972.35521.
10.1103/PhysRevLett.71.1661
CAS PubMed Web of Science® Google Scholar
13 Constantin A. and Escher J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica. (1998) 181, no. 2, 229–243, https://doi.org/10.1007/BF02392586, MR1668586, ZBL0923.76025.
10.1007/BF02392586
Web of Science® Google Scholar
14 Constantin A. and Escher J., Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Communications on Pure and Applied Mathematics. (1998) 51, no. 5, 475–504, MR1604278, ZBL0934.35153.
10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
Web of Science® Google Scholar
15 Himonas A. A., Misiołek G., Ponce G., and Zhou Y., Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Communications in Mathematical Physics. (2007) 271, no. 2, 511–522, https://doi.org/10.1007/s00220-006-0172-4, MR2287915, ZBL1142.35078.
10.1007/s00220-006-0172-4
Web of Science® Google Scholar
16 Jiang Z. and Hakkaev S., Wave breaking and propagation speed for a class of one-dimensional shallow water equations, Abstract and Applied Analysis. (2011) 2011, 15, 2-s2.0-84555190034, https://doi.org/10.1155/2011/647368, 647368.
10.1155/2011/647368
Google Scholar
17 McKean H. P., Breakdown of a shallow water equation, The Asian Journal of Mathematics. (1998) 2, no. 4, 867–874, MR1734131, ZBL0959.35140.
10.4310/AJM.1998.v2.n4.a10
Google Scholar
18 Zhou Y., Wave breaking for a periodic shallow water equation, Journal of Mathematical Analysis and Applications. (2004) 290, no. 2, 591–604, https://doi.org/10.1016/j.jmaa.2003.10.017, MR2033045, ZBL1042.35060.
10.1016/j.jmaa.2003.10.017
Web of Science® Google Scholar
19 Zhou Y., On solutions to the Holm-Staley b-family of equations, Nonlinearity. (2010) 23, no. 2, 369–381, https://doi.org/10.1088/0951-7715/23/2/008, MR2578483.
10.1088/0951-7715/23/2/008
PubMed Web of Science® Google Scholar
20 Escher J., Lechtenfeld O., and Yin Z., Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete and Continuous Dynamical Systems A. (2007) 19, no. 3, 493–513, https://doi.org/10.3934/dcds.2007.19.493, MR2335761, ZBL1149.35307.
10.3934/dcds.2007.19.493
Web of Science® Google Scholar
21 Guo Z., Blow-up and global solutions to a new integrable model with two components, Journal of Mathematical Analysis and Applications. (2010) 372, no. 1, 316–327, https://doi.org/10.1016/j.jmaa.2010.06.046, MR2672529, ZBL1205.35045.
10.1016/j.jmaa.2010.06.046
Web of Science® Google Scholar
22 Guo Z. and Ni L., Persistence properties and unique continuation of solutions to a two-component Camassa-Holm equation, Mathematical Physics, Analysis and Geometry. (2011) 14, no. 2, 101–114, https://doi.org/10.1007/s11040-011-9089-z, MR2802456, ZBL1245.35108.
10.1007/s11040-011-9089-z
Web of Science® Google Scholar
23 Guo Z. and Zhou Y., On solutions to a two-component generalized Camassa-Holm equation, Studies in Applied Mathematics. (2010) 124, no. 3, 307–322, https://doi.org/10.1111/j.1467-9590.2009.00472.x, MR2655943, ZBL1189.35255.
10.1111/j.1467-9590.2009.00472.x
Web of Science® Google Scholar
24 Guo Z., Zhu M., and Ni L., Blow-up criteria of solutions to a modified two-component Camassa-Holm system, Nonlinear Analysis. Real World Applications. (2011) 12, no. 6, 3531–3540, https://doi.org/10.1016/j.nonrwa.2011.06.013, MR2832990, ZBL1231.35200.
10.1016/j.nonrwa.2011.06.013
Web of Science® Google Scholar
25 Zhang P. and Liu Y., Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, International Mathematics Research Notices. (2010) no. 11, 1981–2021, https://doi.org/10.1093/imrn/rnp211, MR2646352, ZBL1231.35184.
10.1093/imrn/rnp211
Web of Science® Google Scholar
26 Zhu M., Blow-up, global existence and persistence properties for the coupled Camassa-Holm equations, Mathematical Physics, Analysis and Geometry. (2011) 14, no. 3, 197–209, https://doi.org/10.1007/s11040-011-9094-2, MR2824603, ZBL1245.35115.
10.1007/s11040-011-9094-2
Web of Science® Google Scholar
27 Zhu M. and Xu J., On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system, Journal of Mathematical Analysis and Applications. (2012) 391, no. 2, 415–428, https://doi.org/10.1016/j.jmaa.2012.02.058, MR2903141.
10.1016/j.jmaa.2012.02.058
Web of Science® Google Scholar
28 Kato T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, 1983, 8, Academic Press, New York, NY, USA, 93–128, Advances in Mathematics, Supplementary Studies, MR759907, ZBL0549.34001.
Web of Science® Google Scholar
29 Zhou Y., Blow-up of solutions to a nonlinear dispersive rod equation, Calculus of Variations and Partial Differential Equations. (2006) 25, no. 1, 63–77, https://doi.org/10.1007/s00526-005-0358-1, MR2183855, ZBL1172.35504.
10.1007/s00526-005-0358-1
Web of Science® Google Scholar
30 Zhu M., Jin L., and Jiang Z., A new blow-up criterion for the DGH equation, Abstract and Applied Analysis. (2012) 2012, 10, 515948, https://doi.org/10.1155/2012/515948, MR2926884, ZBL1242.35198.
10.1155/2012/515948
Google Scholar

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Blowup Phenomena for a Modified Dullin-Gottwald-Holm Shallow Water System

Abstract

1. Introduction

2. Preliminaries

3. Blowup Phenomenon

Acknowledgments

References

References

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Blowup Phenomena for a Modified Dullin-Gottwald-Holm Shallow Water System

Abstract

1. Introduction

2. Preliminaries

3. Blowup Phenomenon

Acknowledgments

References

References

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