Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical ModelsView this Special Issue
Persistence Property and Estimate on Momentum Support for the Integrable Degasperis-Procesi Equation
It is shown that a strong solution of the Degasperis-Procesi equation possesses persistence property in the sense that the solution with algebraically decaying initial data and its spatial derivative must retain this property. Moreover, we give estimates of measure for the momentum support.
Recently, Degasperis and Procesi  consider the following family of third order dispersive conservation laws: where , , , , , and are real constants. Within this family, only three equations that satisfy asymptotic integrability condition up to third order are singled out, namely, the KdV equation the Camassa-Holm equation and a new equation (the Degasperis-Procesi equation, the DP equation, for simplicity) which can be written as (after rescaling) the dispersionless form  It is worth noting that in  both the Camassa-Holm and DP equations are derived as members of a one-parameter family of asymptotic shallow water approximations to the Euler equations: this is important because it shows that (after the addition of linear dispersion terms) both the Camassa-Holm and DP equations are physically relevant; otherwise the DP equation would be of purely theoretical interest.
When and in (1), we recover the Camassa-Holm equation derived physically by Camassa and Holm in  by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime, where represents the free surface above a flat bottom. There is also a geometric approach which is used to prove the least action principle holding for the Camassa-Holm equation, compared with . It is worth pointing out that a fundamental aspect of the Camassa-Holm equation, the fact that it is a completely integrable system, was shown in [5, 6]. Some satisfactory results have been obtained for this shallow water equation recently, we refer the readers to see [7–19].
Although, the DP equation (4) has a similar form to the Camassa-Holm equation and admits exact peakon solutions analogous to the Camassa-Holm peakons , these two equations are pretty different. The isospectral problem for equation (4) is while for Camassa-Holm equation it is where for both cases. This implies that the inside structures of the DP equation (4) and the Camassa-Holm equation are truly different. However, we not only have some similar results [21–23], but also have considerable differences in the scattering/inverse scattering approach, compared with the discussion in [5, 6] and in the paper .
Analogous to the Camassa-Holm equation, (4) can be written in Hamiltonian form and has infinitely many conservation laws. Here we list some of the simplest conserved quantities : where . So they are different from the invariants of the Camassa-Holm equation
Set ; then the operator in can be expressed by Equation (4) can be written as while the Camassa-Holm equation can be written as On the other hand, the DP equation can also be expressed in the following momentum form: This formulation is important to motivate us to consider the measure of momentum support which is the second object of this paper, since we found that (12) is similar to the vorticity equation of the three-dimensional Euler equation for incompressible perfect fluids ( is the speed, and is its vorticity) The stretching term in (13) is similar to the term in (12).
One can follow the argument for the Camassa-Holm equation  to establish the following well posedness theorem for the Degasperis-Procesi equation.
It should be mentioned that due to the form of (10) (no derivative appears in the convolution term), Coclite and Karlsen  established global existence and uniqueness result for entropy weak solutions belonging to the class .
2. Unique Continuation
The purpose of this section is to show that the solution to (10) and its first-order spatial derivative retain algebraic decay at infinity as their initial values do. Precisely, we prove.
Theorem 2. Assume that for some and, is a strong solution of the initial value problem associated with (10), and that satisfies that for some Then uniformly in the time interval .
Notation. We will say that where is a nonnegative constant.
Proof. We introduce the following notations:
Multiplying (10) by with and integrating the result in the -variable, one gets
The first term in (20) is
and for the rest, we have
From the above inequalities, we get
and therefore, by Sobolev embedding theorem and Gronwall’s inequality, there exists a constant such that
taking the limits in (24) (note that and ) from (25) we get
We will now repeat the above arguments using the barrier function
where . Observe that for all we have
Using notation in (18), from (10) we obtain
Hence, as in the weightless case (26), we get
A simple calculation shows that there exists depending only on such that, for any ,
Thus, for any appropriate function one finds that
Combining with (30), we get
where . By Gronwall’s inequality, there exists a constant for any such that
Finally, taking the limit as goes to infinity in (34) we find that for any
From (15), we get as .
Next, differentiating (10) in the -variable produces the equation Again, multiplying (36) by , integrating the result in the -variable, and using integration by parts one gets the inequality and therefore as before Since , we can use (25) and pass to the limit in (39) to obtain from (36) we get We need to eliminate the second derivatives in the second term in (41). Thus, combining integration by parts and (28), we find Since , the argument in (32) also shows that Similarly, we get where .
Then, taking the limit as goes to infinity, we find that for any Since as and (15), we get This completes the proof.
3. Measure of Momentum Support
It is known that, for the Degasperis-Procesi equation, the momentum density with compactly supported initial data will retain this property; that is, is also compactly supported . However, the same argument for is false . Note that a detailed description of solution outside of the support of is given in [26, 27]. Moreover, the exponential behavior of in outside this support is obvious. The comparison of the DP equation and the incompressible Euler equation above implies that the momentum in (12) plays a similar role as the vorticity does in (13). This motivates us to estimate the size of supp for strong solutions. The approach is inspired by the work of Kim  and the recent work .
We first introduce the particle trajectory method. Let be a strong solution of (4) guaranteed by the well posedness Theorem 1. Let be the solution of the following initial value problem: Then, is an increasing diffeomorphism. It is shown [21, 23] that this implies that the support of propagates along the flow. Set to be the support of . Let , and let be given by the following: Moreover, we also want to mention the standard argument on the first Dirichlet eigenvalue problem. Let be an open interval in , and, be the first Dirichlet eigenvalue of the Laplacian on . Then we have It is just and the normalized eigenfunctions are the suitable translations of
Theorem 3. Let be a strong solution of (12). Let be the support of for with its initial being connected.(I) Suppose there exists a positive constant such that for . Then (II) does not change sign or and ; then, for all
Proof. (I) The relation of momenta and gives
Then, we have by (12) and the lower bound of
Therefore, (56), (58), and Gronwall inequality imply that
On the other hand, due to Propositions A.2 and A.3, is Lipschitz and differentiable almost everywhere. Moreover, we have
Then, it follows that
with . So (52) follows from (61) and (59).
(II) If does not change sign, we conclude that solutions of (10) exist globally in time. Equality (56) and the conservation of yield By similar arguments of (I), constant in (61) can be replaced by ; then (54) follows. If (53) is satisfied, we know that the solution of (10) exists globally in time [21, 30]. From (53) and (48), it is easy to get where we denote with by . By direct computation, we have Next, we prove that is decreasing with respect to time. To this end, one gets, by differentiating (64) with respect to and integrating by parts, This implies that Therefore, (54) follows by replacing with in (61).
The following propositions with standard proofs are known in ; we list them here only for convenience of readers.
Proposition A.1. Let , and can be bounded by a constant ; then (a) (b) (c)
Proof. (a) Differentiating (47) with respect to , we obtain
Since is an increasing diffeomorphism, then . Combining the bound of , there holds
This can be solved as (a).
(b) Differentiating (49) with respect to to get then (A.2) is a direct consequence of (A.1).
(c) Equation (49) and the definition of Sobolev norm give that where we have used the change of variable . So (A.3) follows from (A.1).
Proposition A.2. Under the hypothesis of Theorem 3, for ,
Proposition A.3. Under the hypothesis of Theorem 3, for ,
Proof. Let with be a first normalized eigenfunction on , and let be such that its -transport is a normalized first eigenfunction on . For , using the left halves of (A.1) and (A.2) and then the right half of (A.3) we get Hence The other part is similar.
This work was partially supported by ZJNSF, under Grant nos. LQ12A01009 and LQ13A010008, and NSFC, under Grant nos. 11301394,11226176, and 11226172.