Persistence Property and Estimate on Momentum Support for the Integrable Degasperis-Procesi Equation

and Applied Analysis 3 and for the rest, we have 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫ ∞ −∞ u 2p−1 uu x dx 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 = 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫ ∞ −∞ u 2p u x dx 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 󵄩 󵄩 󵄩 󵄩 u x (t) 󵄩 󵄩 󵄩 󵄩∞ ‖u (t)‖ 2p


Introduction
Recently, Degasperis and Procesi [1] consider the following family of third order dispersive conservation laws: where , ,  0 ,  1 ,  2 , and  3 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 [1]   −   + 4  = 3    +   .
It is worth noting that in [2] 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  1 = −3 3 /2 2 and  2 =  3 /2 in (1), we recover the Camassa-Holm equation derived physically by Camassa and Holm in [3] 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 [4].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][8][9][10][11][12][13][14][15][16][17][18][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 [20], 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][22][23], but also have considerable differences in the scattering/inverse scattering approach, compared with the discussion in [5,6] and in the paper [24].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 [20]: where V = (4− 2  ) −1 .So they are different from the invariants of the Camassa-Holm equation Set  = (1 −  2  ); then the operator  −1 in R 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 −3  in (12).One can follow the argument for the Camassa-Holm equation [8] to establish the following well posedness theorem for the Degasperis-Procesi equation.

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  > 0 and  > 3/2,  ∈ ([0, ];   (R)) is a strong solution of the initial value problem associated with (10), and that  0 () = (, 0) satisfies that for some  > 1 Then uniformly in the time interval [0, ].
Notation.We will say that where  is a nonnegative constant.
Proof.We introduce the following notations: Multiplying ( 10) by  2−1 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    ∈  1 and () ∈  1 ∩  ∞ ) from ( 25) we get We will now repeat the above arguments using the barrier function where  ∈ Z + .Observe that for all  we have Using notation in (18), from (10) we obtain Hence, as in the weightless case (26) A simple calculation shows that there exists  0 > 0 depending only on  such that, for any  ∈ Z + , From (15), we get |(, )| = ( − ) as  ↑ ∞.
Next, differentiating (10) in the -variable produces the equation Again, multiplying (36) by  2−1  , ( ∈ Z + ), integrating the result in the -variable, and using integration by parts This completes the proof.

Measure of Momentum Support
It is known that, for the Degasperis-Procesi equation, the momentum density (, ) with compactly supported initial data  0 () will retain this property; that is, (, ) is also compactly supported [21].However, the same argument for (, ) is false [21].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 [28] and the recent work [29].We first introduce the particle trajectory method.Let  ∈ ([0, ],  3 (R)) ∩  1 ([0, ],  2 (R)) be a strong solution of (4) guaranteed by the well posedness Theorem 1.Let  ∈ [0, ], (; , ) be the solution of the following initial value problem: Then, (; ⋅, ) : R → R is an increasing diffeomorphism.It is shown [21,23] that  ( (; , 0) , )  3  (; , 0) =  (, 0) ; this implies that the support of  propagates along the flow.