AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 390132 10.1155/2013/390132 390132 Research Article Persistence Property and Estimate on Momentum Support for the Integrable Degasperis-Procesi Equation http://orcid.org/0000-0002-0752-6836 Guo Zhengguang 1 Jin Liangbing 2 Raja Sekhar T. 1 College of Mathematics and Information Science Wenzhou University Wenzhou Zhejiang 325035 China wzu.edu.cn 2 Department of Mathematics Zhejiang Normal University Jinhua Zhejiang 321004 China zjnu.edu.cn 2013 7 11 2013 2013 24 04 2013 04 10 2013 2013 Copyright © 2013 Zhengguang Guo and Liangbing Jin. 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.

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.

1. Introduction

Recently, Degasperis and Procesi  consider the following family of third order dispersive conservation laws: (1)ut+c0ux+γuxxx-α2uxxt=(c1u2+c2ux2+c3uuxx)x, where α,  γ,  c0,  c1,  c2, and c3 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 (2)ut+ux+uux+uxxx=0, the Camassa-Holm equation (3)ut-uxxt+3uux=2uxuxx+uuxxx, and a new equation (the Degasperis-Procesi equation, the DP equation, for simplicity) which can be written as (after rescaling) the dispersionless form  (4)ut-uxxt+4uux=3uxuxx+uuxxx. 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 c1=-3c3/2α2 and c2=c3/2 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 u(x,t) 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 .

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 (5)Ψx-Ψxxx-λyΨ=0, while for Camassa-Holm equation it is (6)Ψxx-14Ψ-λyΨ=0, where y=u-uxx 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 , 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 : (7)H-1=u3dx,H0=ydx,H1=yvdx,H5=y1/3dx,H7=(yx2y-7/3+9y-1/3)dx, where v=(4-x2)-1u. So they are different from the invariants of the Camassa-Holm equation (8)E(u)=(u2+ux2)dx,    F(u)=(u3+uux2)dx.

Set Q=(1-x2); then the operator Q-1 in can be expressed by (9)Q-1f=G*f=12e-|x-y|f(y)dy. Equation (4) can be written as (10)ut+uux+xG*(32u2)=0, while the Camassa-Holm equation can be written as (11)ut+uux+xG*(u2+12ux2)=0. On the other hand, the DP equation can also be expressed in the following momentum form: (12)yt+yxu=-3yuxy=(1-x2)u. 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 (U is the speed, and ω is its vorticity) (13)ωt+(U·)ω=(ω·)U,divU=0,curlU=ω. The stretching term (ω·)U in (13) is similar to the term -3yux in (12).

One can follow the argument for the Camassa-Holm equation  to establish the following well posedness theorem for the Degasperis-Procesi equation.

Theorem 1 (see [<xref ref-type="bibr" rid="B29">23</xref>]).

Given u(x,t=0)=u0Hs(), s>3/2, then there exist a T and a unique solution u to (4) (also (10)) such that (14)u(x,t)C([0,T);Hs())C1([0,T);Hs-1()).

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 L1()BV().

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 T>0 and  s>3/2, uC([0,T];Hs()) is a strong solution of the initial value problem associated with (10), and that u0(x)=u(x,0) satisfies that for some θ>1(15)|u0(x)|,|xu0(x)|=O(x-θ)as  x. Then (16)|u(x,t)|,|xu(x,t)|=O(x-θ)as  x, uniformly in the time interval [0,T].

Notation. We will say that (17)|f(x)|=O(x-θ)as  xif  limx|f(x)|x-θ=L, where L is a nonnegative constant.

Proof.

We introduce the following notations: (18)F(u)=32u2,(19)M=supt[0,T]u(t)Hs. Multiplying (10) by u2p-1 with pZ+ and integrating the result in the x-variable, one gets (20)-u2p-1(ut+uux+xG*F(u))dx=0. The first term in (20) is (21)-u2p-1utdx=-12pdu2pdtdx=12pddt-u2pdx=u(t)2p2p-1ddtu(t)2p, and for the rest, we have (22)|-u2p-1uuxdx|=|-u2puxdx|ux(t)u(t)2p2p,|-u2p-1xG*F(u)dx|u(t)2p2p-1xG*F(u)(t)2p. From the above inequalities, we get (23)ddtu(t)2pux(t)u(t)2p+xG*F(u)2p, and therefore, by Sobolev embedding theorem and Gronwall’s inequality, there exists a constant M such that (24)u(t)2p(u(0)2p+0txG*F(u)2pdτ)eMt. Since fL1()L() implies (25)limqfq=f, taking the limits in (24) (note that xGL1 and F(u)L1L) from (25) we get (26)u(t)(u(0)+0txG*F(u)dτ)eMt. We will now repeat the above arguments using the barrier function (27)φN(x)={1,x1,xθ,x(1,N),Nθ,xN, where N+. Observe that for all N we have (28)0φN(x)θφN(x)a.e.  x. Using notation in (18), from (10) we obtain (29)(uφN)t+(uφN)ux+φNxG*F(u)=0. Hence, as in the weightless case (26), we get (30)u(t)φN  eMtu(0)φN+eMt0tφNxG*F(u)dτ. A simple calculation shows that there exists C0>0 depending only on θ such that, for any N+, (31)12φN(x)-e-|x-y|1φN(y)dyC0. Thus, for any appropriate function f one finds that (32)|φNxG*f2(x)|=|12φN(x)-sgn(x-y)e-|x-y|f2(y)dy|φN(x)2-e-|x-y|1φN(y)φN(y)f(y)f(y)dy(φN(x)2-e-|x-y|φN(y)dy)φNffC0φNff. Combining with (30), we get (33)u(t)φNC1(u0φN+0tφNudτ), where C1=C1(M;T,)>0. By Gronwall’s inequality, there exists a constant C~ for any t[0,T] such that (34)φNuC~u0φNC~u0·max(1,xθ). Finally, taking the limit as N goes to infinity in (34) we find that for any t[0,T](35)|u(x,t)xθ|C~u0·max(1,xθ).

From (15), we get |u(x,t)|=O(x-θ) as x.

Next, differentiating (10) in the x-variable produces the equation (36)uxt+uuxx+ux2+x2G*(32u2)=0. Again, multiplying (36) by ux2p-1,(p+), integrating the result in the x-variable, and using integration by parts (37)-uuxx(ux)2p-1dx=-u(ux)2p2pdx=-12p-ux(ux)2pdx, one gets the inequality (38)ddtux(t)2p2ux(t)ux(t)2p+x2G*F(u)2p, and therefore as before (39)ux(t)2p(ux(0)2p+0tx2G*F(u)2pdτ)e2Mt. Since x2G=G-δ, we can use (25) and pass to the limit in (39) to obtain (40)ux(t)(ux(0)+0tx2G*F(u)dτ)e2Mt; from (36) we get (41)t(uxφN)+uuxxφN+(uxφN)ux+φNx2G*F(u)=0. We need to eliminate the second derivatives in the second term in (41). Thus, combining integration by parts and (28), we find (42)|-uuxxφN(uxφN)2p-1dx|=|-u(uxφN)2p-1(x(uxφN)-uxφN)dx|=|-u(x((uxφN)2p2p)-uxφN(uxφN)2p-1)dx|κ·(u(t)+xu(t))xuφN2p2p. Since x2G=G-δ, the argument in (32) also shows that (43)|φNx2G*f2(x)|C0φNff. Similarly, we get (44)ux(t)φNC2(ux(0)φN+0tu(τ)φNdτ), where C2=C2(M;T).

Then, taking the limit as N goes to infinity, we find that for any t[0,T](45)|ux(t)xθ|C2(ux(0)xθ+0tu(τ)xθdτ). Since |u(x,t)|=O(x-θ) as x and (15), we get (46)|xu(x,t)|=O(x-θ),as  x. This completes the proof.

3. Measure of Momentum Support

It is known that, for the Degasperis-Procesi equation, the momentum density y(x,t) with compactly supported initial data y0(x) will retain this property; that is, y(x,t) is also compactly supported . However, the same argument for u(x,t) is false . Note that a detailed description of solution u(x,t) outside of the support of y(x,t) is given in [26, 27]. Moreover, the exponential behavior of u in x outside this support is obvious. The comparison of the DP equation and the incompressible Euler equation above implies that the momentum y(x,t) in (12) plays a similar role as the vorticity does in (13). This motivates us to estimate the size of supp y(t,·) for strong solutions. The approach is inspired by the work of Kim  and the recent work .

We first introduce the particle trajectory method. Let uC([0,T],H3())C1([0,T],H2()) be a strong solution of (4) guaranteed by the well posedness Theorem 1. Let s[0,T],q(t;α,s) be the solution of the following initial value problem: (47)dq(t;α,s)dt=u(s+t,q(t;α,s)),s,s+t[0,T],α,q(0;α,s)=α,α. Then, q(t;·,s): is an increasing diffeomorphism. It is shown [21, 23] that (48)y(q(t;x,0),t)qx3(t;x,0)=y(x,0); this implies that the support of y propagates along the flow. Set D(t) to be the support of y(·,t). Let ψL2(D(s)), and let ψtL2(D(s+t)) be given by the following: (49)ψt(q(t;α,s))=ψ(α). Moreover, we also want to mention the standard argument on the first Dirichlet eigenvalue problem. Let Ω be an open interval in , and, λ1(Ω) be the first Dirichlet eigenvalue of the Laplacian on Ω. Then we have (50)λ1(Ω)=inf{ϕL2(Ω)2ϕH01(Ω)  with  ϕL2(Ω)=1}. It is just (π/|Ω|)2 and the normalized eigenfunctions are the suitable translations of (51)±(2|Ω|)1/2sin(πx|Ω|).

Theorem 3.

Let yC([0,T];H1())C1([0,T];L2()) be a strong solution of (12). Let D(t) be the support of y(·,t) for t[0,T] with its initial D(0) being connected.

Suppose there exists a positive constant K such that ux(x,k)>-K for (x,t)×[0,T]. Then (52)|D(0)|e-(exp(5KT/2)y0L2())t|D(t)||D(0)|e(exp(5KT/2)y0L2())t.

y0 does not change sign or (53)y0(x)0,x(-,x0),y0(x)0,x(x0,),

and y0H1()L1(); then, for all t0(54)|D(0)|e-y0L1()t|D(t)||D(0)|ey0L1()t.

Proof.

(I) The relation of momenta y and u gives (55)u(x,t)=12e-|x-ξ|y(ξ,t)dξ,(56)ux(x,t)=12sgn(ξ-x)e-|x-ξ|y(ξ,t)dξ. Then, we have by (12) and the lower bound of ux(57)ddty2(x,t)dx=-5ux(x,t)y2(x,t)dx5Ky2(x,t)dx. Thus (58)ddty(x,t)L225Ky(x,t)L22. Therefore, (56), (58), and Gronwall inequality imply that (59)|ux(x,t)|12y(x,t)L212e5KT/2y0L2. On the other hand, due to Propositions A.2 and A.3, λ1(D(s)) is Lipschitz and differentiable almost everywhere. Moreover, we have (60)-4M1λ1(D(s))ddsλ1(D(s))4M1λ1(D(s)). Then, it follows that (61)e-4M1sλ1(D(0))λ1(D(s))e4M1sλ1(D(0)) with λ1(D(s))=π2/|D(s)|2. So (52) follows from (61) and (59).

(II) If y0H1()L1() does not change sign, we conclude that solutions of (10) exist globally in time. Equality (56) and the conservation of y(x,t)dx yield (62)|ux(x,t)|12y(x,t)L1()=12y0(x)L1(). By similar arguments of (I), constant M1 in (61) can be replaced by y0(x)L1()/2; 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 (63)y(x,t)0,x(-,q(x0,t)),y(x,t)0,x(q(x0,t),), where we denote q(t;x,s) with s=0 by q(x,t). By direct computation, we have (64)|y(x,t)|dx=q(x0,t)y(x,t)dx--q(x0,t)y(x,t)dx. Next, we prove that y(x,t)L1() is decreasing with respect to time. To this end, one gets, by differentiating (64) with respect to t and integrating by parts, (65)ddt|y(x,t)|dx=q(x0,t)yt(x,t)dx--q(x0,t)yt(x,t)dx-2(yu)(q(x0,t),t)=-q(x0,t)(yxu+3yux)dx+-q(x0,t)(yxu+3yux)dx-2(yu)(q(x0,t),t)=-2q(x0,t)yuxdx+2-q(x0,t)yuxdx=u2(q(x0,t),t)-ux2(q(x0,t),t)=q(x0,t)e-ξy(ξ,t)dx-q(x0,t)eξy(ξ,t)dx0. This implies that (66)|ux(x,t)|12y(x,t)L1()12y0(x)L1(). Therefore, (54) follows by replacing M1 with y0(x)L1()/2 in (61).

Appendix

The following propositions with standard proofs are known in ; we list them here only for convenience of readers.

Proposition A.1.

Let s,s+t[0,T],αD(s), and ψH01(D(s));ux can be bounded by a constant M1; then

(a)(A.1)e-M1|t|qα(t;α,s)eM1|t|,

(b)(A.2)|ψ(α)|e-M1|t||(ψt)(q(t;α,s))||ψ(α)|eM1|t|,

(c)(A.3)ψL2(D(s))e-M1|t|/2ψtL2(D(s+t))ψL2(D(s))eM1|t|/2.

Proof.

(a) Differentiating (47) with respect to α, we obtain (A.4)dqtdα=uqqα. Since q(t;·,s): is an increasing diffeomorphism, then qα>0. Combining the bound of ux, there holds (A.5)-M1qαqαtM1qα. This can be solved as (a).

(b) Differentiating (49) with respect to α to get (A.6)ψqtqα=ψ(α), then (A.2) is a direct consequence of (A.1).

(c) Equation (49) and the definition of Sobolev norm give that (A.7)ψtL2(D(s+t))2=D(s+t)ψt(x)2dx=D(s)ψ2(α)qαdα, where we have used the change of variable x=q(t;α,s). So (A.3) follows from (A.1).

Proposition A.2.

Under the hypothesis of Theorem 3, for s,s+t[0,T], (A.8)limt0+supλ1(D(s+t))-λ1(D(s))t4M1λ1(D(s)),limt0-infλ1(D(s+t))-λ1(D(s))t-4M1λ1(D(s)).

Proof.

Let t>0,ϕ1H01(D(s)) with ϕ1L2(D(s))=1 be a first normalized eigenfunction on D(s). Then, for φH01(D(s+t)) with φL2(D(s+t))=1, we have (A.9)λ1(D(s+t))-λ1(D(s))=infφL2(D(s+t))2-ϕ1L2(D(s))2ϕ1tL2(D(s+t))-2(ϕ1t)L2(D(s+t))2-ϕ1L2(D(s))2. Furthermore (A.10)ϕ1tL2(D(s+t))-2(ϕ1t)L2(D(s+t))2=ϕ1tL2(D(s+t))-2D(s)[(ϕ1t)]2qαdαϕ1tL2(D(s+t))-2e3M1tϕ1L2(D(s))2e4M1tϕ1L2(D(s))2. Combing (A.9) and (A.10) together yields (A.11)limt0+supλ1(D(s+t))-λ1(D(s))tlimt0+supe4M1tϕ1L2(D(s))2-ϕ1L2(D(s))2t=4M1λ1(D(s)). The second one follows by similar arguments for t<0.

Proposition A.3.

Under the hypothesis of Theorem 3, for s,s+t[0,T], (A.12)limt0-supλ1(D(s+t))-λ1(D(s))t4M1λ1(D(s)),limt0+infλ1(D(s+t))-λ1(D(s))t-4M1λ1(D(s)).

Proof.

Let ϕ1H01(D(s)) with ϕ1L2(D(s))=1 be a first normalized eigenfunction on D(s), and let ϕ2L2(D(s)) be such that its t-transport is a normalized first eigenfunction on D(s+t). For t>0, using the left halves of (A.1) and (A.2) and then the right half of (A.3) we get (A.13)(ϕ2t)L2(D(s+t))2=D(s+t)[(ϕ2t(x))]2dx=D(s)[(ϕ2t)]2qαdαe-3M1tD(s)[ϕ2(α)]2dα=e-3M1tϕ2L2(D(s))2(ϕ2ϕ2L2(D(s))2)L2(D(s))2e-4M1tϕ2tL2(D(s+t))2λ1(D(s))=e-4M1tλ1(D(s)). Hence (A.14)limt0+infλ1(D(s+t))-λ1(D(s))tlimt0+infe-4M1t-1tλ1(D(s))=-4M1λ1(D(s)). The other part is similar.

Acknowledgments

This work was partially supported by ZJNSF, under Grant nos. LQ12A01009 and LQ13A010008, and NSFC, under Grant nos. 11301394,11226176, and 11226172.

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