PERSISTENCE PROPERTIES AND UNIQUE CONTINUATION OF SOLUTIONS OF THE CAMASSA-HOLM EQUATION

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1 PERSISTENCE PROPERTIES AND UNIQUE CONTINUATION OF SOLUTIONS OF THE CAMASSA-HOLM EQUATION A. ALEXANDROU HIMONAS, GERARD MISIO LEK, GUSTAVO PONCE, AND YONG ZHOU Abstract. It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution. 1. Introduction This work is concerned with the nonperiodic Camassa-Holm equation 1.1 t u t 2 xu + 3u x u 2 x u 2 xu u 3 xu =, t, x R. This equation appears in the context of hereditary symmetries studied by Fuchssteiner Fokas [FF]. It was first written explicitly, derived physically as a water wave equation by Camassa Holm [CH], who also studied its solutions. Equation 1.1 is remarkable for its properties such as infinitely many conserved integrals, bi-hamiltonian structure or its non-smooth travelling wave solutions known as peakons see formula 1.9. It was also derived as an equation for geodesics of the H 1 -metric on the diffeomorphism group, see [Mi]. For a discussion of how it relates to the theory of hereditary symmetries see [F]. The inverse scattering approach to the Camassa-Holm equation has also been developed in several works, for example see [CH], [CoMc], [Mc1], [BSS], references therein. A considerable amount of work has been devoted to the study of the corresponding Cauchy problem in both nonperiodic periodic cases. Among these results, of relevance to the present paper will be the fact that 1.1 is locally well-posed in Hadamard s sense in H s R for any s > 3/2, see for example [LO], [R], [D]. The long time behaviour of solutions has been studied conditions which guarantee their global existence their finite blow up have been deduced. In particular, in [Mc1] a necessary sufficient condition was established on the initial datum to guarantee finite time singularity formation for the corresponding strong solution. For further results in this direction we refer to [Mc1], [CoE] the survey article [Mo] references therein. For well-posedness results in the periodic case we refer to [HM1], [Mi], [DKT], where the equation is studied in its integral-differential form see 1.2 below as an ODE on the space of diffeomorphisms of the circle. A recent result demonstrating that the solution map u u 1

2 2 A. ALEXANDROU HIMONAS, GERARD MISIO LEK, GUSTAVO PONCE, AND YONG ZHOU for the Camassa-Holm equation is not locally uniformly continuous in Sobolev spaces can be found in [HM2]. Also the Camassa-Holm equation has been studied as an integrable infinite-dimensional Hamiltonian system, several works have been devoted to several aspect of its scattering setting, see [CH], [CoMc], [Mc1], [BSS] references therein. It is convenient to rewrite the equation in its formally equivalent integral-differential form 1.2 t u + u x u + x G u xu 2 =, where Gx = 1 2 e x. Our first objective here is to formulate decay conditions on a solution, at two distinct times, which guarantee that u is the unique solution of equation 1.1. The idea of proving unique continuation results for nonlinear dispersive equations under decay assumptions of the solution at two different times was motivated by the recent works [EKPV1], [EKPV2] on the nonlinear Schrödinger the k-generalized Korteweg-de Vries equations respectively. In the recent works [Co], [He] [Z] it was shown that u cannot preserve compact support in a non-trivial time interval i.e. for t [, ɛ], ɛ > except if u. However, this result does not preclude the possibility of the solution having compact support at a later time. In fact, in [Z] the question concerning the possibility of a smooth solution of 1.1 having compact support at two different times was explicitly stated. In particular, our first result, Theorem 1.1, gives a negative answer to this question. Theorem 1.1. Assume that for some T > s > 3/2 1.3 u C[, T ] : H s R is a strong solution of the IVP associated to the equation 1.2. If u x = ux, satisfies that for some α 1/2, u x oe x, x u x Oe αx as x, there exists t 1, T ] such that 1.5 ux, t 1 oe x as x, then u. Notation We shall say that fx Oe ax as x if lim x fx e ax = L, fx oe ax as x if lim x fx e ax =.

3 PROPAGATION SPEED FOR THE C-H EQUATION 3 Remarks a Theorem 1.1 holds with the corresponding decay hypothesis in stated for x <. b The time interval [, T ] is the maximal existence time interval of the strong solution. This guarantees that the solution is uniformly bounded in the H s -norm in this interval see 2.12, that our solution is the strong limit of smooth ones such that the integration by parts in the proof see 2.21, 2.29 can be justified. The proof of Theorem 1.1 will be a consequence of the following result concerning some persistence properties of the solution of the equation 1.2 in L -spaces with exponential weights. Theorem 1.2. Assume that for some T > s > 3/2 1.6 u C[, T ] : H s R is a strong solution of the IVP associated to the equation 1.2 that u x = ux, satisfies that for some θ, u x, x u x Oe θx as x. Then 1.8 ux, t, x ux, t Oe θx as x, uniformly in the time interval [, T ]. The following result establishes the optimality of Theorem 1.1 tells us that a strong non-trivial solution of 1.2 corresponding to data with fast decay at infinity will immediately behave asymptotically, in the x-variable at infinity, as the peakon solution 1.9 v c x, t = c e x ct, t >. Theorem 1.3. Assume that for some T > s > 3/2 1.1 u C[, T ] : H s R is a strong solution of the IVP associated to the equation 1.2 that u x = ux, satisfies that for some α 1/2, u x Oe x, x u x Oe αx as x for some α 1/2, 1. Then 1.12 ux, t Oe x as x, uniformly in the time interval [, T ]. In the case when the solution ux, t possesses further regularity its data u has stronger decay properties we shall give a more precise description of its behavior at infinity in the space variable.

4 4 A. ALEXANDROU HIMONAS, GERARD MISIO LEK, GUSTAVO PONCE, AND YONG ZHOU As it was noted in both [Co] [Z] in the case of compactly supported initial data u the difference hx, t of the solution its second derivative, i.e hx, t = 1 2 xux, t, remains compactly supported. Thus, if u is supported in the interval [a, b] in its lifespan, one has that hx, t has compact support in the time interval [ηa, t, ηb, t], for the definition of η, see Theorem 1.4. Assume that for some T > s > 5/ u C[, T ] : H s R is a strong solution of the IVP associated to the equation 1.2. a If u x = ux, has compact support, then for any t, T ] { c + t e x, for x > ηb, t, 1.15 ux, t = c t e x, for x < ηa, t. b If for some µ > 1.16 j xu Oe 1+µ x as x j =, 1, 2, then for any t, T ] 1.17 hx, t = 1 2 x ux, t Oe 1+µ x as x, 1.18 lim x ± e±x ux, t = c ± t, where in 1.15, 1.18 c +, c denote continuous non-vanishing functions, with c + t > c t < for t, T ]. Furthemore, c + is strictly increasing function, while c is strictly decreasing. Theorem 1.4 tells us that, as long as it exists, the solution ux, t is positive at infinity negative at minus infinity regardless of the profile of a fast-decaying data u. Finally, as a consequence of some of the estimates obtained in the proofs of the previous results we shall show that any strong solution corresponding to a data with compact support blows up in finite time. For details on the structure of the blow-up we refer to [Mc1], [Mc2] references therein. Other blow up results are discussed in [Mo]. Corollary 1.1. Assume that for some T > s > 5/ u C[, T ] : H s R

5 PROPAGATION SPEED FOR THE C-H EQUATION 5 is a strong solution associated to the equation 1.2 with initial data ux, = u x having compact support. Then the solution ux, t blows up in finite time, i.e. there exists a time interval [, T such that u C[, T : H s R 1.2 T x ut L dt =. 2. Proof of the results First, assuming the result in Theorem 1.2 we shall prove Theorem 1.1. Proof of Theorem 1.1. Integrating equation 1.2 over the time interval [, t 1 ] we get 2.1 ux, t 1 ux, + 1 By hypothesis we have u x ux, τdτ ux, t 1 ux, oe x as x. From 1.4 Theorem 1.2 it follows that 2.3 so u x ux, τdτ Oe 2αx as x, u x ux, τdτ oe x as x. x G u xu 2 x, τdτ =. We shall show that if u, then the last term in 2.1 is Oe x but not oe x. Thus, we have t1 x G u xu 2 x, τ dτ = x G u xu 2 x, τ dτ 2.5 = x G ρx, where by 1.4 Theorem ρx Oe 2αx, so that ρx oe x as x. Therefore 2.7 x G ρx = 1 2 = 1 2 e x x sgnx y e x y ρy dy e y ρy dy ex e y ρy dy. x

6 6 A. ALEXANDROU HIMONAS, GERARD MISIO LEK, GUSTAVO PONCE, AND YONG ZHOU From 2.6 it follows that 2.8 e x e y ρy dy = o1e x e 2y dy o1e x oe x, if ρ one has that 2.9 x x x e y ρy dy c, for x 1. Hence, the last term in satisfies 2.1 x G ρx c 2 e x, for x 1 which combined with yields a contradiction. Thus, ρx consequently u, see 2.5. Proof of Theorem 1.3. This proof is similar to that given for Theorem 1.1 therefore it will be omitted. We proceed to prove Theorem 1.2. Proof of Theorem 1.2. We introduce the following notations 2.11 F u = u 2 + xu 2, M = sup t [,T ] ut H s. Multiplying the equation 1.2 by u 2p 1 with p Z + integrating the result in the x-variable one gets 2.13 The estimates u 2p 1 t u dx + u 2p 1 t u dx = 1 2p Hölder s inequality in 2.13 yield 2.16 u 2p 1 u x u dx + u 2p 1 x G F u dx =. d dt ut 2p = ut 2p 1 d 2p dt ut 2p u 2p 1 u x u dx xut ut 2p 2p d dt ut 2p x ut ut 2p + x G F ut 2p

7 therefore, by Gronwall s inequality 2.17 ut 2p u 2p + Since f L 2 R L R implies PROPAGATION SPEED FOR THE C-H EQUATION lim q f q = f, x G F uτ 2p dτ e Mt. taking the limits in 2.17 notice that x G L 1 F u L 1 L from 2.18 we get 2.19 ut u + x G F uτ dτ e Mt. Next, differentiating 1.2 in the x-variable produces the equation 2.2 t x u + u xu 2 + x u 2 + xg 2 u xu 2 =. Again, multiplying the equation 2.2 by x u 2p 1 p Z + integrating the result in the x-variable using integration by parts 2.21 u xu 2 x u 2p 1 x u 2p dx = u x dx = 1 x u x u 2p dx 2p 2p one gets the inequality d 2.22 dt xut 2p 2 x ut x ut 2p + xg 2 F ut 2p therefore as before 2.23 x ut 2p x u 2p + xg 2 F uτ 2p dτ e 2Mt. Since xg 2 = G δ, we can use 2.18 pass to the limit in 2.23 to obtain 2.24 x ut x u + xg 2 F uτ dτ e 2Mt. We shall now repeat the above arguments using the weight 1, x, 2.25 ϕ N x = e θx, x, N, e θn, x N where N Z +. Observe that for all N we have 2.26 ϕ Nx ϕ N x a.e. x R. Using notation in 2.11, from equation 1.2 we obtain 2.27 t u ϕ N + u ϕ N x u + ϕ N x G F u =,

8 8 A. ALEXANDROU HIMONAS, GERARD MISIO LEK, GUSTAVO PONCE, AND YONG ZHOU while from 2.2 we get 2.28 t x u ϕ N + u 2 xu ϕ N + x u ϕ N x u + ϕ N 2 xg F u =, We need to eliminate the second derivatives in the second term in Thus, combining integration by parts 2.26 we find 2.29 u xu 2 ϕ N x u ϕ N 2p 1 dx = = u x u ϕ N 2p 1 x x u ϕ N x u ϕ Ndx x u ϕ N 2p u x dx 2p 2 ut + x ut x u ϕ N 2p 2p Hence, as in the weightless case , we get u x u ϕ N x u ϕ N 2p 1 dx 2.3 utϕ N + x utϕ N e 2Mt uϕ N + x uϕ N + e 2Mt ϕn x G F uτ + ϕ N xg 2 F uτ dτ. A simple calculation shows that there exists c >, depending only on θ, 1 see such that for any N Z ϕ N x e x y 1 ϕ N y dy c. Thus, for any appropriate function f one sees that ϕ N x G f 2 x = 1 2 ϕ Nx sgnx y e x y f 2 y dy ϕ Nx e x y 1 ϕ N y ϕ Nyfyfy dy 1 ϕ N x e x y 1 2 ϕ N y dy ϕ N f f c ϕ N f f. Since xg 2 = G δ the argument in 2.32 also shows that 2.33 ϕn xg 2 f 2 x c ϕ N f f.

9 PROPAGATION SPEED FOR THE C-H EQUATION 9 Thus, inserting into 2.3 using it follows that there exits a constant c = cm; T > such that utϕ N + x utϕ N c uϕ N + x uϕ N c c uτ + x uτ uτϕn + x uτϕ N dτ uϕ N + x uϕ N + uτϕn + x uτϕ N dτ. Hence, for any N Z + any t [, T ] we have utϕ N + x utϕ N c uϕ N + x uϕ N 2.35 c ue θx + x ue θx. Finally, taking the limit as N goes to infinity in 2.35 we find that for any N Z + any t [, T ] 2.36 sup t [,T ] ute θx + x ute θx c ue θx + x ue θx which completes the proof of Theorem 2. It remains to prove Theorem 1.4. Proof of Theorem 1.4. A simple calculation shows that the solution u of equation 1.1 satisfies the identity xu η x η 2 = 1 2 xu, it has a mechanical interpretation as conservation of spacial angular momentum. Here η = ηx, t is the flow of u, that is dηx, t = uηx, t, t, 2.38 dt ηx, = x, so that by the assumption the stard ODE theory t ηt is a smooth curve of C 1 -diffeomorphisms of the line, close to the identity map defined on the same time interval as u see [Mi] for details in the periodic case. From 2.37 we then have 2.39 ux, t = 1 2 where e x y hy, tdy = 1 2 e x x 2.4 hx, t = 1 2 xux, t = 1 2 xu η 1 x, t x ηη 1 x, t, t 2. e y hy, tdy ex e y hy, tdy, x

10 1 A. ALEXANDROU HIMONAS, GERARD MISIO LEK, GUSTAVO PONCE, AND YONG ZHOU Let us first prove part a. Thus, from 2.4 it follows that if u has compact support in x in the interval [a, b], then so does h, t in the interval [ηa, t, ηb, t], for any t [, T ]. Moreover, defining 2.41 E + t = ηb,t ηa,t one has from 2.4 that e y hy, t dy E t = ηb,t ηa,t e y hy, t dy, 2.42 ux, t = 1 2 e x hx, t = 1 2 e x E + t, x > ηb, t, 2.43 ux, t = 1 2 e x hx, t = 1 2 ex E t, x < ηa, t. Hence, it follows that for x > ηb, t 2.44 ux, t = x ux, t = 2 xux, t = 1 2 e x E + t, for x < ηa, t 2.45 ux, t = x ux, t = 2 xux, t = 1 2 ex E t. Next, integration by parts, 2.44, 2.45, the equation in 1.1 yield the identities E + = de + t dt = = e y hy, dy = e y u ydy + e y u x u dy + e y u ydy e y x u ydy = e y 2 xu x u dy e y 2 xu ydy e y x F udy = e y x u x u u x u e y u 2 + xu = e y u 2 + xu 2 2 dy >. e y F udy Therefore, in the life-span of the solution ux, t, E + t is an increasing function. Thus, from 2.46 it follows that E + t > for t, T ]. Similarly, it is easy to see that E t is decreasing with E =, therefore E t < for t, T ]. Taking c ± t = 1 2 E ±t we obtain 1.15.

11 PROPAGATION SPEED FOR THE C-H EQUATION 11 Next, let us consider part b. Since hx, t = 1 2 xux, t satisfies the equation 2.48 t hx, t + ux, t x hx, t = 2 x ux, thx, t, x, t R [, T ], an argument similar to that given in the proof of Theorem 1.2 shows that 2.49 sup hte 1+µ x c he 1+µ x, t [,T ] with c depending only on M in 2.12 T, that for any θ, j xut Oe θ x as x for j =, 1, 2. Thus, the definitions in 2.41 make sense with the integrals extended to the whole real line the computations in can be carried out in the same fashion. Finally, using 2.49 in 2.39 we obtain Proof of Corollary. 1.1 Suppose that the support of u is contained in some interval [a, b]. According to a result of McKean [Mc2] the corresponding solution ux, t persists for all time only if either h x = 1 2 xu x is of one sign or h x when a x < x o h x when x o < x b for some x o in a, b. Otherwise there exists T < such that x ut L as t T. Assuming that the solution is global in time computing as in 2.46 we find that b a e x h x dx = b a e x h x dx =. This implies that h x must change sign. Furthermore, since e x is strictly increasing we have the inequalites x e xo h x dx > a xo a e x h x dx = b x o e x h x dx > e xo b x h x dx similarly xo xo b b e xo h x dx < e x h x dx = e x h x dx < e xo h x dx. a a x o x o These two sets of inequalities cannot both hold so T < is the maximal existence time of the solution u. In order to prove 1.2, we can combine 2.16, to get e 2 R t xuτ dτ ut + x ut u + x u + for any t < T. Since + e R τ xur dr x G F uτ + p 2 xg F uτ dτ x G F u c x u u + x u

12 12 A. ALEXANDROU HIMONAS, GERARD MISIO LEK, GUSTAVO PONCE, AND YONG ZHOU we have e 2 R t 2 xg F u c x u u + x u xuτ dτ ut + x ut u + x u + + c Applying Gronwall s inequality again we obtain x uτ e 2 R τ xur dr uτ + x uτ dτ. ut + x ut u + x u exp c x uτ dτ valid for any t < T. This estimate implies that if T x ut dt were finite then the solution u could be extended beyond time T. The corollary follows. Acknowledgments The authors would like to thank Prof. D. Holm for useful comments concerning this work. G. P. was supported by an NSF grant. Y. Z. was supported by NSFC under grant no Shanghai Rising-Star Program 5QMX1417. References [BSS] Beals, R., Sattinger, D., Szmigielski, J., Multipeakons the classical moment problem, Adv. Math , no. 2, pp [CH] Camassa, R. Holm, D., An integrable shallow water equation with peaked solutions, Phys. Rev. Lett , pp [Co] Constantin, A., Finite propagation speed for the Camassa-Holm equation, J. Math. Phys , no 2, pp. 4 [CoE] Constantin, A. Escher, J., Global existence blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci , no. 2, pp [CoMc] Constantin, A. McKean, H., A shallow water equation on the circle, Comm. Pure Appl. Math , pp [CoS] Constantin, A. Strauss, W., Stability of peakons, Comm. Pure Appl. Math. 53 2, pp [D] Danchin, R., A few remarks on the Camassa-Holm equation, Differential Integral Equations 14 21, pp [DKT] De Lellis, C., Kappeler, T. Topalov, P., Low-regularity solutions of the periodic Camassa- Holm equation, pre-print. [EKPV1] Escauriaza, L., Kenig, C. E., Ponce, G., Vega, L., On unique continuation of solutions of Schrödinger equations, to appear in Comm. PDE [EKPV2] Escauriaza, L., Kenig, C. E., Ponce, G., Vega, L., On uniqueness properties of solutions of the k-generalized KdV equations, pre-print. [FF] Fuchssteiner, B. Fokas, A., Symplectic structures, their Backlund transformations hereditary symmetries, Phys. D /1982, pp

13 PROPAGATION SPEED FOR THE C-H EQUATION 13 [F] Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations: generalization of the Camassa-Holm equation, Physica D 95, 1996, pp [He] Henry, D., Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 25, pp [HM1] Himonas, A. Misio lek G., The Cauchy problem for an integrable shallow water equation, Differential Integral Equations 14 21, pp [HM2] Himonas, A. Misio lek G., High-frequency smooth solutions well-posedness of the Camassa-Holm equation, Int. Math. Res. Not , pp [LO] Li, Y. Olver, P., Well-posedness blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations 162 2, pp [Mc1] McKean, H., Breakdown of the Camassa-Holm equation, Comm. Pure Appl. Math , pp [Mc2] McKean, H., Breakdown of a shallow water equation, Asian J. Math , pp [Mi] Misio lek, G., Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal , pp [Mo] Molinet, L., On well-posedness results for the Camassa-Holm equation on the line: A survey, J. Nonlin. Math. Phys , pp [R] Rodriguez-Blanco, G., On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal , pp [Z] Zhou, Y., Infinite propagation speed for a shallow water equation, pre-print. A. Alexrou Himonas Department of Mathematics University of Notre Dame Notre Dame, IN USA himonas.1@nd.edu Gerard Misio lek Department of Mathematics University of Notre Dame Notre Dame, IN USA gmisiole@nd.edu Gustavo Ponce Department of Mathematics University of California Santa Barbara, CA 9316 USA ponce@math.ucsb,edu

14 14 A. ALEXANDROU HIMONAS, GERARD MISIO LEK, GUSTAVO PONCE, AND YONG ZHOU Yong Zhou Department of Mathematics East China Normal University Shangai 262 China Institute des Hautes Éudes Scientifiques 35, route de Chartres F-9144 Bures-sur-Yvette France