On the Well-posedness and Stability of Peakons for a Generalized Camassa-Holm Equation

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.1 (006) No.3, pp On the Well-posedness and Stability of Peakons for a Generalized Camassa-Holm Equation Sevdzhan Hakkaev 1, Kiril Kirchev 1 Faculty of Mathematics and Informatics, Shumen University 971 Shumen, Bulgaria Institute of Mathematics and Informatics, Bulgarian Academy of Sciences 1133 Sofia, Bulgaria (eceived 1 April 006, accepted 3 May 006) Abstract: We establish the local well-posedness for the generalized Camassa-Holm equation. We also prove that the peaked solitary wave solutions are orbitally stable. Keywords: local well-posedness; peakon solutions; stability 1 Introduction In this paper we consider the equation { ut u xxt + (a(u)) x = ( 1 b (u)u x + b(u)u xx ) x u(x, 0) = u 0 (x), (1.1) where a, b : are given C 3 -functions. For a(u) = ku + 3 u and b(u) = u, Eq.(1.1) becomes the well-known Camassa-Holm equation u t u xxt + ku x + 3uu x = u x u xx + uu xxx (1.) which can be derived [] as a model for unidirectional propagation of shallow water waves over a flat bottom, where u(x, t) is the fluid velocity at time t 0 in the special direction. Eq. (1.) was first obtained [1] as a bi-hamiltonian generalization of the Korteweg-de Vries equation. The Camassa-Holm equation (1.) has global solutions and also solutions which blow-up in finite time [5, 1,, 3]. It admits solitary wave solutions, i.e. solutions of the form u(x, t) = ϕ(x ct) which travel with fixed speed c vanishing at infinity. In the case k > 0 the solutions are smooth solitary waves and their stability is studied in [10], using the method of Grillakis, Shatah and Strauss ([13]). In the integrable case k = 0 the solitary waves are of the form u(x, t) = cϕ(x ct), where ϕ(x) = exp( x ). They are peaked waves and their stability is studied in [9]. ecently, the following generalization of the Camassa-Holm equation [ ] g(u) u t u xxt + ku x + = u x u xx + uu xxx (1.3) has been studied. The well-posedness and blow-up problem are considered in [1, ]. Solitary wave solutions to Eq. (1.3) and their properties are investigated in [18]. Setting a(u) = ku + g(u) and b(u) = u in (1.1), we obtain the equation (1.3). It seems to us that the equation (1.1) is a better generalization of Camassa-Holm, because it has conservation laws E(u), F (u) (see Section 3 below) which are more convenient for studying of the stability problem. In the paper [11] of 1 Corresponding author: shakkaev@fmi.shu-bg.net Copyright c World Academic Press, World Academic Union IJNS /019 x

2 140 International Journal of Nonlinear Science, Vol.1(006), No.3, pp the authors the well-posedness in the case a(u) = p+ up+1 and b(u) = u p is investigated, by the use of the method of parabolic regularization. Moreover, the orbital stability and instability of solitary wave solutions are studied. The aim of this paper is to establish the well-posedness for more general a(u) and b(u) and stability of peaked solitary wave solutions of Eq. (1.1). The local well-posedness is obtained by applying Kato s semigroup approach. The local well-posedness results for the equations (1.) and (1.3) [5, 17, ] are special cases of our result. The stability of solitary wave solutions when a(u) = p+ up+1 and b(u) = u p is investigated using the same lines of ideas as in [9]. In this case the solitary waves are of the form u(x, t) = c 1 p ϕ(x ct), where ϕ(x) = exp( x ). They are peaked waves and can only be understood as weak solutions. Throughout this paper, we denote by * the convolution. Let s and (, ) s denote the norm and inner product in the Sobolev space H s (), respectively; [A, B] denotes the commutator of two linear operators A and B ; L(Y, X) denotes the space of all bounded linear operators from Y to X (L(x), if X = Y ). Local well-posedness In this section we consider the problem of local existence and uniqueness of solutions to Eq. (1.1). Our strategy is to use Kato s theory for quasilinear evolution equations. Consider the abstract quasilinear equation dv dt + A(v)v = f(v), t 0, v(0) = v 0. (.1) Let X and Y be real Hilbert spaces with norms X and Y, respectively, and assume that Y is continuously and densely embedded in X. Let Q : Y X be a topological isomorphism. (H1) A(y) L(Y, X) for y X, A(y) G(X, 1, β) ( i.e. A(y) is quasi-m-accretive ) and for any > 0, there exists µ 1 such that (A(y) A(z))w X µ 1 y z X w y for all y, z, w Y with y Y, z Y. (H) QA(y)Q 1 = A(y) + B(y), where B(y) L(X) is bounded uniformly for y Y bounded, and for any > 0, there exists µ such that (B(y) B(z))w X µ y z Y w X for y, z Y, w X with y Y, z y. (H3) f : Y X and extends also to a map from X in X and there exists µ 3, µ 4, such that for all y, z Y with y Y, z Y. The result of Kato is the following f(y) f(z) Y µ 3 y z Y, f(y) f(z) X µ 4 y z X Theorem.1 ([14]) Assume (H1)-(H3). Then for any > 0, there exist T > 0 such that for any v 0 Y with v 0 Y there exists a unique solution v of (1) with v C([0, T ]; Y ) C 1 ([0, T ]; X) Moreover, the map v 0 v(, v 0 ) is continuous from Y to C([0, T ]; Y ) C 1 ([0, T ]; X). We write the Eq. (1.1) in the equivalent form { ut + b(u)u x = (1 x) 1 (b(u)u x ) (1 x) 1 x [ 1 b (u)u x + a(u) ] u(x, 0) = u 0 (x), t 0, x. (.) The following results are useful for our further considerations. IJNS for contribution: editor@nonlinearscience.org.uk

3 S Hakkaev,K Kirchev: On the Well-posedness and Stability of Peakons for a Generalized Lemma.1 ([14]) Let r and t be real numbers such that r < t r. Then hg t c h r g t, if r > 1, hg r+t 1 where c is a positive constant depending on r and t. c h r g t, if r < 1, Lemma. ([16]) Let f H s, s > 3. Then Λ r [ Λ r+t+1, M f ] Λ t L(L ) c f s, r, t s 1, where M f is the operator of multiplication by f, and c is a constant depending only on r, t. Lemma.3 ([8]) Let F C m+ () with F (0) = 0. Then for every r ( 1, m] we have that F (u) r F ( u ) u r, u H r, where F is a monotonic increasing function depending only on F and r. Our main result in this section is the following theorem Theorem. Assume that a, b C m+3 (), m and a(0) = 0. Given u 0 H s, 3 exists a maximal T > 0 and a unique solution u to Eq. (.) such that < s < m, there u = u(, u 0 ) C([0, T ); H s ) C 1 ([0, T ]; H s 1 ). Moreover, the solution depends continuously on the initial data. Proof: Let [ ] 1 A(u) = b(u) x, f(u) = (1 x) 1 (b(u)u x ) (1 x) 1 x b (u)u x + a(u), Y = H s, X = H s 1 and Q = Λ = (1 x) 1. Obviously, Q is an isomorphism of H s onto H s 1. With the above notations the Eq. (.) might be written in the following form u t + A(u) = f(u) (.3) First we will prove that A(u) is quasi-m-accretive and A(u) L(H s, H s 1 ). Due to u H s and s > 3, it follows that u, u x L and u L < u s, u x L < u s. Since H s 1 is a Hilbert space, A(u) G(H s 1, 1, β) if and only if (1) given > 0, there exists β such that for u s, (A(u)y, y) s 1 β y s 1. () A(u) is the infinitesimal generator of a C 0 - semigroup on H s 1 for some λ > β. Using the equality Λ s 1 (b(u) x y) = [Λ s 1, b(u)] x y + b(u)(λ s 1 x y) we have (A(u)y, y) s 1 = (Λ s 1 (b(u) x y), Λ s 1 y) 0 = ([Λ s 1, b(u)] x y, Λ s 1 y) 0 + (b(u)(λ s 1 x y), Λ s 1 y) 0 = ([Λs 1, b(u)] x y, Λ s 1 y) 0 1 ( xb(u), (Λ s 1 y) ) 0 ([Λ s 1, b(u)] x Λ 1 s Λ s 1 y, Λ s 1 y) xb(u) L Λ s 1 y 0 [Λ s 1, b(u)]λ s L(L ) Λ s 1 y xb(u) L Λ s 1 y 0. IJNS homepage:

4 14 International Journal of Nonlinear Science, Vol.1(006), No.3, pp From Lemma.. with r = 0, t = s, we obtain (A(u)y, y) s 1 c b(u) s y s xb(u) L y s 1 [ c b(u) s + 1 ] b(u) s y s 1 c b( u L ) u s y s 1 c b( u s ) u s y s 1 c b() y s 1. Setting β = c b(), the proof of (1) is complete. Next, we prove (). Let S = Λ s 1. Note that S is an isomorphism on H s 1 onto L, and H s 1 is continuously and densely embedded in L as s > 3. Define A 1 (u) = SA(u)S 1, B 1 (u) = A 1 (u) A(u). Let y L and u H s. From Lemma.. with r = 0, t = s and Lemma.3. we have that B 1 (u)y 0 = [Λ s 1, b(u) x ]Λ 1 s y 0 = [Λ s 1, b(u)]λ 1 s x y 0 [Λ s 1, b(u)]λ s L(L ) Λ 1 x y 0 c b(u) s y 0 c b( u L ) u s y 0 Hence B 1 (u) L(L ). In [14] it is proved that A(u) G(L, 1, β ), with β 1 sup a (u)u x. By a perturbation theorem for semigroup we have that A 1 (u) G(L, 1, β ) and from Theorem 5.8 in [[19], 4.5] H s 1 is A(u) admissible and from Theorem 5.5 in [[19], 4.5] A(u) is the infinitesimal generator of a C 0 -semigroup on H s 1. Let u, v, w H s, s > 3. Since Hs 1 is a Banach algebra, then (A(u) A(v))w s 1 = (b(u) b(v)) x w s 1 b(u) b(v) s 1 x w s 1 b() u v s 1 w s Taking v = 0 in the above inequality we obtain A(u) L(H s, H s 1 ). Define B(u) = [Λ 1, b(u) x ]Λ 1 L(H s 1 ). Let u, v H s, > 0, u s, v s and w H s 1. We have (B(u) B(v))w s 1 = Λ s 1 ([Λ 1, b(u) x ]Λ 1 [Λ 1, b(v) x ]Λ 1 )w 0 = Λ s 1 [Λ 1, (b(u) b(v)) x ]Λ 1 w 0 = Λ s 1 [Λ 1, b(u) b(v)] x Λ 1 w 0 Λ s 1 [Λ 1, b(u) b(v)]λ 1 s L(L ) Λ s x w 0 By applying Lemma. with r = 1 s, t = s 1, we obtain (B(u) B(v))w s 1 c b(u) b(v) s w s 1 c b() u v s w s 1. Taking v = 0 in the above inequality we obtain B(u) L(H s 1 ). Next we will prove the following inequalities (a) f(u) f(v) s µ 3 u v s (b) f(u) f(v) s 1 µ 4 u v s 1. IJNS for contribution: editor@nonlinearscience.org.uk

5 S Hakkaev,K Kirchev: On the Well-posedness and Stability of Peakons for a Generalized Let u, v H s, s > 3 and > 0, u s, v s. Setting f 1 (u) = (1 x) 1 (b(u)u x ) = x (1 x) 1 g(u) f (u) = x (1 x) 1 ( 1 b (u)u ) x f 3 (u) = x (1 x) 1 (a(u)) where g (u) = b(u), then f(u) = f 1 (u) + f (u) + f 3 (u). By applying Lemma.3, we get and f 1 (u) f 1 (v) s c g(u) g(v) s 1 c b() u v s f 3 (u) f 3 (v) s c a(u) a(v) s 1 cã 1 () u v s. Since s > 3 and Hs 1 is a Banach algebra, then f (u) f (v) s c b (u)u x b (v)v x s 1 c ( b (u)(u x v x) s 1 + v x(b (u) b (v) s 1 ) Again using Lemmas.3, we obtain c( b (u) b (0) s 1 x (u v) s 1 x (u + v) s 1 + b (0) x (u v) s 1 x (u + v) s 1 + v x s 1 b (u) b (v) s 1 f (u) f (v) s c(( b 1 () + b (0) ) u v s u + v s + v s 1 b () u v s 1 ) ( ( ) ) c b1 () + b (0) + 3 b () u v s ). Combining above inequalities we obtain (a). Next, we prove (b). Let u, v H s, s > 3. Analogously of (a), we have and Next, f 1 (u) f 1 (v) s 1 c g(u) g(v) s b() u v s 1. f 3 (u) f 3 (v) s 1 c a(u) a(v) s ã 1 () u v s 1. f (u) f (v) s 1 c b (u)u x b (v)v x s c ( b (u)(u x v x) s + v x(b (u) b (v)) s ) From Lemma.1., with r = s 1, t = s, and Lemmas.3, we have f (u) f (v) s 1 c ( b (u) x (u + v) s 1 x (u v) s + v x s 1 b (u) b (v) s ) c( b (u) b (0) s 1 u + v s u v s 1 + b (0) u + v s u v s 1 + v b s 1 () u v s ) ( ) c b1 () + b (0) + 3 b1 () u v s 1. Applying Kato s theory to abstracting quasilinear evolution equation of hyperbolic type ( see [14] ) we can obtain the local well-posedness of Eq. (1.1) in H s, for 3 < s m. The solution u belongs to This completes the proof of Theorem. C([0, T ); H s ) C 1 ([0, T ); H s 1 ) IJNS homepage:

6 144 International Journal of Nonlinear Science, Vol.1(006), No.3, pp Stability of peakons In this section we consider the stability of peaked solitary wave solutions of Eq. (1.1) in the case a(u) = p+ up+1, b(u) = u p : ( ) (p + )(p + 1) 1 u t u xxt + u p u x = up 1 u x + u p u xx (3.1) x Equation (3.1) has the following conservation laws (see [11]) E(u) = (u + u x)dx, F (u) = (u p+ + u p u x)dx. (3.) The peaked solitary wave solutions can only be understood as weak solutions of Eq. (3.1). Definition 3.1 If u C([0, T ]; H s ) C 1 ([0, T ]; H s 1 ). with s > 3 is a solution to (3.1), then u(x, t) is called strong solution to (3.1) (or (1.1)). Note that if p(x) = 1 exp( x ), x, then (1 x) 1 g = p g for all g L () and p (u u xx ) = u. Setting ( G(u) = B(u) + p B(u) + 1 ) b (u)u x + a(u) where B (u) = b(u), equation (1.1) can be written as the conservation law u t + G(u) x = 0, u(x, 0) = u 0 (x), t > 0, x. Definition 3. Let u 0 H 1 be given. A function u : [0, T ] : is called weak solution to (3.1), if u L loc ([0, T ]; H1 ) satisfies the identity T 0 (uψ t + G(u)ψ x )dxdt + u 0 (x)ψ(x, 0)dx = 0 for all ψ C0 ([0, T ] ) that are restrictions to [0, T ) of a continuously differentiable function on with compact support contained in ( T, T ). Proposition 3.1 (i) Every strong solution is a weak solution. (ii) If u is a weak solution and u C([0, T ]; H s ) C 1 ([0, T ]; H s 1 ), with s > 3, then it is a strong solution. (iii) All nontrivial travelling wave solutions of (3.1) are not strong solutions. (iv) There exist peaked solitary wave solutions of (3.1), which are weak solutions. Proof: The proof of (i) and (ii) in the case p = 1 was proved in [5]. By mimicking their proof, the results for p > 1 can be obtained. Assume that there exists a nontrivial travelling wave u(x, t) = ϕ(x ct) H 3, which is a strong solution of (1). Then, obviously, ϕ is a solution of the following ordinary differential equation cϕ + cϕ + Since ϕ(ξ) 0, as ξ, after integration of Eq. (3.3) we obtain ( ) (p + )(p + 1) 1 ϕ p ϕ = ϕp 1 ϕ + ϕ p ϕ. (3.3) cϕ + cϕ + p + ϕp+1 = 1 ϕp 1 ϕ +ϕ p ϕ. IJNS for contribution: editor@nonlinearscience.org.uk

7 S Hakkaev,K Kirchev: On the Well-posedness and Stability of Peakons for a Generalized Next, multiplying by ϕ, we obtain Integrating once more gives cϕϕ + cϕ ϕ + p + ϕp+1 ϕ = 1 ϕp 1 ϕ 3 +ϕ p ϕ ϕ. (ϕ ϕ )(c ϕ p ) = 0. (3.4) Since ϕ belong to H 3, this is impossible. The solution of (3.4) is peaked solitary wave u(x, t) = c 1 p ϕ(x ct), where ϕ(ξ) = exp( ξ ). The main result in this section is the following theorem. Theorem 3.1 For every ε > 0, there is a δ > 0 such that if u C([0, T ]; H 1 ) is a solution to (3.1) with then u(, 0) c 1 p ϕ 1 < δ, u(, t) c 1 p ϕ( ξ(t)) 1 < ε for t (0, T ], where ξ(t) is any point where the function u(, t) attains its maximum. For the proof of Theorem 3.1., we proceeds as in [9]. For simplicity we take c = 1. Note that ϕ(ξ) is continuous on with peak at ξ = 0. Moreover, by simple computation we have E(ϕ) = and F (ϕ) = 4 p+. Lemma 3.1 For every u H 1 and ξ Proof. See[9]. E(u) E(ϕ) = u ϕ( ξ) 1 + 4u(ξ) 4. Lemma 3. Let u H 1 and M = max{u(x)} Then x p p + M p+ E(u)M p + F (u) 0. Proof. Let M be taken at x = ξ and define the function { u ux, x < ξ g(x) = u + u x, x > ξ In [9] it is calculated that Next, we calculate u p g (x)dx = = ξ g (x)dx = E(u) M (3.5) u p (u u x ) dx + u p (u + u x)dx + ξ ξ u p (u + u x ) dx u p+1 u x dx + = F (u) p + up+ ξ + p + up+ = F (u) 4 p + M p+. On the other hand, F (u) 4 p + M p+ max x up (x) This completes the proof of Lemma ξ + g (x)dx = M P (E(u) M ). ξ u p+1 u x dx IJNS homepage:

8 146 International Journal of Nonlinear Science, Vol.1(006), No.3, pp Lemma 3.3 Let u H 1. If u ϕ 1 < δ, then (a) E(u) E(ϕ) δ(δ + ) (b) F (u) F (ϕ) (δ+ ) p+ p. Proof: (a) is proved in [9]. From (3.5) we have M E(u) and sup u(x) 1 E(u) 1 = 1 u 1 (3.6) x From (3.6), u L 1 u 1 1 (δ + ) and using that ϕ L = 1, we obtain [ F (u) F (ϕ) = u p (u + u x) ϕ p (ϕ + ϕ ) ] dx (u p ϕ p )(u + u x)dx + ϕ p (u + u x ϕ ϕ )dx u p ϕ p L u 1 + u ϕ 1 + (u ϕ)ϕ + (u x ϕ x )ϕ x dx ( u p ϕ p L δ + ) δ + + δ + ϕ 1 u ϕ 1 ( ) ( u ϕ L u p 1 L + u p L δ + ) + δ + δ [ ( δ δ p 1 ( δ p ( + 1) + + 1) + + 1] δ + ) + δ + δ This completes the proof of Lemma 3.3. Lemma 3.4 Let u H 1 be a solution of (1). For sufficiently small δ if u ϕ 1 < δ. M 1 δ Proof: From Lemma 3.3. and for sufficiently small δ, E is near, F is near 4 p+ and and M p F E = p p +, (3.7) M p+ p + p E(u)M p + p + F (u) 0. (3.8) p Consider the polynomial (y) = y p+ p+ p E(u)yp + p+ p F (u). In the case E(u) = E(ϕ) = and F (u) = F (ϕ) = 4 p+ it takes the form 0 (y) = y p+ p + p yp + p. We have 0 (1) = 0 (1) = 0 and 0 (1) 0. Therefore y = 1 is a double root of 0(y). Moreover 0 (y) is decreasing in the interval (0, 1) and increasing in (1, + ). Hence 0 (y) has no roots other than y = 1, for y > 0. For δ sufficiently small, Lemma 3.3. and inequalities (3.7) and (3.8) show that must have two roots near 1 and M must lie between this two roots. We will show that these two roots are closer to y = 1. Consider the polynomial 1 (y) = y p+ p + p (δ + [ ) y p + p + 4 p p + + (δ + ] ) p+. p IJNS for contribution: editor@nonlinearscience.org.uk

9 S Hakkaev,K Kirchev: On the Well-posedness and Stability of Peakons for a Generalized The graph of 1 (y) on (0, + ) lies below the graph of (y). By a direct computation (we can take δ < 1)) we have 1 (1) < 0 and 1 (1 + ( δ) 1 8 p + (8 3 )p 4 ) δ C 1p δ 3 p and 1 (1 δ) 1 p ( 8 p + (8 3 )p 4 ) δ C p δ 3. Hence, there is δ 0 = δ 0 (p) such that for 0 < δ < δ 0, 1 (1 + δ) > 0 and 1 (1 δ) > 0. This completes the proof of Lemma 3.4. Proof of Theorem 3.1 Let u C([0, T ), H 1 ) be a solution of (3.1) and suppose we are given ε > 0. Since E and F are conservation laws for Eq. (3.1), we have E(u(, t)) = E(u 0 ), F (u(, t)) = F (u 0 ), t [0, T ]. (3.9) Taking δ sufficiently small and δ < 1 81 ε4 we apply Lemma 3.3. to u 0. By (3.9) the hypotheses of Lemma 3.4. are satisfied for u(, t). Hence From Lemma 3.1, we conclude u(ξ(t), t) 1 9 ε, t [0, T ]. (3.10) u(, t) ϕ( ξ(t) 1 = E(u 0 ) E(ϕ) + 4 4u(ξ(t), t) ε. Acknowledgements This work was partially supported by Grant MM-1403/04 MESC and by Scientefic esearch Grant of Shumen University. eferences [1] J. Bona,. Smith: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. oy. Soc. (London). 78, (1975) []. Camassa, D. Holm: An integrable shallow water equation with peaked solitons.phys. ev. Letters. 71, (1993) [3]. Camassa, D. Holm, J. Hyman: A new integrable shallow water equation. Adv. Appl. Mech. 31,1-33(1994) [4] A. Constantin: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble). 50,31-36(000) [5] A. Constantin, J. Escher: Global existence and blow-up for a shallow water equation. Annali Sc. Norm. Sup. Pisa. 6,303-38(1998) [6] A. Constantin, J. Escher: On the Cauchy problem for a family of quasilinear hyperbolic equations.comm. Partial Diff. Equations. 3(7,8), (1998) [7] A. Constantin, L. Molinet: Orbital stability of solitary waves for a shallow water equation.physica D.157,75-89(001) [8] A. Constantin, L. Molinet: The initial value problem for a generalized Boussinesq equation. Differential and Integral Equations. 15, (00) IJNS homepage:

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