Proceedings of WASCOM 2005, pages World Sci. Publishing, 2006

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1 Proceedings of WASCOM 2005, pages World Sci. Publishing, 2006 A SEMIGROUP OF SOLUTIONS FOR THE DEGASPERIS-PROCESI EQUATION G. M. COCLITE AND K. H. KARLSEN Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO 0316 Oslo, Norway giusepc@math.uio.no, kennethk@math.uio.no We prove the existence of a strongly continuous semigroup of solutions associated with the Cauchy problem for the Degasperis-Procesi equation with initial conditions in L 2 L Introduction Our aim is to investigate well-posedness in the class L 2 L 4 for the Degasperis-Procesi equation 5 t u 3 txxu + 4u x u = 3 x u 2 xxu + u 3 xxxu, (t, x) (0, ) R, (1) endowed with the initial condition u 0 : where we assume that u(0, x) = u 0 (x), x R, (2) u 0 L 2 (R) L 4 (R). (3) Degasperis, Holm, and Hone 7 proved the exact integrability of (1) by constructing a Lax pair. In addition, they displayed a relation to a negative flow in the Kaup-Kupershmidt hierarchy by a reciprocal transformation and derived two infinite sequences of conserved quantities along with a bi-hamiltonian structure. They also showed that the Degasperis-Procesi Partially supported by the BeMatA program of the Research Council of Norway. Current address: Department of Mathematics, University of Bari, Via E. Orabona 4, I Bari, Italy. Supported by an Outstanding Young Investigators Award from the Research Council of Norway. 1

2 2 equation possesses non-smooth solutions that are superpositions of multipeakons and described the integrable finite-dimensional peakon dynamics, which were compared with the multipeakon dynamics of the Camassa- Holm equation. An explicit solution was also found in the perfectly antisymmetric peakon-antipeakon collision case. Lundmark and Szmigielski 8 presented an inverse scattering approach for computing n-peakon solutions to (1). Mustafa 9 proved that smooth solutions to (1) have infinite speed of propagation, namely they loose instantly the property of having compact support. Regarding well-posedness (in terms of existence, uniqueness, and stability of solutions) of the Cauchy problem for (1), Yin has studied this within certain functional classes in a series of recent papers 11,12,13,14. In 4 the authors, proved the well-posedness of the Cauchy problem (1)- (2) in the functional setting L 1 BV. Moreover, in Section 4 of the same paper they showed the existence of solutions under the assumption (3). Here we want to complete the proof of the well-posedness in the functional framework L 2 L 4, in the sense that we show the uniqueness and stability of the solutions of the Cauchy problem (1)-(2) under the assumption (3). Motivated by the fact that, at least formally, (1) is equivalent to the elliptic-hyperbolic system t u + u x u + x P = 0, 2 xxp + P = 3 2 u2, (4) following 4, we have the definitions Definition 1.1. We say that u : [0, ) R R is a weak solution of the Cauchy problem (1)-(2) provided u L ( (0, ); L 2 (R) ) and u satisfies (4)- (2) in the sense of distributions. Moreover, we call u entropy weak solution of (4)-(2) if: u is a weak solution of (1)-(2), u L ((0, T ); L 4 (R)), T > 0, and for any convex C 2 entropy η : R R with η bounded there holds t η(u)+ x q(u)+η (u) x P u 0, in the sense of distributions on (0, ) R, where q : R R is defined by q (u) = η (u) u and P u solves 2 xxp u +P u = 3 2 u2. Of course the definition of entropy weak solution cames form the strong analogy between (1) and the Burgers equation, indeed (4) can be rewritten as a Burgers equation with a nonlocal source. Our main results are collected in the following theorem: Theorem 1.1. There exists a strongly continuous semigroup of entropy weak solutions associated to the Cauchy problem (1)-(2). More precisely, there exists a map S : [0, ) ( L 2 (R) L 4 (R) ) L ((0, ); L 2 (R)),

3 3 with the following properties: for each u 0 L 2 (R) L 4 (R) the map u(t, x) = S t (u 0 )(x) is an entropy weak solution of (1)-(2) and it is strongly continuous with respect to the initial condition in the following sense: if u 0,n u 0 in L 2 (R) then S(u 0,n ) S(u 0 ) in L ((0, T ); L 2 (R)), for every {u 0,n } n N L 2 (R) L 4 (R), u 0 L 2 (R) L 4 (R), T > Viscous Approximations and Compactness Our approach is based on proving the compactness of a family of smooth functions {u ε } ε>0 solving the following viscous problems (see 2 ): t u ε + u ε x u ε + x P ε = ε xxu 2 ε, t > 0, x R, xxp 2 ε + P ε = 3 2 u2 ε, t > 0, x R, (5) u ε (0, x) = u ε,0 (x), x R. We shall assume that u 0,ε H 1 (R) BV (R), u 0,ε u 0 in L 2 (R) L 4 (R). (6) We know from Theorem that for each ε > 0 there exists a unique smooth solution u ε C([0, ); H 1 (R)) C([0, ); BV (R)) to (5). The starting point of our argument is the following compactness result for (5) (see Theorem 4 4 ). Lemma 2.1. Assume (3) and (6). Then there exist an infinitesimal sequence {ε k } k N (0, ) and an entropy weak solution u L ((0, ); L 2 (R)) L ((0, T ); L 4 (R)), T > 0, to (1)-(2) such that u εk u, in L p ((0, T ) R), 1 p < 4, T > 0. The proof is based on ε uniform bounds of u ε in L ((0, ); L 2 (R)) and L ((0, T ); L 4 (R)), T > 0, (see Lemma and Lemma ) and on a compensated compactness argument 10. In order to construct the semigroup of solutions we need to prove that the vanishing viscosity limit (see Lemma 2.1) is unique and depends continuously on the initial condition. The key point of our argument is the L 2 stability estimate stated in the next section. 3. Vanishing Viscosity: L 2 Stability Estimate Let v, w C([0, ); H 1 (R)) C([0, ); BV (R)) be the smooth solutions

4 4 to (see Theorem ) t v + v x v + x V = λ xxv, 2 xxv 2 + V = 3 2 v2, v(0, x) = v 0 (x), respectively, where we shall assume t w + w x w + x W = µ xxw, 2 xxw 2 + W = 3 2 w2, w(0, x) = w 0 (x), λ, µ > 0, v 0, w 0 H 1 (R) BV (R). (7) Theorem 3.1. Assume (7). Then for any t > 0, v(t, ) w(t, ) L 2 (R) e α(t)/2 v 0 w 0 L 2 (R) (8) + e α(t)/2 µ λ ( ) v0 L2 (R) + w 0 L 2 (R), [ ( ) 2 α(t) :=t ( ) 2 x v 0 L1 (R) + x w 0 ] L1 (R) v 0 L 2 (R) + w 0 L 2 (R) + +t 2 24( )( 2 x v 0 L1 (R) + x w 0 L1 (R) v0 L 2 (R) + w 0 L (R)) 2 +t ( v 0 L 2 (R) + w 0 L 2 (R) ) Our approach, as in 1,2, is based on the following homotopy argument. Let 0 θ 1. The function ω θ interpolates between the functions v and w. More precisely, denote by ω θ the smooth solution of the initial value problem t ω θ + ω θ x ω θ + x Ω θ = (θµ + (1 θ)λ) xxω 2 θ, xxω 2 θ + Ω θ = 3 2 ω2 θ, (9) ω θ (0, x) = θw 0 (x) + (1 θ)v 0 (x). Clearly ω 0 = v, ω 1 = w. Indeed θ ω θ (t, x) is a curve joining v(t, x) and w(t, x), and, for each t 0 v(t, ) w(t, ) L2 (R) dist L2 (R)( v(t, ), w(t, ) ) lengthl2 (R)( ωθ (t, ) ). Employing the Implicit Function Theorem, as in Lemma and Lemma 3.2 2, we have the following: Lemma 3.1. Assume (7). The curve θ [0, 1] ω θ (t, ) H 1 (R) BV (R) is of class C 1. In particular, we infer length L2 (R)( ωθ (t, ) ) = 1 0 θ ω θ (t, ) L2 (R)dθ, t 0.

5 5 Using the notation z θ := θ ω θ, Z θ := θ Ω θ and differentiating the equations in (9) with respect to θ, we have t z θ + z θ x ω θ + ω θ x z θ + x Z θ = (θµ + (1 θ)λ) xxz 2 θ + (µ λ) xxω 2 θ, xxz 2 θ + Z θ = 3ω θ z θ, z θ (0, x) = w 0 (x) v 0 (x). Finally, L 2 energy estimates on z θ (t, ) similar to the ones in 1,2 give (8). 4. Proof of Theorem 1.1 Finally we are ready for proving Theorem 1.1. The first step consists in the existence of the semigroup. Lemma 4.1. There exists a strongly continuous semigroup of solutions associated with the Cauchy problem (1)-(2) S : [0, ) (L 2 (R) L 4 (R)) L ((0, ); L 2 (R)), namely, for each u 0 L 2 (R) L 4 (R) the map u(t, x) = S t (u 0 )(x) is a weak solution of (1)-(2). Clearly, this lemma is a direct consequence of the following one and of the ones in the previous section. Lemma 4.2. Let {ε n } n N, {µ n } n N (0, ), ε n, µ n 0, and u, v L ((0, ); L 2 (R)) L ((0, T ); L 4 (R)), T > 0, be such that u εn u, u µn v, strongly in L p ((0, T ) R), T > 0, 1 p < 4, then u = v. Proof. Let t > 0. From Theorem 3.1, we have that ( u εh (t, ) u µh (t, ) L2 (R) e A(t,ε h,µ h ) u 0,εh u 0,µh L2 (R)+ ε h µ h ), min{ε h, µ h } with 0 A(t, ε h, µ h ) δt + δt m 2 h min{ε h, µ h } + δt 2 m h min{ε h, µ h } + δt 3 min{ε h, µ h }, where δ depends only on sup k ( u 0,εk L2 (R) + u 0,µk L2 (R)) and m h := x u 0,εh L1 (R) + x u 0,µh L1 (R). Choosing suitable subsequences as in Lemma we get u = v. The third and last step is the stability of the semigroup. Lemma 4.3. The semigroup S defined on [0, ) L 2 (R) L 4 (R) satisfies the stability property stated in Theorem 1.1.

6 6 Proof. Fix ε > 0 and denote by S ε the semigroup associated with the viscous problem (5). Choose {u 0,n } n N L 2 (R) L 4 (R), u 0 L 2 (R) L 4 (R) such that u 0,n u 0 in L 2 (R). The initial data u 0,ε,n, u 0,ε H 1 (R) BV (R) satisfy condition (6). Finally, write u ε,n := S ε (u 0,ε,n ), u n := S(u 0,n ), u := S(u 0 ). Employing Lemma 2.1, we get with u n (t, ) u(t, ) L2 (R) = lim ε 0 u ε,n (t, ) u ε (t, ) L2 (R), t > 0. Using Theorem 3.1, we get the inequality u ε,n (t, ) u ε (t, )) L2 (R) e A(t,ε,n) u 0,ε,n u 0,ε L2 (R), 0 A(t, ε, n) δ 1 t + δ 1t m 2 ε,n + δ 1t 2 m ε,n + δ 1t 3 ε ε ε, where δ 1 depends only on sup ε>0 u 0,ε L 2 (R) and m ε,n := x u 0,ε,n L 1 (R) + x u 0,ε,n L1 (R). Now, choosing a suitable sequence {ε n } n N as in Lemma the claim follows. References 1. G. M. Coclite and H. Holden, J. Math. Anal. Appl. 308, 221 (2005). 2. G. M. Coclite, H. Holden, and K. H. Karlsen, Discrete Contin. Dynam. Systems 13, 659 (2005). 3. G. M. Coclite, H. Holden, and K. H. Karlsen, to appear on SIAM J. Math. Anal. 4. G. M. Coclite and K. H. Karlsen, to appear on J. Funct. Anal. 5. A. Degasperis and M. Procesi, Symmetry and perturbation theory (Rome, 1998), World Sci. Publishing, River Edge, NJ, 23 (1999). 6. A. Degasperis, D. D. Holm, and A. N. W. Hone, Nonlinear physics: theory and experiment, II (Gallipoli, 2002), World Sci. Publishing, River Edge, NJ, 37 (2003). 7. A. Degasperis, D. D. Holm, and A. N. I. Khon, Teoret. Mat. Fiz. 133, 170 (2002). 8. H. Lundmark and J. Szmigielski, IMRP Int. Math. Res. Pap., 53 (2005). 9. O. G. Mustafa, J. Nonlinear Math. Phys. 12, 10 (2005). 10. M. E. Schonbek, Comm. Partial Differential Equations 7, 95 (1982). 11. Z. Yin, J. Math. Anal. Appl. 283, 129 (2003). 12. Z. Yin, Illinois J. Math. 47, 649 (2003). 13. Z. Yin, Indiana Univ. Math. J. 53, 1189 (2004). 14. Z. Yin, J. Funct. Anal. 212, 182 (2004).