Stability of Peakons for the Degasperis-Procesi Equation

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the entirely different form of conservation laws for these two equations [2, 12]. Furthermore, the Camassa-Holm equation is a re-expression of geodesic flow on the diffeomorphism group [9] or on the Bott-Virasoro group [27], while no such geometric derivation of the Degasperis-Procesi equation is available. The remainder of the paper is organized as follows. In Section 2, we recall the local well-posedness of the Cauchy problem of equation (1.5), the precise blow-up scenario of strong solutions, and several useful results which are crucial in the proof of stability theorem for equation (1.5) from [29, 30]. Section 3 is devoted to the proof of the stability result (Theorem 1). Notation. As above and henceforth, we denote by ∗ convolution with respect to the spatial variable x ∈ R. We use ‖ · ‖ L p to denote the norm in the Lebesgue space L p (R) (1 ≤ p ≤ ∞), and ‖ · ‖ H s, s ≥ 0 for the norm in the Sobolev spaces H s (R). 2 Preliminaries In the present section, we discuss the issue of well-posedness. The local existence theory of the initial-value problem is necessary for our study of nonlinear stability. We briefly collect the needed results from [23, 29, 30]. Denote p(x) := 1 2 e−|x| , x ∈ R, then (1 − ∂x) 2 −1 f = p ∗ f for all f ∈ L 2 (R) and p ∗ (u − u xx ) = u. Using this identity, we can rewrite the DP equation (1.5) as follows: ( ) 3 u t + uu x + ∂ x p ∗ 2 u2 = 0, t > 0, x ∈ R. (2.1) The local well-posedness of the Cauchy problem of equation (1.5) with initial data u 0 ∈ H s (R), s > 3 2 can be obtained by applying Kato’s theorem [22, 29]. As a result, we have the following well-posedness result. Lemma 2.1 [29] Given u 0 ∈ H s (R), s > 3 2 , there exist a maximal T = T (u 0 ) > 0 and a unique solution u to equation (1.5) (or equation (2.1)), such that u = u(·, u 0 ) ∈ C([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)). Moreover, the solution depends continuously on the initial data, i.e. the mapping u 0 ↦→ u(·, u 0 ) : H s (R) → C([0, T ); H s (R)) ∩ C 1 ([0, T ); H s−1 (R)) is continuous and the maximal time of existence T > 0 can be chosen to be independent of s. The following two lemmas show that the only way that a classical solution to (1.5) may fail to exist for all time is that the wave may break. Lemma 2.2 [29] Given u 0 ∈ H s (R), s > 3 2 , blow up of the solution u = u(·, u 0) in finite time T < +∞ occurs if and only if lim inf { inf [u x(t, x)]} = −∞. t↑T x∈R 7
Lemma 2.3 [23] Assume u 0 ∈ H s (R), s > 3 2 . Let T be the maximal existence time of the solution u to equation (1.5). Then we have ‖u(t, x)‖ L ∞ ≤ 3‖u 0 (x)‖ 2 L 2t + ‖u 0(x)‖ L ∞, ∀t ∈ [0, T ]. Now consider the following differential equation { qt = u(t, q), t ∈ [0, T ), q(0, x) = x, x ∈ R. (2.2) Applying classical results in the theory of ordinary differential equations, one can obtain the following two results on q which are crucial in the proof of global existence and blow-up solutions. Lemma 2.4 [30] Let u 0 ∈ H s (R), s ≥ 3, and let T > 0 be the maximal existence time of the corresponding solution u to equation(1.5). Then the equation (2.2) has a unique solution q ∈ C 1 ([0, T ) × R, R). Moreover, the map q(t, ·) is an increasing diffeomorphism of R with (∫ t ) q x (t, x) = exp u x (s, q(s, x))ds > 0, ∀(t, x) ∈ [0, T ) × R. 0 Lemma 2.5 [30] Let u 0 ∈ H s (R), s ≥ 3, and let T > 0 be the maximal existence time of the corresponding solution u to equation (2.2). Setting y := u − u xx , we have y(t, q(t, x))q 3 x(t, x) = y 0 (x), ∀(t, x) ∈ [0, T ) × R. The next two lemmas clearly show that the solution of equation (1.5) is affected by the shape of the initial profiles, not the smoothness and size of the initial profiles. Lemma 2.6 [23] Let u 0 ∈ H s (R), s > 3 2 . Assume there exists x 0 ∈ R such that { y0 (x) = u 0 (x) − u 0,xx (x) ≥ 0 if x ≤ x 0 , y 0 (x) = u 0 (x) − u 0,xx (x) ≤ 0 if x ≥ x 0 , and y 0 changes sign. Then, the corresponding solution to equation(1.5) blows up in a finite time. Lemma 2.7 [23] Assume u 0 ∈ H s (R), s > 3 2 and there exists x 0 ∈ R such that { y0 (x) ≤ 0 if x ≤ x 0 , y 0 (x) ≥ 0 if x ≥ x 0 . Then equation (1.5) has a unique global strong solution u = u(., u 0 ) ∈ C([0, ∞); H s (R)) ∩ C 1 ([0, ∞); H s−1 (R)). Moreover, E 2 (u) = ∫ R yv dx is a conservation law, where y = (1 − ∂2 x)u and v = (4 − ∂ 2 x) −1 u, and for all t ∈ R + we have (i) u x (t, ·) ≥ −|u(t, ·)| on R, (ii) ‖u‖ 2 1 ≤ 6‖u 0 ‖ 4 L 2 t 2 + 4‖u 0 ‖ 2 L 2 ‖u 0 ‖ L ∞t + ‖u 0 ‖ 2 1. 8
Page 1 and 2: Stability of Peakons for the Degasp
Page 3 and 4: These three cases exhaust in the co
Page 5 and 6: Let us denote E 2 (u) = ‖u‖ 2 X
Page 7: introduce a new idea. By constructi
Page 11 and 12: 3 Proof of stability In this primar
Page 13 and 14: To estimate the first term, by inte
Page 15 and 16: □ Without changing the integral i
Page 17 and 18: A similar argument also shows that
Page 19 and 20: (iii) can be obtained from (i) and
Page 21 and 22: [10] A. Constantin and L. Molinet,

Stability of Peakons for the Degasperis-Procesi Equation

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