A Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.9(2010) No.3,pp A Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation Caixia Shen 1, Li-meng Xia 2 1,2 Faculty of Science, Jiangsu University (Received 2 January, accepted 21 May 2010) Abstract: In this paper, we study the existence of low-regularity non-continuous solutions of the periodic Degasperis-Procesi equation. A class of shock wave solution are given. Keywords: periodic; Degasperis-Procesi equation; solution 1 Introduction In this paper we study the initial value problem of the Degasperis-Procesi equation (DP) u t + uu x = x (1 2 x) 1 ( 3 2 u2 ) u(0, x) = u 0 (x), on the circle S = R/Z. For smooth functions equation (1) can be written in the more familiar form u t u txx + 4uu x 3u x u xx uu xxx = 0. In [2], the authors concern the existence and the uniqueness of solutions of Camassa-Holm equations using the ODE approach, and it is applied for F-W equations in [5]. Naturally, we find that this method is also usable for DP equations and the same result can be obtained. How ever, such result doesn t contain any non-continuous solution. Also with the ODE method, we concern the existence of some non-continuous solutions of (DP). Suppose that = R/2Z. As usual H s p = H s p(, R) denotes the Sobolev space H s p = {f = k ˆf k e 2kπix : f s,p < } where and ( f s,p = Λ s f L p := Λ s f(x) p dx ) 1 p ˆΛ s f(k) =< k > s ˆf(k), < k >:= (1 + k 2 ) 1 2, particularly, according to [1, heorem 7.48], the norm f s,p in Hp s for 1 s := 1 + σ < 2 and 1 p < is equivalent to ( f p1,p + f (x) f (y) p ) 1/p. For s = 0, we often write L p instead of Hp. s Further we introduce for a continuous 2-periodic function f : R the norm f W 1, := f + f x and the space W 1, = Lip = Lip() Lip() := {f : R : f W 1, < }. his project is supported by the Postdoctoral Foundation and Science Foundation of Jiangsu University ( , 07JDG035, 07JDG038). Corresponding author. address: shencaixia@tom.com Copyright c World Academic Press, World Academic Union IJNS /361

2 368 International Journal of NonlinearScience,Vol.9(2010),No.3,pp Definition 1: Define the set B ρ,r as B ρ,r := {f Lip : f < ρ, f x < r} and denote by B ρ,r H s p and by L H s p the corresponding subsets of H s p endowed with the topolohy induced by the norm of H s p. Definition 2: For odd u 0 on [ 1 2, 1 2 ] define its 2-periodization eu 0, and 0 y 1 2, eu 0 (x ± 2) = eu 0 (x) = eu 0 ( x), eu 0 (y) = u(y), eu 0 (1 y) = u(y) = u( y) contained in H s p(). Definition 3: For any u = u(t, x) define on R, define u = u(t, x) = u(t, x ± 1) such that u(t, y) = u(t, y), y [ 1 2, 1 2 ), then u is 1-periodic on S Our Main Result is stated as following: heorem 1 For any odd u 0 on [ 1 2, 1 2 ] such that u 0 L + u 0 L <, then eu 0 OE s p, hence there exists a 2-periodic solution eu(t, x) of the Cauchy problem of (DP) with initial value eu 0, so u(t, x) = eu(t, x) is the 1-periodic solution with initial value u 0 (on S, determined by x [ 1 2, 1 2 )). 2 Preliminaries In this section we collect some preliminary results needed. For s, p 1, consider the linear space Ep s := Hp s Lip supplied with the norm u E s p := u s,p + u W 1,. Lemma 2. Ep s is a Banach space and OE s p = {f E s p f(x) = f( x)} is a Banach subspace. Lemma 3. (Moser s estimate)for s > 0, 1 < p <, we have uv s,p C( u L v s,p + u s,p v L ) (cf. [4,Ch. 13, 10, Corollary 10.6]). Corollary 1. For s 0, 1 < p <, both H s p L and E s p are algebras with respect to the pointwise multiplication of functions. For s, p 1, denote by D s p the Banach manifold of transformations with elements : (x) = x + f(x) for all f OE s p such that ess inf f(x) x > 1. Lemma 4. Let 1 s < 2 and 1 p <. hen D s p defines a homeomorphism : with inverse 1 D s p. Proof. At first, is an orientation preserving homeomorphism of. 1 ((x)) = x = (x) f( 1 ((x))), IJNS for contribution: editor@nonlinearscience.org.uk

3 C. X. Shen, L. M. Xia: A Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation 369 since f is 2-periodic odd, then is odd and (2 + x) = 2 + (x) as well as 1, so g = f 1 is 2-periodic odd. According to [2], g is contained in E s p, so g OE s p. Lemma 5. Let 1 s < 2, 1 p <. hen u E s p for any u E s p and D s p. More over, if u OE s p, then u OE s p. Proof. For any D s p : x x + f(x), f(x) E s p, and any u E s p, u is still continuous(hence absolutely continuous). Since (u ) = u. u W 1, = u + u 2 u W 1,. Using the Hölder inequality we have, with s = 1 + σ, u ((x)) (x) u ((y)) (y) p 2 p (x) p u ((x)) u ((y)) p +2 p u ((y)) p (x) (y) p = 2 p (I + II), estimate I and II, respectively. I = where g(x) = 1 (x) x, so we have u E s p. If u E s p is odd, then (x) p u ((x)) u ((y)) p (x) p 1 (y) 1 u ((x)) u ((y)) p (x) (y) 1+σp (x) (y) 1+σp x y 1+σp d(x)d(y) (1 + f W 1, ) p(σ+1) (1 + g W 1, ) u ((x)) u ((y)) p (x) (y) 1+σp d(x)d(y) (1 + f W 1, ) p(σ+1) (1 + g W 1, ) u p s,p < II = = u ((y)) p (x) (y) p u ((y)) p f (x) f (y) p u W 1, f p s,p < u(( x)) = u( (x)) = u((x)) hence u is also odd. Lemma 6. Let 1 s < 2, 1 < p <. hen for any D s p, 0 < ρ and 0 < r < the right translation B ρ,r H s p H s p, u u is continuous. Lemma 7. Let 1 s < 2, 1 p <. For any sequence (u k ) k 1 E s p with u k u in E s p and any sequence ( k ) k 1 in D s p so that k in H s p with and D s p one has sup k 1 ( k W 1, + 1 W 1, ) < k u k k u in Hp. s Lemma 8. For any sequence ( k ) k 1 in Dp s and any Dp s with k in Dp, s it follows that 1 k 1 in Hp s and he proof of [Lemma 6-8]is similar to that in [2]. sup( k W 1, + 1 k W 1, ) <. k 1 IJNS homepage:

4 370 International Journal of NonlinearScience,Vol.9(2010),No.3,pp he vector field F For (, v) D s p OE s p(1 s < 2, 1 < p < ) define the vector field F by (, v) F (, v) := (v, f(, v)) OE s p OE s p where f(, v) := {(1 2 x) 1 x ( 3 2 (v 1 ) 2 )} First note that 1 D s p and v 1 E s p, hence 3 2 (v 1 ) 2 E s p. As the range of the operator (1 2 x) 1 x acting on E s p is still contained in E s p, we get that (1 2 x) 1 x (v 1 ) 2 E s p, hence F (, v) = (v, f(, v)) E s p E s p, more over, if v OE s p, then v 1 is still odd, so x ( 3 2 (v 1 ) 2 ) is odd as well as hus we have x (1 2 x) 1 ( 3 2 (v 1 ) 2 ). F (, v) = (v, f(, v)) OE s p OE s p, In this section we prove that F is a C 1 -vector field. First let us rewrite f(, v) in the following way ( ) 3 f(, v) = (R Λ 2 x R 1) 2 v2 where Λ 2 = (1 2 x) and R v denotes the right translation v of v E s p by element D s p. Write P := R P R 1 for any given operator P acting on E s p. Note that (R ) 1 = R 1 and (Λ ) 1 = (Λ 1 ). Let t (t) D s p be a C 1 -path passing through 0 = with d dt t=0(t) = w E s p. he directional derivative of P in direction w is defined (at least in a formal way) as where D w (P )(f) := d dt t=0(p (t) f). Similar to the proof in [2], we have the following results. Lemma 9. (i) ( x ) = 1 x ; (ii) D w ( x ) = w ( ) 2 x. Lemma 10. he operators (1 ± x ) : Ep s Hp s 1 L are bijective and (1 ± x ) 1 f = ±(F ±(0) ± Moreover, for any D s p given, the linear mapping is bounded. Lemma 11. he mapping x 0 (y)e ±(y) f(y)dy)e (y) 1 F ± (0) = ±(e ±1 1) 1 (y)e ±(y) f(y)dy. (1 ± x ) 1 0 : H s 1 p L E s p is continuous. Where D s p L(E s p, L(H s 1 p D ( ) (1 ± x ) 1 L, E s p)), D ( ) (1 ± x ) 1 : w D w (1 ± x ) 1 IJNS for contribution: editor@nonlinearscience.org.uk

5 C. X. Shen, L. M. Xia: A Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation 371 Proposition 1. he mapping D s p L(H s 1 L, E s p), (1 ± x ) 1 is C 1. Lemma 12. Write h(, v) = 3 2 v2. hen (i) d dt t=0h((t), v) = 0; (ii) d dt t=0h(, v + tw) = 3vw. Proposition 2. he map h, defined on Dp s Ep s takes values in Ep s Hp s 1 L. Viewed as a map h : Dp s Ep s Hp s 1 L, h is C 1. Proposition 3.he mapping F : Dp s OEp, s (, v) (v, f(, v)) defines a C 1 -vector field in a neighborhood of (id, 0) Dp s OEp. s 4 ODE on D s p OE s p Consider the following ODE on D s p OE s p with initial data where 1 s < 2 and 1 < p < and { = v v = f(, v) { (0) = id v(0) = v 0 E s p f(, v) := {(1 2 x) 1 x ( 3 2 (v 1 ) 2 )} By Proposition 3, F (, v) = (v, f(, v)) is a C 1 -vector field in an open neighborhood of (id, 0) in D s p OE s p(not of (id, 0) in D s p E s p). Hence the standard existence and uniqueness theorem for ODEs can be applied (see e.g. [6], IV, 1). heorem 2. Let 1 s < 2 and 1 < p <. hen there exists a neighborhood U(0) of zero in OE s p and > 0 such that the initial value problem (45)-(46) has a unique C 1 -solution Moreover, the map is C 1. With the notations of heorem 2, we have Corollary 5. he map is a C 1 -map. (, v) : (, ) D s p OE s p, t ((t), v(t)). (, ) U(0) D s p OE s p, (t, v 0 ) ((t, v 0 ), v(t, v 0 )) U(0) C 1 ((, ), D s p OE s p), v 0 ((, v 0 ), v(, v 0 )) 5 he initial value problem he Degasperis-Procesi equation on is given by { ut + uu x = x (1 x) 2 1 ( 3 2 u2 ) u(0, x) = u 0 Let 1 s < 2 and 1 < p <, then a solution (, v) C 1 ((, ), D s p OE s p) of { = v v = f(, v) IJNS homepage:

6 372 International Journal of NonlinearScience,Vol.9(2010),No.3,pp with initial data { (0) = id v(0) = u 0 OE s p gives rise to a solution u(t, x) := v(t, 1 (t)(x)) of (DP). A direct computation shows that t u(t) := v(t) (t) 1 H s p satisfies the Degasperis-Procesi equation u t + uu x = (1 2 x) 1 ( 3 2 u2 ) pointwise for any t (, ) and x. As t u(t) is continuous in H s p it follows that t u t (t) H s 1 p L. Hence we have shown that u C 0 ((, ), B ρ,r Hs p) C 1 ((, ), H s 1 p L ), where ρ, r are chosen depending on 0 < <. For any solution u = u(t, x) of Degasperis-Procesi equation on above, u 2 is 1-periodic continuous on S and u satisfying the Degasperis-Procesi equation for any x Z If x = n, then in then sense of distributional functions, the Degasperis-Procesi equation still holds. Hence u is an 1-periodic distributional solution. Particularly, if u(0, 1 2 ) = 0, then this solution is not continuous. How ever, for these solutions, u(t, x)dx is a conserved quantity, i.e., is independent S of < t <. (In particular, u(t, x)dx 0.) S Denoted by [f](t, x) = f(t, x + 0) f(t, x 0). heorem 3. u is a non-continuous weak solution. Proof. Since the line of discontinuity of u is x = 1 2 (on S) and [ 1 2 u2 ] = 0, hence the condition of non-continuous solution: [u]s = [ 1 2 u2 ] holds, where s is dx dt on the line of discontinuity. heorem 4. u is a shock wave solution. Proof. Since u 2 is continuous, u t + uu x + x ( 1 2 e x ( 3 2 u2 )) = 0 for any x = 1 2 and it is easy to show that [u t + uu x + x ( 1 2 e x ( 3 2 u2 ))] is finite, hence for any positive φ(x) Cc, we have { t (p( u)) + x (q( u)) p (u) x (e x ( 3 2 u2 ))}φdx = 0 S for any C 2 convex entropy p(u), q(u) such that q (u) = p (u)u, so the entropy condition is satisfied. An example: Suppose that u 0 = sinh(x) c on [ 1 2, 1 2 ) with c > 0, then it is odd and not continuous on S. As we know, there is a solution sinh(x) u(t, x) = cosh( 1 2 )(t + c/ cosh( 1 2 )) for all t > c/ cosh( 1 2 ) and x S. If c < 0, then t < c/ cosh( 1 2 ). Hence for any c = 0, we can choose 0 = c / cosh 1 2. References [1] R. Adams, Sobolev spaces, Pure and Applied Mathematics, vol.65, Academic Press, [2] C. Lellis,. Kappeler,P. opalov,low-regularity solutions of the periodic,communications in Partial Differential Equations, 32(1): [3] V. Arnold, Sur lag eom etrie differentielle des groupes de Lie de dimension infinitr et ses applications `a l hydrodynamique des fluids parfaits, Ann. Inst. Fourier, 16(1):(1996), IJNS for contribution: editor@nonlinearscience.org.uk

7 C. X. Shen, L. M. Xia: A Class of Shock Wave Solutions of the Periodic Degasperis-Procesi Equation 373 [4] M. aylor Partial Differential Equations III. Nonlinear Equations, Applied Mathe- matical Sciences 117, Springer-Verlag, [5] L. ian, Y. Chen, X. Jiang, L. Xia, Low-regularity solutions of the periodic Fornberg-Whitham equation, Journal Of Mathematical Physics, Volume 50, Issue 7,July [6] S. Lang, Differential manifolds, Addison-Wesley series in mathematics, IJNS homepage: