Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

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1 Wang and Wang Boundary Value Problems (07) 07:86 RESEARCH Open Access Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion Junfang Wang* and Zongmin Wang * Correspondence: wangjunfang0309@6.com School of Mathematics and Statistics, Lanzhou University, LanZhou, Gansu , P.R. China Full list of author information is available at the end of the article Abstract The goal of this paper is two-fold. Firstly, by using the Fourier restriction norm method and the fixed point theorem, we prove that the Cauchy problem for a generalized Ostrovsky equation 3 x ut β x3 u + x (u3 ) γ u = 0, β > 0, γ > 0, is locally well-posed in Hs (R) with s 4. Secondly, we prove that the Cauchy problem for a generalized Ostrovsky equation is not well-posed in Hs (R) with s < 4 in the sense that the solution map is C 3. MSC: 35G5 Keywords: generalized Ostrovsky equation with positive dispersion; Cauchy problem; sharp well-posedness Introduction In this paper, we are concerned with the Cauchy problem for a generalized Ostrovsky equation with positive dispersion, 3 3 γ u = 0, x ut β x u + x u 3 γ > 0, β R. (.) Here u(x, t) represents the free surface of the liquid and the parameter γ > 0 measures the effect of rotation. (.) describes the propagation of internal waves of even modes in the ocean; for instance, see the work of Galkin and Stepanyants [, Leonov [, and Shrira [3, 4. The parameter β determines the type of dispersion, more precisely, when β < 0, (.) denotes the generalized Ostrovsky equation with negative dispersion; when β > 0, (.) denotes the generalized Ostrovsky equation with positive dispersion. When γ = 0, (.) reduces to the modified Korteweg-de Vries equation which has been investigated by many authors; for instance, see [5. Kenig et al. [9 proved that the Cauchy problem for the modified KdV equation is locally well-posed in H s (R) with s 4. Kenig et al. [0 proved that the Cauchy problem for the modified KdV equation is illposed in H s (R) with s < 4. Colliander et al. [6 proved that the Cauchy problem for the The Author(s) 07. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Wangand Wang Boundary Value Problems (07) 07:86 Page of modified KdV equation is globally well-posed in H s (R)withs > and globally well-posed 4 in H s (T) withs.guo[7 andkishimoto[ proved that the modified KdV equation is globally well-posed in H 4 (R) with the aid of the I method and some new spaces. Now we give a brief review of the Ostrovsky equation, u t β 3 x u + 3 x ( u ) γ x u =0, γ > 0. (.) Equation (.) was proposed by Ostrovsky in [ as a model for weakly nonlinear long waves in a rotating liquid, by taking into account the Coriolis force, to describe the propagationofsurfacewavesintheoceaninarotatingframeofreference.theparameterβ determines the type of dispersion, more precisely, β < 0 (negative dispersion) for surface and internal waves in the ocean or surface waves in a shallow channel with an uneven bottom and β > 0 (positive dispersion) for capillary waves on the surface of liquid or for oblique magneto-acoustic waves in plasma [, 3 5. Some authors have investigated the stability of the solitary waves or soliton solutions of (.); for instance, see [6 8. Many people have studied the Cauchy problem for (.), for instance, see [7, The result of [3, 5, 3 showed that s = 3 is the critical regularity index for (.). Coclite and di Ruvo [3, 33 have investigated the convergence of the Ostrovsky equation to 4 the Ostrovsky-Hunter one and the dispersive and diffusive limits for Ostrovsky-Hunter type equation. Recently, Li et al. [34 proved that the Cauchy problem for the Ostrovsky equation with negative dispersion is locally well-posed in H 3 4 (R). Levandosky and Liu [6 studied the stability of solitary waves of the generalized Ostrovsky equation, [ ut βu xxx + ( f (u) ) x x = γ u, x R, (.3) where f is a C function which is homogeneous of degree p in the sense that it satisfies sf (s)=pf (s). Levandosky [8 studied the stability of ground state solitary waves of (.4) with homogeneous nonlinearities of the form f (u)=c u p + c u p u, c, c R, p. Equation (.) can be written in the following form: u t β 3 x u + 3 x ( u 3 ) γ x u = 0. (.4) Let w(x, t)=β u(x, β t), then w(x, t)isthesolutionto w t w xxx + 3 x( w 3 ) γβ w =0. Without loss of generality, we can assume that β = γ =. Motivated by [35, firstly, by using the X s,b spaces introduced by [36 40anddeveloped in [8, 4, 4 and the Strichartz estimates established in [9, 43, we prove that (.3)with initial data u(x,0)=u 0 (x) (.5) is locally well-posed in H s (R)withs 4, β >0,γ > 0; secondly, we prove that the problems (.3), (.5) are not quantitatively well-posed in H s (R) withs < 4, β 0,γ >0.Thus,our result is sharp.

3 Wangand Wang Boundary Value Problems (07) 07:86 Page 3 of We introduce some notations before giving the main result. Throughout this paper, we assume that C is a positive constant which may vary from line to line and 0 < ɛ <0 4. A B means that B A 4 B. A B means that A >4 B. ψ(t) is a smooth function supportedin[,andequalsin[,.weassumethatf u is the Fourier transformation of u with respect to both space and time variables and F u is the inverse transformation of u with respect to both space and time variables, while F x u denotes the Fourier transformation of u with respect to the space variable and F x u denotes the inverse transformation of u with respect to the space variable. Let I R, χ I (x)=ifx I; χ I (x)=0ifx does not belong to I.Let =+, φ(ξ)=ξ 3 + ξ, σ = τ + φ(ξ), σ j = τ j + φ(ξ j ) (j =,,3). The space X s,b is defined by X s,b = { u S ( R ) : u Xs,b = ξ s τ + φ(ξ) b F u(ξ, τ) L τξ (R ) < }. The space X T s,b denotes the restriction of X s,b onto the finite time interval [ T, T andis equipped with the norm u X T s,b = inf { w Xs,b : w X s,b, u(t)=w(t)for T t T }. The main results of this paper are as follows. Theorem. Let s and β >0and γ >0.Then the problems (.4), (.5) are locally wellposed in H s (R). More precisely, for u 0 H s (R), there exist a T >0and a unique solution 4 u C([ T, T; H s (R)). Remark The result of Theorem. is optimal in the sense of Theorem.. Theorem. Let s < and β >0and γ >0.Then the problems (.4), (.5) are not wellposed in H s (R) in the sense that the solution map is C 3 4. The rest of the paper is arranged as follows. In Section, wegivesomepreliminaries. In Section 3, we establish a trilinear estimate. In Section 4, we prove Theorem.. In Section 5, we prove Theorem.. Preliminaries In this section, we give Lemmas.-.4. Lemma. Let 0<ɛ < 0 8 and F (P a f )(ξ)=χ { ξ a} (ξ)f f (ξ) with a and F (D b x f )(ξ)= ξ b F f (ξ) with b R. Then we have u L 6 xt C u X0, +ɛ, (.) D 6x P a u L 6 xt C u X0, +ɛ, (.) u L 4 xt C u X0, 3 (.3) 4 ( +ɛ).

4 Wangand Wang Boundary Value Problems (07) 07:86 Page 4 of FortheproofofLemma., we refer the reader to (.7) and (.) of [9. Lemma. Let φ(ξ)=ξ 3 + ξ and F ( I s (u, v) ) (ξ, τ)= φ (ξ ) φ (ξ ) s F u (ξ, τ )F u (ξ, τ ) dξ dτ. ξ=ξ +ξ τ=τ +τ Then we have I (u, u ) L xt C u j. (.4) X0, +ɛ FortheproofofLemma., we refer the reader to Lemma.5 of [43. Lemma.3 Let T (0, ) and b (, 3 ). Then, for s R and θ [0, 3 b), we have ηt (t)s(t)φ Xs,b (R ) CT b φ H s (R), t η T(t) S(t τ)f(τ) dτ CT θ F Xs,b +θ (R ). Xs,b (R ) 0 FortheproofofLemma.3, we refer the reader to [8, 39, 44. Lemma.4 Let a j R (j =,,3)and 3 a j 0.Then we have ( ) 3 a j + 3 a j ) (a 3j + aj [ =3(a + a )(a + a 3 )(a + a 3 ) 3 3 a j( 3 a j). (.5) Proof By using the following two identities: ( ) 3 ( a j a 3 j ) =3(a + a )(a + a 3 )(a + a 3 ), ( 3 a j )(a a + a a 3 + a a 3 ) a j =(a + a )(a + a 3 )(a + a 3 ),

5 Wangand Wang Boundary Value Problems (07) 07:86 Page 5 of which can be found in [6, we have ( ) 3 a j + 3 a j ( ) 3 = a j ) (a 3j + aj [ a 3 j a j 3 a j [ ( 3 =3(a + a )(a + a 3 )(a + a 3 ) a j)(a a + a a 3 + a a 3 )+ 3 3 a j( 3 a j) [ (a + a )(a + a 3 )(a + a 3 ) =3(a + a )(a + a 3 )(a + a 3 ) 3 a j( 3 a j) [ =3(a + a )(a + a 3 )(a + a 3 ) 3 3 a j( 3 a. j) a j Thus, (.5) is valid. This ends the proof of Lemma.4. 3 The trilinear estimate In this section, by using Lemmas.-.,wegivetheproofofLemma3.. Lemma 3. Let u j X s, +ɛ with s and j =,,3.Then we have 4 ) 3 x u j C u j Xs,. (3.) Xs, +ɛ +ɛ Proof To prove (3.), by duality, it suffices to prove that ) [ 3 ū(x, t) x u j dxdt C u j Xs, R u. (3.) X s, +ɛ ɛ Let F(ξ, τ)= ξ s σ ɛ F u(ξ, τ), F j (ξ j, τ j )= ξ j s σ j +ɛ F u j (ξ j, τ j ) (j =,,3). (3.3) To obtain (3.), from (3.3), it suffices to prove that R ξ=ξ +ξ +ξ 3 τ=τ +τ +τ 3 C F L ξ ξ s F(ξ, τ) 3 F j(ξ j, τ j ) σ ɛ 3 ξ j s σ j +ɛ dξ dτ dξ dτ dξ dτ F j L ). (3.4)

6 Wangand Wang Boundary Value Problems (07) 07:86 Page 6 of Without loss of generality, by using the symmetry, we assume that ξ ξ ξ 3 and F(ξ, τ) 0, F j (ξ j, τ j ) 0(j =,).Wedefine { = (ξ, τ, ξ, τ, ξ, τ) R 6, ξ = { = (ξ, τ, ξ, τ, ξ, τ) R 6, ξ = { 3 = (ξ, τ, ξ, τ, ξ, τ) R 6, ξ = { 4 = (ξ, τ, ξ, τ, ξ, τ) R 6, ξ = ξ j, τ = ξ j, τ = ξ j, τ = ξ j, τ = } τ j, ξ 3 ξ ξ 64, } τ j, ξ 64, ξ 4 ξ, } τ j, ξ 64, ξ ξ, ξ ξ 3, } τ j, ξ 64, ξ ξ ξ 3. Obviously, {(ξ, τ, ξ, τ, ξ, τ) R 6, ξ = 3 ξ j, τ = 3 τ j, ξ 3 ξ ξ } 4 j.let and K(ξ, τ, ξ, τ, ξ, τ)= I = R ξ= 3 ξ j τ= 3 τ j ξ ξ s σ ɛ 3 σ j +ɛ (3.5) K(ξ, τ, ξ, τ, ξ, τ)f(ξ, τ) 3 F j (ξ j, τ j ) dξ dτ dξ dτ dξ dτ. ().Inthissubregion,wehave K(ξ, τ, ξ, τ, ξ, τ) C σ ɛ 3 σ. (3.6) j +ɛ By using (3.6) and the Cauchy-Schwartz inequality and the Plancherel identity and the Hölder inequality as well as (.), we have I C R ξ= 3 ξ j τ= 3 τ j C F(ξ, τ) σ ɛ L F(ξ, τ) 3 F j(ξ j, τ j ) σ ɛ 3 σ j dξ +ɛ dτ dξ dτ dξ dτ ξ= 3 ξ j τ= 3 τ j ( ) C F L F Fj σ j +ɛ 3 F j(ξ j, τ j ) L 3 σ j dξ +ɛ dτ dξ dτ L 6 xt ) C F L F j L ).

7 Wangand Wang Boundary Value Problems (07) 07:86 Page 7 of ().Inthissubregion,since φ (ξ ) φ (ξ ) =3 ξ ξ + 3 ξ 3ξ ξ ξ C ξ and ξ ξ,wehave K(ξ, τ, ξ, τ, ξ, τ) C ξ σ ɛ 3 σ j +ɛ C ξ ξ + 3ξ C ξ σ ɛ 3 σ = C φ (ξ ) φ (ξ ) j +ɛ σ ɛ 3 σ. (3.7) j +ɛ By using (3.7) and the Cauchy-Schwartz inequality and the Plancherel identity and the Hölder inequality as well as (.3)-(.4), since 3 4 ( + ɛ)< ɛ,wehave I C R ξ= 3 ξ j τ= 3 τ j φ (ξ ) φ (ξ ) F(ξ, τ) 3 F j(ξ j, τ j ) ( ) ( ( C F F σ ɛ I F L 4 xt ( ) F F3 σ 3 +ɛ C F L L 4 xt F j L ). σ ɛ 3 σ j +ɛ dξ dτ dξ dτ dξ dτ F σ +ɛ ) ( )), F F σ +ɛ L xt (3) 3.Inthissubregion,since φ (ξ ) φ (ξ 3 ) =3 ξ ξ ξ 3ξ ξ 3 ξ 3 C ξ, we have K(ξ, τ, ξ, τ, ξ, τ) C ξ σ ɛ 3 σ j +ɛ C ξ ξ 3 + 3ξ CC ξ 3 σ ɛ 3 σ C φ (ξ ) φ (ξ 3 ) j +ɛ σ ɛ 3 σ. (3.8) j +ɛ By using (3.8) and the Cauchy-Schwartz inequality and the Plancherel identity and the Hölder inequality as well as (.3)-(.4), since 3 4 ( + ɛ)< ɛ,wehave I C R ξ= 3 ξ j τ= 3 τ j φ (ξ ) φ (ξ 3 ) F(ξ, τ) 3 F j(ξ j, τ j ) ( ) ( ( C F F σ ɛ I F L 4 xt ( ) F F σ +ɛ C F L L 4 xt F j L ). σ ɛ 3 σ j +ɛ dξ dτ dξ dτ dξ dτ F σ +ɛ ) ( )), F F3 σ 3 +ɛ L xt

8 Wangand Wang Boundary Value Problems (07) 07:86 Page 8 of (4) 4.Inthissubregion,sinces 4 and ξ ξ ξ 3,wehave K(ξ, τ, ξ, τ, ξ, τ) C ξ s σ ɛ 3 σ j +ɛ C 3 ξ j 6 σ ɛ 3 σ j +ɛ. (3.9) By using (3.9) and the Cauchy-Schwartz inequality and the Plancherel identity and the Hölder inequality as well as (.), since 3 4 ( + ɛ)< ɛ,wehave I C R ξ= 3 ξ j τ= 3 τ j C F σ ɛ C F L L F(ξ, τ) 3 ξ j 6 F j (ξ j, τ j ) σ ɛ 3 σ j +ɛ dξ dτ dξ dτ dξ dτ ( ) D 6 x P F Fj σ j +ɛ F j L ). L 6 xt ) This completes the proof of Lemma Proof of Theorem. In this section, we use Lemmas.3, 3. to prove Theorem.. The solution to (.3), (.5) can be formally rewritten as follows: u(t)=e t( 3 x x ) u We define t (u)=ψ(t)e t( 3 x x ) u 0 + ( t 3 ψ T 0 e (t s)( 3 x x ) x ( u 3 ) ds. (4.) ) t By taking advantaging of Lemmas.3, 3.,wederivethat ( ) (u) Xs, C u 0 H s (R) + C t t ψ +ɛ T 0 0 e (t s)( 3 x x ) x ( u 3 ) ds. (4.) e (t s)( 3 x x ) ( x u 3 ) ds Xs, +ɛ C u 0 H s (R) + CT ɛ x ( u 3 ) ds Xs, +ɛ C u 0 H s (R) + CT ɛ u 3 X. (4.3) s, +ɛ We define B = {u X s, +ɛ : u X C u s, 0 H +ɛ s (R)}. Byusing(4.3), by choosing T sufficiently small such that 4C 3 T ɛ u 0 H s <,wehave (u) Xs, C u 0 H s (R) + CT ɛ( ) 3 C u 0 H s (R) C u0 H s (R), (4.4) +ɛ

9 Wangand Wang Boundary Value Problems (07) 07:86 Page 9 of thus, (u) is a mapping on B. By using a proof similar to (4.4), by choosing T sufficiently small such that 4C 3 T ɛ u 0 H s <,weobtain (u ) (u ) Xs, +ɛ CT ɛ[ u X s, +ɛ + u Xs, +ɛ u Xs, +ɛ + u X s, +ɛ u u Xs, +ɛ u u Xs, +ɛ, (4.5) thus, (u) is a contraction mapping on the closed ball B. Consequently, have a fixed point u and the Cauchy problem for (.) possesses a local solution on [ T, T. The uniqueness of the solution is obvious. This completes the proof of Theorem.. 5 Proof of Theorem. In this section, inspired by [5, 35, 45, we present the proof of Theorem..Wewillprove Theorem. by contradiction. We assume that the solution map of (.4), (.5) isc 3 in H s (R) withs < 4.Then,from Theorem 3 of [35, we have sup B 3 (u 0 ) H s C u 0 3 H s (5.) t [0,T for u 0 H s (R). Here B (u 0 )=e t( 3 x x ) u 0, (5.) B 3 (u 0 )= 3 t We consider the initial data 0 e (t τ)( 3 x x ) x (( B (u 0 ) ) 3) dτ. (5.3) { r r u 0 (x)=r N s e inx e ixξ dξ + e inx By using a direct computation, we have 0 r } e ixξ dξ, r N = O(), N. (5.4) F x u 0 (ξ)=cr N s{ χ [ N, N+r (ξ)+χ [N+r,N+r (ξ) }. Here χ I denotes the characteristic function of a set I R.Obviously, u 0 H s (R). (5.5) We define I := [ N, N + r andi := [N + r, N +r and := I I. By using a direct computation, we have F x B u 0 (ξ)=ce itφ(ξ) F x u 0 (ξ). (5.6)

10 Wangand Wang Boundary Value Problems (07) 07:86 Page 0 of Combining (5.6) with the definition of B 3 (u 0 ), we have B 3 (u 0 )(x, t)=cg. (5.7) Here where g = Cr 3 N 3s H(ξ, ξ, ξ 3 )= We define ξ ξ ξ 3 ( ξ j )e ix 3 ξ j H(ξ, ξ, ξ 3 ) dξ dξ dξ 3, (5.8) eit(φ(ξ )+φ(ξ )+φ(ξ 3 )) e itφ( 3 ξ j ) φ(ξ )+φ(ξ )+φ(ξ 3 ) φ( 3 ξ j). (5.9) ( θ := φ(ξ )+φ(ξ )+φ(ξ 3 ) φ ξ j ). (5.0) From Lemma.4,wehave θ = 3 [ (ξ + ξ )(ξ + ξ 3 )(ξ + ξ 3 ) [ 3 3 ξ j( 3 ξ. (5.) j) To estimate g H s (R), we need to consider the following three cases: Case : ξ j I (j =,,3), Case : ξ j I (j =,,3), Case 3: ξ j I (j =,), ξ 3 I or ξ I, ξ j I (j =,3) or ξ j I (j =,), ξ 3 I or ξ I, ξ j I (j =,3). We assume that g H s (R) corresponding to cases,, 3 are denoted by L, L, L 3,respectively. Case. In this case, we have θ N 3 and ξ + ξ + ξ 3 N.Sincer N = O(), we have L Cr 3 N 3s N s r 5 N CN s 5. (5.) Case. In this case, we have θ N 3 and ξ + ξ + ξ 3 N.Sincer N = O(), we have L Cr 3 N 3s N s r 5 N CN s 5. (5.3) Case 3. In this case, we have θ r N and ξ + ξ + ξ 3 N as well as H t. Since r N = O(), we have L 3 C t r 3 N 3s N s r 5 N C t N s+. (5.4)

11 Wangand Wang Boundary Value Problems (07) 07:86 Page of Combining (5.), (5.5) with(5.)-(5.4), we have t N s+ L 3 L L sup B 3 (u 0 ) H s C u 0 3 Hs C. (5.5) t [0,T For fixed t >0,whens < 4,letN,wehave t N s+ +, and this contradicts (5.5). This ends the proof of Theorem.. Acknowledgements The second author is supported by the Young core Teachers Program of Henan Normal University and 5A0033. Competing interests The authors declare that they have no competing interests. Authors contributions All authors read and approved the final manuscript. Author details School of Mathematics and Statistics, Lanzhou University, LanZhou, Gansu , P.R. China. College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan , P.R. China. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 8 June 07 Accepted: 9 December 07 References. Galkin, VN, Stepanyants, YA: On the existence of stationary solitary waves in a rotating fluid. J. Appl. Math. Mech. 55, (99). Leonov, A: The effect of the Earth s rotation on the propagation of weak nonlinear surface and internal long oceanic waves. Ann. N.Y. Acad. Sci. 373, (98) 3. Shrira, V: Propagation of long nonlinear waves in a layer of a rotating fluid. Izv., Atmos. Ocean. Phys. 7, 76-8 (98) 4. Shrira, V: On long essentially non-linear waves in a rotating ocean. Izv., Atmos. Ocean. Phys., (986) 5. Bourgain, J: Periodic Korteweg de Vries equation with measures as initial data. Sel. Math. New Ser. 3,5-59 (997) 6. Colliander, J, Keel, M, Staffilani, G, Takaoka, H, Tao, T: Sharp global well-posedness for KdV and modified KdV on R and T.J.Am.Math.Soc.6, (003) 7. Guo, ZH: Global well-posedness of the Korteweg-de Vries equation in H 3/4 (R). J. Math. Pures Appl. 9, (009) 8. Kenig, CE, Ponce, G, Vega, L: The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 7, - (993) 9. Kenig, CE, Ponce, G, Vega, L: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, (993) 0. Kenig, CE, Ponce, G, Vega, L: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 06, (00). Kishimoto, N: Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Differ. Integral Equ., (009). Ostrovskii, LA: Nonlinear internal waves in a rotating ocean. Okeanologia 8, 8-9 (978) 3. Benilov, ES: On the surface waves in a shallow channel with an uneven bottom. Stud. Appl. Math. 87, -4 (99) 4. Gilman, OA, Grimshaw, R, Stepanyants, YA: Approximate and numerical solutions of the stationary Ostrovsky equation. Stud. Appl. Math. 95, 5-6 (995) 5. Grimshaw, R: Evolution equations for weakly nonlinear long internal waves in a rotating fluid. Stud. Appl. Math. 73, -33 (985) 6. Levandosky, S, Liu, Y: Stability of solitary waves of a generalized Ostrovsky equation. SIAM J. Math. Anal. 38, (006) 7. Liu, Y, Varlamov, V: Stability of solitary waves and weak rotation limit for the Ostrovsky equation. J. Differ. Equ. 03, (004) 8. Levandosky, S: On the stability of solitary waves of a generalized Ostrovsky equation. Anal. Math. Phys., (0) 9. Gui, GL, Liu, Y: On the Cauchy problem for the Ostrovsky equation with positive dispersion. Commun. Partial Differ. Equ. 3, (007) 0. Guo, BL, Huo, ZH: The global attractor of the damped forced Ostrovsky equation. J. Math. Anal. Appl. 39, (007). Huo, Z, Jia, YL: Low-regularity solutions for the Ostrovsky equation. Proc. Edinb. Math. Soc. 49, (006). Isaza, P: Unique continuation principle for the Ostrovsky equation with negative dispersion. J. Differ. Equ. 5, (03)

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