Global existence of small-norm solutions in the reduced Ostrovsky equation
aa r X i v : . [ n li n . S I] A p r Global existence of small-norm solutionsin the reduced Ostrovsky equation
Roger Grimshaw
Department of Mathematical Sciences, Loughborough University,Loughborough, LE11 3TU, UK
Dmitry Pelinovsky
Department of Mathematics, McMaster University,Hamilton, Ontario, L8S 4K1, Canada
Abstract : We use a novel transformation of the reduced Ostrovsky equation to the integrableTzitz´eica equation and prove global existence of small-norm solutions in Sobolev space H ( R ).This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reducedOstrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking. The reduced Ostrovsky equation ( u t + uu x ) x = u, (1)is the zero high-frequency dispersion limit ( β →
0) of the Ostrovsky equation( u t + uu x + βu xxx ) x = u. (2)The evolution equation (2) was originally derived by Ostrovsky [17] to model small-amplitudelong waves in a rotating fluid of finite depth. Local and global well-posedness of the Ostrovskyequation (2) in energy space H ( R ) was studied in recent papers [10, 12, 21, 26].Corresponding rigorous results for the reduced Ostrovsky equation (1) are more complicated.Local solutions exist in Sobolev space H s ( R ) for s > [20]. But for sufficiently steep initial data u ∈ C ( R ), local solutions break in a finite time [2, 8, 14] in the standard sense of finite-timewave breaking that occurs in the inviscid Burgers equation u t + uu x = 0.However, a proof of global existence for sufficiently small initial data has remained an openproblem up to now. In a similar equation with a cubic nonlinear term (called the short-pulseequation ), the proof of global existence was recently developed with the help of a bi-infinitesequence of conserved quantities [18]. These global solutions for small initial data coexist withwave breaking solutions for large initial data [13]. Global existence and scattering of small-normsolutions to zero in the generalized short-pulse equation with quartic and higher-order nonlinearterms follow from the results of [20].Rather different sufficient conditions on the initial data for wave breaking were obtainedrecently in [9] on the basis of asymptotic analysis and supporting numerical simulations (similarnumerical simulations can be found in [2]). It was conjectured in [9] that initial data u ∈ C ( R )with 1 − u ′′ ( x ) > x ∈ R generate global solutions of the reduced Ostrovsky equation (1),whereas a sign change of this function on the real line inevitably leads to wave breaking in finitetime.This paper is devoted to the rigorous proof of the first part of this conjecture, that is, globalsolutions exist for all initial data u ∈ H ( R ) such that 1 − u ′′ ( x ) > x ∈ R . Note herethat if u ∈ H ( R ), then u ∈ C ( R ), hence the function 1 − u ′′ ( x ) is continuous for all x ∈ R x → ±∞ . The second part of the conjecture is also discussed and a weakerstatement in line with this conjecture is proven.Integrability of the reduced Ostrovsky equation was discovered first by Vakhnenko [23]. In aseries of papers [16, 24, 25], Vakhnenko, Parkes and collaborators found and explored a trans-formation of the reduced Ostrovsky equation to the integrable Hirota–Satsuma equation withreversed roles of the variables x and t . As a particular application of the power series expansions[19], one can generate a hierarchy of conserved quantities for the reduced Ostrovsky equation (1).This hierarchy includes the first two conserved quantities E = Z R u dx, E − = Z R (cid:20) ( ∂ − x u ) + 13 u (cid:21) dx, (3)where the anti-derivative operator is defined by the integration of u ( x, t ) in x subject to thezero-mass constraint R R u ( x, t ) dx = 0.Higher-order conserved quantities E − , E − , and so on involve higher-order anti-derivatives,which are defined under additional constraints on the solution u . Hence, these conserved quantitiesare not related to the H s -norms for positive s and play no role in the study of global well-posednessof the reduced Ostrovsky equation (1) in Sobolev space H s ( R ) for s > . Note in passing thatthe global well-posedness of the regular Hirota–Satsuma equation in the energy space H ( R ) wasconsidered recently in [6].However, a different transformation has recently been discovered for the reduced Ostrovskyequation (1). This transformation is useful to generate a bi-infinite sequence of conserved quan-tities, which are more suitable for the proof of global existence.The alternative formulation of the integrability scheme for the reduced Ostrovsky equationstarts with the work of Hone and Wang [7], where the reduced Ostrovsky equation (1) wasobtained as a short-wave limit of the integrable Degasperis–Procesi equation . As a result of theasymptotic reduction, these authors obtained the following Lax operator pair for the reducedOstrovsky equation (1) in the original space and time variables: (cid:26) λψ xxx + (1 − u xx ) ψ = 0 ,ψ t + λψ xx + uψ x − u x ψ = 0 , (4)where λ is a spectral parameter. Note that the function 1 − u xx arises naturally in the third-ordereigenvalue problem (4) in the same way as the function m = u − u xx arises in another integrable Camassa–Holm equation to determine if the global solutions or wave breaking will occur in theCauchy problem [4, 5].More recently, based on an earlier study of Manna & Neveu [15], Kraenkel et al. [11] founda transformation between the reduced Ostrovsky equation (1) and the integrable
Bullough–Doddequation , which is also widely known as the
Tzitz´eica equation after its original derivation in 1910[22]. In new characteristic variables Y and T (see section 2), the Tzitz´eica equation can be writtenin the form, ∂ V∂T ∂Y = e − V − e V . (5)Note that the Tzitz´eica equation is similar to the sine–Gordon equation in characteristic coordi-nates, which arises in the integrability scheme of the short-pulse equation [18]. Similarly to thesine–Gordon equation, the Tzitz´eica equation has a bi-infinite sequence of conserved quantities,which was discovered in two recent and independent works [1, 3]. Among those, we only need thefirst two conserved quantities Q = Z R (cid:0) e V + e − V − (cid:1) dY, Q = Z R (cid:18) ∂V∂Y (cid:19) dY, (6)2hich were obtained from the power series expansions [1]. The conserved quantities (6) are relatedto the conserved quantities of the reduced Ostrovsky equation (1) in original physical variables E = Z R h (1 − u xx ) / − i dx, E = Z R ( u xxx ) (1 − u xx ) / dx. (7)Note that the conserved quantities (7) also appeared in the balance equations derived in [11].In Section 2, we shall use the conserved quantities E in (3) and Q , Q in (6), as well as thereduction to the Tzitz´eica equation (5), to prove our main result, which is, Theorem 1
Assume u ∈ H ( R ) such that − u ′′ ( x ) > for all x ∈ R . Then, the reducedOstrovsky equation (1) admits a unique global solution u ∈ C ( R + , H ( R )) . It is natural to expect that the finite-time wave breaking occurs for any u ∈ H ( R ) such that1 − u ′′ ( x ) changes sign for some x ∈ R . As shown in [9] in a periodic setting, this criterion ofwave breaking is sharper than the previous criteria of wave breaking in [8, 14]. Although we arenot able to give a full proof of this sharp criterion in the present work, we shall prove the followingweaker statement in Section 3: Theorem 2
Assume that u ∈ H ( R ) is given and there is a finite interval [ X − , X + ] and a point X ∈ ( X − , X + ) such that − u ′′ ( x ) < , x ∈ ( X − , X + ) , (8) and u ′ ( x ) < , x ∈ ( X − , X ) , u ′ ( x ) > , x ∈ ( X , X + ) , (9) whereas − u ′′ ( x ) ≥ for all x ≤ X − and x ≥ X + . Then, a local solution u ∈ C ([0 , t ) , H ( R )) of the reduced Ostrovsky equation (1) breaks in a finite time t ∈ (0 , ∞ ) in the sense lim sup t ↑ t k u ( · , t ) k H ( R ) = ∞ if u x ( x − ( t ) , t ) < and u x ( x + ( t ) , t ) > hold for all t ∈ [0 , t ) along the characteristics x = x ± ( t ) originating from x ± (0) = X ± . A prototypical example of the initial data for the reduced Ostrovsky equation on the infiniteline is the first Hermite function u ( x ) = xe − ax , where a > u ′′ ( x ) shows that u ′′ ( x ) < for all x ∈ R if a ∈ (0 , a ), where a = e −√ − √ ≈ . . In this case, Theorem 1 implies global existence of solutions for such initial data. When a > a ,condition (8) is satisfied. In addition, u ( x ) has a global minimum at x = − √ a so that condition(9) is satisfied for a > a ∗ = e ≈ . a ∗ > a .) Theorem 2 implies wavebreaking in a finite time provided that additional constraints are satisfied, that is, u x ( x − ( t ) , t ) < u x ( x + ( t ) , t ) > x = x ± ( t ) originating from x ± (0) = X ± . Although we strongly believe that these additionalconstraints as well as condition (9) are not needed for the statement of Theorem 2, we were notable to lift out these technical restrictions.The initial function u ( x ) = xe − ax for a > a ∗ is shown on Fig. 1, where the points X − , X + ,and X introduced in Theorem 2 are also shown.3
101 X − X X + xu (x) Figure 1: An example of the initial condition u ( x ) = xe − ax with a = 0 .
1, where positions of X − , X + , and X are shown. We introduce characteristic coordinates for the reduced Ostrovsky equation (1) [9, 14, 23]: x = X + Z T U ( X, T ′ ) dT ′ , t = T, u ( x, t ) = U ( X, T ) . (10)The coordinate tranformation is one-to-one and onto if the Jacobian φ ( X, T ) = 1 + Z T U X ( X, T ′ ) dT ′ , (11)which is positive for T = 0 because φ ( X,
0) = 1, remains positive for all (
X, T ) ∈ R × [0 , T ],where T > u = U XT φ = φ T T φ , (12)whereas the transformation formulas (10) and (11) yield the relations u x = U X φ = φ T φ , (13)and u xx = 1 φ (cid:18) U X φ (cid:19) X = φ T X φ − φ X φ T φ . (14)4ext, in accordance with [11], we introduce the variable f = (1 − u xx ) / . (15)If u satisfies the reduced Ostrovsky equation (1), then f satisfies the balance equation f t + ( uf ) x = { u − u xt − ( uu x ) x } x f / = 0 . (16)In characteristic coordinates (10), we set f ( x, t ) = F ( X, T ), use equation (13), and rewrite thebalance equation (16) in the equivalent form(
F φ ) T = 0 ⇒ F ( X, T ) φ ( X, T ) = F ( X ) , (17)where F ( X ) = F ( X, F ( X, T ): ∂ ∂T ∂X log( F ) = − ∂ ∂T ∂X log( φ ) = 13 φ ( F −
1) = 13 F ( X )( F − F − ) . (18)We shall now consider the Cauchy problem for the reduced Ostrovsky equation (1) with initialdata u ∈ H ( R ). By the local well-posedness result [20], there exists a unique local solution of thereduced Ostrovsky equation in class u ∈ C ([0 , t ] , H ( R )) for some t >
0. By Sobolev embeddingof H ( R ) into C ( R ), the function f ( x ) := (1 − u ′′ ( x )) / is continuous, bounded, and satisfies f ( x ) → | x | → ∞ .To prove Theorem 1, we further require that f ( x ) > x ∈ R , which means from theabove properties that inf x ∈ R f ( x ) >
0. Because x = X for t = T = 0, we have F ∈ C ( R ) suchthat inf X ∈ R F ( X ) >
0. In this case, the transformation from X to Y defined by Y := − Z X F ( X ′ ) dX ′ (19)is one-to-one and onto for all X ∈ R , because the Jacobian of the transformation is − F ( X ) < F ( X ) → | X | → ∞ . The change of variable, F ( X, T ) = e − V ( Y,T ) , (20)transforms the evolution equation (18) to the integrable Tzitz´eica equation (5).We can now transfer the well-posedness result for local solutions of the reduced Ostrovskyequation (1) to local solutions of the Tzitz´eica equation (5). Lemma 1
Assume u ∈ H ( R ) such that − u ′′ ( x ) > for all x ∈ R . Let V ( Y ) := −
13 log(1 − u ′′ ( x )) , Y := − Z x (1 − u ′′ ( x ′ )) / dx ′ . There exists a unique local solution of the Tzitz´eica equation (5) in class V ∈ C ([0 , T ] , H ( R )) for some T > such that V ( Y,
0) = V ( Y ) .Proof. We rewrite transformations (15) and (20) into the equivalent form, u xx ( x, t ) = 13 (cid:0) − f ( x, t ) (cid:1) = 13 (cid:16) − e − V ( Y,T ) (cid:17) . V ( Y, T ) = −
13 log (1 − u xx ( x, t )) . For local solutions of the reduced Ostrovsky equation (1) in class u ∈ C ([0 , t ] , H ( R )), we have u xx ∈ C ([0 , t ] , H ( R )) for some t > u xx → | x | → ∞ . We have further assumed thatsup x ∈ R u ′′ ( x ) < , which implies that there is T ∈ (0 , t ) such that sup x ∈ R u xx ( x, t ) < for all t ∈ [0 , T ]. Under the same condition, the transformation from X to Y is one-to-one and onto forall X ∈ R . Therefore, V is well-defined for all ( Y, T ) ∈ R × [0 , T ] and V ( Y, T ) → | Y | → ∞ .By construction, V is a solution of the Tzitz´eica equation (5) and V ( Y,
0) = V ( Y ). It remains toshow that V is in class V ∈ C ([0 , T ] , H ( R )).The variables V and u xx are related by V = u xx G ( u xx ), where G ( u xx ) := log(1 − u xx )( − u xx ) . Both the function G and its first derivative G ′ remain bounded in L ∞ norm as long assup x ∈ R u xx ( x, t ) < , which is satisfied for all t ∈ [0 , T ]. Note that G ( z ) is analytic in z if | z | < , but we only needboundedness of G ( z ) and G ′ ( z ), which is achieved if z < .Next recall the transformations (10) and (19) for any function W ( Y, T ) = w ( x, t ), k W ( · , T ) k L = Z R W ( Y, T ) dY = 13 Z R W ( Y, T ) F ( X ) dX = 13 Z R w ( x, t ) F ( X ) φ ( X, T ) dx = 13 Z R w ( x, t ) f ( x, t ) dx. Therefore, k V ( · , T ) k L ≤ √ k G ( u xx ( · , t )) k L ∞ k f ( · , t ) k L ∞ k u xx ( · , t ) k L , which remains bounded as long as k u ( · , t ) k L ∞ and k u xx ( · , t ) k L remain bounded. Similarly, wecan prove that k V Y ( · , T ) k L remains bounded as long as k u ( · , t ) k L ∞ and k u xxx ( · , t ) k L remainbounded. Thus, we have V ∈ C ([0 , T ] , H ( R )) for some T > (cid:3) Remark 1
The Jacobian of the transformation from ( X, T ) to ( x, t ) is given by (11) and con-trolled by the relation (13). Since φ ( X,
0) = 1 and φ ( X, T ) = exp (cid:18)Z T u x ( x ( X, T ) , T ) dT (cid:19) , (21) we can see that there is T > such that φ ( X, T ) > for all ( X, T ) ∈ R × [0 , T ] . Because φ ( X, T ) = F ( X ) F ( X, T ) = F ( X ) e V ( Y,T ) , (22) the condition φ ( X, T ) > remains true as long as V ( Y, T ) remains bounded in L ∞ -norm. emma 2 Let V ∈ C ([0 , T ] , H ( R )) for some T > be a unique local solution of the Tzitz´eicaequation (5). Then, in fact, V ∈ C ( R + , H ( R )) .Proof. We shall use Q and Q in (6). The quantities are well-defined for a local solution inclass V ∈ C ([0 , T ] , H ( R )) and conserves in time for the Tzitz´eica equation (5), according to thestandard approximation arguments in Sobolev spaces.To be able to use Q for the control of k V ( · , T ) k L , we note that the function H ( V ) :=2 e V + e − V − V = 0 with H (0) = H ′ (0) = 0 and H ′′ ( V ) = 2 e V + 4 e − V ≥ , V ∈ R . Therefore, H ( V ) ≥ V for all V ∈ R , so that k V k H = k V k L + k V Y k L ≤ Q + Q . By a standard continuation technique, a local solution in class V ∈ C ([0 , T ] , H ( R )) is uniquelycontinued into a global solution in class V ∈ C ( R + , H ( R )). (cid:3) It remains to transfer results of Lemmas 1 and 2, as well as the L conservation of E in (3)for the proof of Theorem 1. Proof of Theorem 1.
It follows from the proof in Lemma 1 that u xx = V g ( V ), where g ( V ) := 1 − e − V V .
Both the function g and its first derivative g ′ remain bounded as long as V remains bounded.By Lemma 2, V ∈ C ( R + , H ( R )) and hence F ( X, T ) > X, T ) ∈ R × R + . Therefore, φ ( X, T ) > X, T ) ∈ R × R + , so that the transformation (10) is one-to-one and onto forall ( X, T ) ∈ R × R + . Using the bounded functions g and g ′ , we hence have u xx ∈ C ( R + , H ( R )).Finally, conservation of E in (3) and the elementary Cauchy–Schwarz inequality, k u x k L ≤ k u k L k u xx k L , implies that u ∈ C ( R + , H ( R )). This argument completes the proof of Theorem 1. (cid:3) We utilize the characteristic coordinates (10) and consider the evolution of the Jacobian φ definedby (11). Recall that φ ( X,
0) = 1 whereas F ( X,
0) = F ( X ) = (1 − u ′′ ( X )) / . By conservation(17), assumption (8), and local existence in class u ∈ C ([0 , t ] , H ( R )), we have F ( X, T ) < X ∈ ( X − , X + ) at least for small T ≥
0, whereas F ( X, T ) ≥ X ≤ X − and X ≥ X + .Using conservation (17) and evolution (18) for F , we obtain the evolution equation for φ ( X, T ): ∂ ∂T ∂X log( φ ) = 13 φ (cid:18) − F ( X ) φ (cid:19) . (23)Integrating this equation in T with the initial condition φ ( X,
0) = 1, we obtain ∂φ∂X = 13 φ ( X, T ) Z T φ ( X, T ′ ) (cid:18) − F ( X ) φ ( X, T ′ ) (cid:19) dT ′ . (24)Because the right-hand side of (24) is positive for all X ∈ ( X − , X + ), the function φ ( X, T ) ismonotonically increasing for all X ∈ ( X − , X + ) at least for small T ≥
0. Moreover, we obtain thefollowing inequality. 7 emma 3
Let ψ ( X, T ) := R T φ ( X, T ′ ) dT ′ . Under assumption (8) of Theorem 2, we have ∂ψ∂X ≥ ψ ( X, T ) , X ∈ ( X − , X + ) , (25) as long as the solution remains in class u ∈ C ([0 , t ] , H ( R )) .Proof. Because F ( X ) < X ∈ ( X − , X + ), we have from (24): ∂φ∂X ≥ φ ( X, T ) Z T φ ( X, T ′ ) dT ′ = 16 ∂∂T (cid:18)Z T φ ( X, T ′ ) dT ′ (cid:19) . Integrating this inequality in T , we obtain the assertion of the lemma. (cid:3) It follows from Lemma 3 that ∂∂X (cid:18) − ψ (cid:19) ≥ ⇒ ψ ( X, T ) ≥ ψ ( ξ, T )6 − ( X − ξ ) ψ ( ξ, T ) , X ∈ ( ξ, X + ) , (26)for any ξ ∈ ( X − , X + ), which may depend on T . Therefore, ψ ( X, T ) becomes infinite near X = X + provided that ( X + − ξ ) ψ ( ξ, T ) > T ∈ [0 , T ), for which the solution is defined.To ensure that this is inevitable under assumptions of Theorem 2, we prove the following result. Lemma 4
Under assumptions (8) and (9) of Theorem 2, there exists a C function ξ ( T ) and T -independent constants ξ ± such that φ ( ξ ( T ) , T ) = 1 , ξ (0) = X ∈ ( X − , X + ) , and ξ ( T ) ∈ [ ξ − , ξ + ] ⊂ ( X − , X + ) for all T ≥ , as long as the solution remains in class u ∈ C ([0 , t ] , H ( R )) with U X ( X − , T ) < and U X ( X + , T ) > .Proof. Under assumption (9), the function φ T | T =0 = U X | T =0 = u ′ ( X ) changes sign at X = X from being negative for X ∈ ( X − , X ) to being positive for X ∈ ( X , X + ). Therefore, we candefine ξ (0) = X and consider the level curve φ ( ξ ( T ) , T ) = 1. It follows from the definition (11)that the function φ ( X, T ) is continuously differentiable in X and T as long as the solution remainsin class u ∈ C ([0 , t ] , H ( R )) with dξdT = − φ T ( ξ ( T ) , T ) φ X ( ξ ( T ) , T ) = − U X ( ξ ( T ) , T ) φ X ( ξ ( T ) , T ) . (27)Equation (24) implies that φ X ( ξ ( T ) , T ) > ξ ( T ) remains in the interval ( X − , X + ).The differential equation (27) hence implies that if U X ( X − , T ) < U X ( X + , T ) > T ≥
0, for which the solution is defined, then there exists T -independent constants ξ ± such that ξ ( T ) ∈ [ ξ − , ξ + ] ⊂ ( X − , X + ). (cid:3) Remark 2
Since ξ (0) = X is the point of minimum of U ( X,
0) = u ( X ) and φ X ( X,
0) = 0 , itfollows from equation (27) that ξ ′ (0) = − U XT ( X ,
0) + ξ ′ (0) U XX ( X , φ XT ( X , T ) ⇒ ξ ′ (0) = − u ( X )2 u ′′ ( X ) . This equation shows that ξ ′ (0) > if u ( X ) < and ξ ′ (0) < if u ( X ) > . Therefore, it is notapriori clear if ξ ( T ) can reach X − or X + in a finite time. The restrictions u X ( X − , T ) < and u X ( X + , T ) > serve as a sufficient condition that ξ ( T ) does not reach X − and X + in a finitetime, for which the solution is defined. Proof of Theorem 2.
We use estimate (26) with ξ ( T ) defined by Lemma 4. Then, we have Z T φ ( X, T ′ ) dT ′ ≥ T − ( X − ξ ( T )) T , X ∈ ( ξ ( T ) , X + ) . (28)By Lemma 4, there are T -independent constants ξ ± such that ξ ( T ) ∈ [ ξ − , ξ + ] ⊂ ( X − , X + ) as longas U X ( X − , T ) < U X ( X + , T ) >
0. The lower bound in (28) diverges at a point X ∈ ( ξ + , X + )if T > X + − ξ + . However, divergence of R T φ ( X, T ′ ) dT ′ implies divergence of φ ( X, T ) for some X ∈ ( ξ + , X + ) also in a finite time T ∈ (0 , ∞ ). Then, equation (21) shows that u x ( x, t ) cannotbe bounded if φ ( X, T ) becomes infinite for some X ∈ ( ξ + , X + ) and some T = T , hence the norm k u ( · , T ) k H ( R ) diverges as T ↑ T . This argument completes the proof of Theorem 2. (cid:3) Remark 3
Based on the asymptotic analysis and numerical simulations of [9], we anticipate thatdivergence of φ ( X, T ) near X = X + is related to the vanishing of φ ( X, T ) near X = X − , such thatequation (13) with U X ( X − , T ) < would imply that u x diverges in a finite time near x = x − ( t ) .However, the best that can be obtained from equation (24) is φ ( X − , T ) ≤ φ ( X, T ) e − α ( X ) T , α ( X ) := 12 / Z XX − | F ( X ′ ) | dX ′ , X ∈ [ X − , X + ] . (29) This upper bound is obtained from the minimization of the integrand in (24) as follows: φ + | F ( X ) | φ ≥ / | F ( X ) | . If X = ξ ( T ) ∈ ( X − , X + ) with φ ( ξ ( T ) , T ) = 1 , the bound (29) only gives an exponential decay of φ ( X − , T ) to zero as T → ∞ . The same difficulty appears in our attempts to use bound (29) inestimate (26). Acknowledgement:
D.P. appreciates support and hospitality of the Department of Mathe-matical Sciences of Loughborough University. The research was supported by the LMS VisitingScheme Program.
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