Global existence of small-norm solutions in the reduced Ostrovsky equation

We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitz\'eica equation and prove global existence of small-norm solutions in Sobolev space $H^3(R)$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.


Introduction
The reduced Ostrovsky equation is the zero high-frequency dispersion limit (β → 0) of the Ostrovsky equation The evolution equation (2) was originally derived by Ostrovsky [17] to model small-amplitude long waves in a rotating fluid of finite depth. Local and global well-posedness of the Ostrovsky equation (2) in energy space H 1 (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 > 3 2 [20]. But for sufficiently steep initial data u 0 ∈ C 1 (R), local solutions break in a finite time [2,8,14] in the standard sense of finite-time wave 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 open problem up to now. In a similar equation with a cubic nonlinear term (called the short-pulse equation), the proof of global existence was recently developed with the help of a bi-infinite sequence of conserved quantities [18]. These global solutions for small initial data coexist with wave breaking solutions for large initial data [13]. Global existence and scattering of small-norm solutions to zero in the generalized short-pulse equation with quartic and higher-order nonlinear terms follow from the results of [20].
Rather different sufficient conditions on the initial data for wave breaking were obtained recently in [9] on the basis of asymptotic analysis and supporting numerical simulations (similar numerical simulations can be found in [2]). It was conjectured in [9] that initial data u 0 ∈ C 2 (R) with 1 − 3u ′′ 0 (x) > 0 for all 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 finite time.
This paper is devoted to the rigorous proof of the first part of this conjecture, that is, global solutions exist for all initial data u 0 ∈ H 3 (R) such that 1 − 3u ′′ 0 (x) > 0 for all x ∈ R. Note here that if u 0 ∈ H 3 (R), then u 0 ∈ C 2 (R), hence the function 1 − 3u ′′ 0 (x) is continuous for all x ∈ R and approaches 1 as x → ±∞. The second part of the conjecture is also discussed and a weaker statement in line with this conjecture is proven. Integrability of the reduced Ostrovsky equation was discovered first by Vakhnenko [23]. In a series of papers [16,24,25], Vakhnenko, Parkes and collaborators found and explored a transformation of the reduced Ostrovsky equation to the integrable Hirota-Satsuma equation with reversed 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 where the anti-derivative operator is defined by the integration of u(x, t) in x subject to the zero-mass constraint R u(x, t)dx = 0. Higher-order conserved quantities E −1 , E −2 , and so on involve higher-order anti-derivatives, which are defined under additional constraints on the solution u. Hence, these conserved quantities are not related to the H s -norms for positive s and play no role in the study of global well-posedness of the reduced Ostrovsky equation (1) in Sobolev space H s (R) for s > 3 2 . Note in passing that the global well-posedness of the regular Hirota-Satsuma equation in the energy space H 1 (R) was considered recently in [6].
However, a different transformation has recently been discovered for the reduced Ostrovsky equation (1). This transformation is useful to generate a bi-infinite sequence of conserved quantities, which are more suitable for the proof of global existence.
The alternative formulation of the integrability scheme for the reduced Ostrovsky equation starts with the work of Hone and Wang [7], where the reduced Ostrovsky equation (1) was obtained as a short-wave limit of the integrable Degasperis-Procesi equation. As a result of the asymptotic reduction, these authors obtained the following Lax operator pair for the reduced Ostrovsky equation (1) in the original space and time variables: where λ is a spectral parameter. Note that the function 1−3u xx arises naturally in the third-order eigenvalue 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 the Cauchy problem [4,5]. More recently, based on an earlier study of Manna & Neveu [15], Kraenkel et al. [11] found a transformation between the reduced Ostrovsky equation (1) and the integrable Bullough-Dodd equation, which is also widely known as the Tzitzéica equation after its original derivation in 1910 [22]. In new characteristic variables Y and T (see section 2), the Tzitzéica equation can be written in the form, Note that the Tzitzéica equation is similar to the sine-Gordon equation in characteristic coordinates, which arises in the integrability scheme of the short-pulse equation [18]. Similarly to the sine-Gordon equation, the Tzitzéica 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 the first two conserved quantities which were obtained from the power series expansions [1]. The conserved quantities (6) are related to the conserved quantities of the reduced Ostrovsky equation (1) in original physical variables 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 0 in (3) and Q 1 , Q 2 in (6), as well as the reduction to the Tzitzéica equation (5), to prove our main result, which is, Then, the reduced Ostrovsky equation (1) admits a unique global solution u ∈ C(R + , H 3 (R)).
It is natural to expect that the finite-time wave breaking occurs for any u 0 ∈ H 3 (R) such that 1 − 3u ′′ 0 (x) changes sign for some x ∈ R. As shown in [9] in a periodic setting, this criterion of wave breaking is sharper than the previous criteria of wave breaking in [8,14]. Although we are not able to give a full proof of this sharp criterion in the present work, we shall prove the following weaker statement in Section 3: and A prototypical example of the initial data for the reduced Ostrovsky equation on the infinite line is the first Hermite function u 0 (x) = xe −ax 2 , where a > 0 is a parameter. A straightforward computation of the maximum of u ′′ In this case, Theorem 1 implies global existence of solutions for such initial data. When a > a 0 , condition (8) is satisfied. In addition, u 0 (x) has a global minimum at x = − 1 √ 2a so that condition (9) is satisfied for a > a * = e 72 ≈ 0.0378. (Note that a * > a 0 .) Theorem 2 implies wave breaking in a finite time provided that additional constraints are satisfied, that is, u x (x − (t), t) < 0 and u x (x + (t), t) > 0 hold for all times before the wave breaking time along the characteristics x = x ± (t) originating from x ± (0) = X ± . Although we strongly believe that these additional constraints as well as condition (9) are not needed for the statement of Theorem 2, we were not able to lift out these technical restrictions.
The initial function u 0 (x) = xe −ax 2 for a > a * is shown on Fig. 1, where the points X − , X + , and X 0 introduced in Theorem 2 are also shown.

Proof of Theorem 1
We introduce characteristic coordinates for the reduced Ostrovsky equation (1) [9,14,23]: The coordinate tranformation is one-to-one and onto if the Jacobian which is positive for T = 0 because φ(X, 0) = 1, remains positive for all (X, We note that the original equation (1) yields the relation whereas the transformation formulas (10) and (11) yield the relations and Next, in accordance with [11], we introduce the variable If u satisfies the reduced Ostrovsky equation (1), then f satisfies the balance equation In characteristic coordinates (10), we set f (x, t) = F (X, T ), use equation (13), and rewrite the balance equation (16) in the equivalent form where F 0 (X) = F (X, 0). Using equations (14) and (15), we obtain the evolution equation for F (X, T ): We shall now consider the Cauchy problem for the reduced Ostrovsky equation (1) with initial data u 0 ∈ H 3 (R). By the local well-posedness result [20], there exists a unique local solution of the reduced Ostrovsky equation in class u ∈ C([0, t 0 ], H 3 (R)) for some t 0 > 0. By Sobolev embedding of H 3 (R) into C 2 (R), the function f 0 (x) := (1 − 3u ′′ 0 (x)) 1/3 is continuous, bounded, and satisfies f 0 (x) → 1 as |x| → ∞.
To prove Theorem 1, we further require that f 0 (x) > 0 for all x ∈ R, which means from the above properties that inf x∈R f 0 (x) > 0. Because x = X for t = T = 0, we have F 0 ∈ C(R) such that inf X∈R F 0 (X) > 0. In this case, the transformation from X to Y defined by is one-to-one and onto for all X ∈ R, because the Jacobian of the transformation is − 1 3 F 0 (X) < 0 and F 0 (X) → 1 as |X| → ∞. The change of variable, transforms the evolution equation (18) to the integrable Tzitzéica equation (5).
We can now transfer the well-posedness result for local solutions of the reduced Ostrovsky equation (1) to local solutions of the Tzitzéica equation (5).
There exists a unique local solution of the Tzitzéica equation (5) Proof. We rewrite transformations (15) and (20) into the equivalent form,

The inverse of this transformation is
For local solutions of the reduced Ostrovsky equation (1) in class u ∈ C([0, t 0 ], H 3 (R)), we have u xx ∈ C([0, t 0 ], H 1 (R)) for some t 0 > 0 and u xx → 0 as |x| → ∞. We have further assumed that sup x∈R u ′′ 0 (x) < 1 3 , which implies that there is T 0 ∈ (0, t 0 ) such that sup x∈R u xx (x, t) < 1 3 for all t ∈ [0, T 0 ]. Under the same condition, the transformation from X to Y is one-to-one and onto for all X ∈ R. Therefore, V is well-defined for all (Y, T ) ∈ R × [0, T 0 ] and V (Y, T ) → 0 as |Y | → ∞. By construction, V is a solution of the Tzitzéica equation (5) and V (Y, 0) = V 0 (Y ). It remains to show that V is in class V ∈ C([0, T 0 ], H 1 (R)).
The variables V and u xx are related by V = u xx G(u xx ), where Both the function G and its first derivative G ′ remain bounded in L ∞ norm as long as which is satisfied for all t ∈ [0, T 0 ]. Note that G(z) is analytic in z if |z| < 1 3 , but we only need boundedness of G(z) and G ′ (z), which is achieved if z < 1 3 . Next recall the transformations (10) and (19) for any function W (Y, T ) = w(x, t), Therefore, which remains bounded as long as u(·, t) L ∞ and u xx (·, t) L 2 remain bounded. Similarly, we can prove that V Y (·, T ) L 2 remains bounded as long as u(·, t) L ∞ and u xxx (·, t) L 2 remain bounded. Thus, we have V ∈ C([0, T 0 ], H 1 (R)) for some T 0 > 0.

Remark 1
The Jacobian of the transformation from (X, T ) to (x, t) is given by (11) and controlled by the relation (13). Since φ(X, 0) = 1 and we can see that there is the condition φ(X, T ) > 0 remains true as long as V (Y, T ) remains bounded in L ∞ -norm.
Proof. We shall use Q 1 and Q 2 in (6). The quantities are well-defined for a local solution in class V ∈ C([0, T 0 ], H 1 (R)) and conserves in time for the Tzitzéica equation (5), according to the standard approximation arguments in Sobolev spaces.
To be able to use Q 1 for the control of V (·, T ) L 2 , we note that the function H(V ) := 2e V + e −2V − 3 is convex near V = 0 with H(0) = H ′ (0) = 0 and Therefore, H(V ) ≥ V 2 for all V ∈ R, so that By a standard continuation technique, a local solution in class V ∈ C([0, T 0 ], H 1 (R)) is uniquely continued into a global solution in class V ∈ C(R + , H 1 (R)).
It remains to transfer results of Lemmas 1 and 2, as well as the L 2 conservation of E 0 in (3) for the proof of Theorem 1.
Proof of Theorem 1. It follows from the proof in Lemma 1 that Both the function g and its first derivative g ′ remain bounded as long as V remains bounded. By Lemma 2, V ∈ C(R + , H 1 (R)) and hence F (X, T ) > 0 for all (X, T ) ∈ R × R + . Therefore, φ(X, T ) > 0 for all (X, T ) ∈ R × R + , so that the transformation (10) is one-to-one and onto for all (X, T ) ∈ R × R + . Using the bounded functions g and g ′ , we hence have u xx ∈ C(R + , H 1 (R)).
Finally, conservation of E 0 in (3) and the elementary Cauchy-Schwarz inequality, H 3 (R)). This argument completes the proof of Theorem 1.
Using conservation (17) and evolution (18) for F , we obtain the evolution equation for φ(X, T ): Integrating this equation in T with the initial condition φ(X, 0) = 1, we obtain Because the right-hand side of (24) is positive for all X ∈ (X − , X + ), the function φ(X, T ) is monotonically increasing for all X ∈ (X − , X + ) at least for small T ≥ 0. Moreover, we obtain the following inequality.
Proof. Because F 0 (X) < 0 for all X ∈ (X − , X + ), we have from (24): Integrating this inequality in T , we obtain the assertion of the lemma. It follows from Lemma 3 that 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 ) > 6 is preserved for all T ∈ [0, T 0 ), for which the solution is defined.

Lemma 4 Under assumptions
Proof. Under assumption (9), the function φ T | T =0 = U X | T =0 = u ′ 0 (X) changes sign at X = X 0 from being negative for X ∈ (X − , X 0 ) to being positive for X ∈ (X 0 , X + ). Therefore, we can define ξ(0) = X 0 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 remains in class u ∈ C([0, t 0 ], H 3 (R)) with Equation (24) implies that φ X (ξ(T ), T ) > 0 as long as ξ(T ) remains in the interval (X − , X + ). The differential equation (27) hence implies that if U X (X − , T ) < 0 and U X (X + , T ) > 0 for all T ≥ 0, for which the solution is defined, then there exists T -independent constants ξ ± such that Remark 2 Since ξ(0) = X 0 is the point of minimum of U (X, 0) = u 0 (X) and φ X (X, 0) = 0, it follows from equation (27) that . This equation shows that ξ ′ (0) > 0 if u 0 (X 0 ) < 0 and ξ ′ (0) < 0 if u 0 (X 0 ) > 0. Therefore, it is not apriori clear if ξ(T ) can reach X − or X + in a finite time. The restrictions u X (X − , T ) < 0 and u X (X + , T ) > 0 serve as a sufficient condition that ξ(T ) does not reach X − and X + in a finite time, for which the solution is defined.
With the help of Lemmas 3 and 4, we complete the proof of Theorem 2.
Remark 3 Based on the asymptotic analysis and numerical simulations of [9], we anticipate that divergence of φ(X, T ) near X = X + is related to the vanishing of φ(X, T ) near X = X − , such that equation (13) with U X (X − , T ) < 0 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 This upper bound is obtained from the minimization of the integrand in (24) as follows: 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) in estimate (26).