Abstract

We consider the Cauchy problem for the Ostrovsky-Hunter equation , ,  , , where . Define . Suppose that is a pseudodifferential operator with a symbol such that , , and . For example, we can take . We prove the global in time existence and the large time asymptotic behavior of solutions.

1. Introduction

We consider the Cauchy problem for the generalized Ostrovsky-Hunter equation where , We assume that is a pseudodifferential operator with a symbol such that with Also we suppose that and . For example, we can choose Denote by the symbol of the linear part of (1). The constant is a positive root of Our strategy of the proof of the main result is similar to the one used in [1]. We translate (1) into the ordinary differential equation by using the evolution operator related to the linear problem; then we divide the nonlinear term into resonance and nonresonance parts. Nonresonance part has an oscillating term which yields better time decay through the integration by parts; however the factor gives us a singularity at ; see (37) for details. This is the reason why we assume the additional condition on the symbol

We define the evolution group , where the multiplication factor , It is well known that the operator is a useful tool for obtaining the -time decay estimates of solutions and has been used widely for studying the asymptotic behavior of solutions to various nonlinear dispersive equations. We have where , and the antiderivative is defined by the Fourier transform such that Note that the commutators are true , , , , where However, it seems that does not work well on the nonlinear terms. In order to avoid the derivative loss, when estimating the norm instead of the operators we apply the modified dilation operator defined by Note that acts well on the nonlinear terms as the first-order differential operator and it almost commutes with : . Also and are related via the identity where Note that In order to get the estimate of , we will show the a priori estimates of , , and Different point compared to the previous works is to consider the estimate of since contains the term with an additional explicit time growth.

When , then (1) was introduced in [2] for modelling the small-amplitude long waves in a rotating fluid of finite depth. Therefore (1) with is called the Ostrovsky equation. It was studied by many authors (see, e.g., [3–5] and references cited therein). When , (1) is called the reduced Ostrovsky equation. Equation (1) has some conservation quantities, when , One of them is the zero mass conservation law which is obtained by integrating in spaceunder the restriction Rewrite (1) asMultiplying both sides of (8) by , integrating in space, using (7), we obtainwhich is the conservation of the momentum. The same approach as in deriving (9) will be used for the high frequency part in order to avoid the derivative loss, when proving the existence of solutions of (1).

Local well-posedness for the Ostrovsky equation was shown in [5] in the case of the initial data by using the parabolic regularization technique and limiting arguments. Their method works also for the case of the generalized nonlinearity and also generalized reduced Ostrovsky equation (1), since the dispersive effects were not used in the proof. Thanks to the high frequency part , the solutions to the linear equation obtain a smoothing property. By using this property, in [3], the local well-posedness for the Ostrovsky equation was shown under the condition The method of [3] depends on the linear part of the equation and also works for the nonlinearities of a general order. In [4, 6–8] the local well-posedness for the Ostrovsky equation was treated by the Fourier restriction norm method of [9] and in [4] the local well-posedness was shown. We note here that the Sobolev space is considered as critical regularity concerning the Korteweg-de Vries equation.

Global well-posedness in the energy class was obtained for the Ostrovsky equation in [3] through the energy conservation law, when the initial data and After their work, the global well-posedness in was proved in [4, 6] due to the -conservation law. The global well-posedness, in the negative order Sobolev space , was shown in [8] by using the method of [10].

We now turn to the case of the reduced Ostrovsky equation. The local well-posedness was shown in the space in paper [11] and after that in in [12]. Their methods work also in the case of the general nonlinear dispersive equations with different nonlinearities. We also refer to [13, 14] for the local well-posedness in the class However there are few works on the global well-posedness for the reduced Ostrovsky equation due to the lack of the smoothing property. The global well-posedness for reduced Ostrovsky equation (1) with and cubic nonlinearity (which is called the short pulse equation) was obtained in [15], when the initial data whereas for the quadratic nonlinearity (which is called the reduced Ostrovsky equation or the Ostrovsky-Hunter equation; see [16, 17]), it was shown in [18] when the initial data for all . The time decay properties of solutions to the corresponding linear problem can be studied if we assume that the initial data decay rapidly at infinity. So the global existence was shown in [12], for the nonlinearity with an integer , when the initial data are small and sufficiently regular: In [1, 19, 20], we considered the large time asymptotics for reduced Ostrovsky equation (1) with and some conditions on the order of nonlinearity.

To state our results precisely we introduce Notation and Function Spaces. We denote the Lebesgue space by , where the norm for and ess. for . The weighted Sobolev space is , , , and We also use the notations , shortly, if they do not cause any confusion. Let be the space of continuous functions from an interval to a Banach space Different positive constants might be denoted by the same letter . We define the free evolution group , where the multiplication factor .

We are now in a position to state our main result.

Theorem 1. Assume that the initial data are real-valued with a sufficiently small norm . Then there exists a unique global solution of Cauchy problem (1) satisfying the time decay estimate Moreover there exists a unique modified final state such that the asymptoticsis valid for uniformly with respect to , where is a small constant and

2. Factorization Technique

We now introduce the factorization formulas for (1). We have for the free evolution group , where the multiplication factor , Denote the Heaviside function for and for Then for the real-valued function we find where the dilation operator . Note that there is a unique stationary point in the integral , which is defined by the root of the equation for all Thus and we introduce the so-called scaling operator and the multiplication factor Note that, in the case of , then is defined by ; namely, for . Hence for , ; see [1]. Therefore is the scaling operator if the symbol is homogeneous.

By the definition of , its inverse operator is defined by Then we have where the phase function and the operator We have , , and Hence Also we decompose the inverse operator Since , then where and the operator Define the new dependent variable . Since , where , applying the operator to (1) we get Then since we find the following representation: Note that for Thus we obtain the following equation for the new dependent variable : where

Now we explain how to use (37) for estimating uniformly with respect to . For the real-valued solution , we have ; hence it is sufficient to consider the case only. From Lemmas 2 and 3, we find that the last two terms of the right-hand side of (37) are the remainders. We need to consider the first and the second terms of the right-hand side of (37). Due to the oscillating factor , integrating by parts with respect to time, we will show that the first term of (37) is also a remainder, since is bounded in view of the conditions for the symbol

We organize the rest of our paper as follows. In Section 3, we state main estimates for the decomposition operators and related to the evolution group . We prove a priori estimates of solutions in Section 4. Section 5 is devoted to the proof of Theorem 1.

3. Preliminaries

3.1. Two Kernels

Define the kernel where , the phase function , , the cut-off function is such that for or and for , and the Heaviside function for and for We change ; then we get To compute the asymptotics of the kernel for large we apply the stationary phase method (see [21, 22], p. 163):for , where the stationary point is defined by By virtue of formula (40) with , , , we get In particular we have the estimate Also we define the kernel We change ; then we get By virtue of formula (40) with , , , we obtain In particular we have the estimate

3.2. Estimates in the Uniform Norm

In the next lemma we estimate the operator in the uniform norm. Denote , , , , , and

Lemma 2. Let Then the estimates are valid for all

Proof. We write for For the first summand we integrate by parts via the identitywith , , to get Using the estimates in the domain , we find Changing we have Thus we obtain for all , , and
To estimate the second integral we integrate by parts via the identitywith , , to get Using the estimate in the domain , or , we obtain Changing we obtain For , and For Hence In the same manner changing we get For , and For Hence for Thus we have for all , , and
For the case of we integrate by parts using identity (53): Using the estimate in the domain , , we obtain Then as above we get for all , , and Lemma 2 is proved.

By Lemma 2, we have the estimate We next consider the operator .

Lemma 3. The estimates are valid for all , where

Proof. We find for In the first integral using the identity with ,   , we integrate by parts Then using the identity we get Applying the estimates and , , and for , we find Hence In the second integral , using the identity with we integrate by parts Then usingwe get Then using the estimates in the domain or , we get Therefore Next we consider Using the identity with we integrate by parts Then using formula (82), we get Then using the estimates in the domain and , we get Therefore Lemma 3 is proved.

3.3. Estimates for Derivatives

Denote , such that Since and then by the Riesz interpolation theorem (see [23], p. 52) we have for We now estimate the derivative .

Lemma 4. The estimate is true for all ,  , where

Proof. Since and , we have for So we need to estimate where , and For the first summand using we have for Consider the second summand where Changing we get We can rotate the contour of integration , since we see that for , for , and for , and hence Then by the Young inequality we obtain To estimate we integrate by parts via identity (47) Using the estimates we obtain Since changing we get In the case of , the same estimate is obtained easier than the case of the positive line. Lemma 4 is proved.

We need estimate of .

Lemma 5. The estimate is true for all , where

Proof. Since integrating by parts we get with where Using the estimate we find for the first summand and for the second summand Consider the third summand where Changing , , we get We can rotate the contour of integration , since we see that for , for , and for , and hence Then by the Young inequality we obtain Next we estimate We integrate by parts via identity (53) to get Using the estimates , we obtain Changing we find Hence we get In the last integral we integrate by parts via identity (47) Using the estimates , we obtain Since changing we get In the case of , the same estimate is obtained easier than the case of the positive line. Lemma 5 is proved.

3.4. Asymptotics for the Nonlinearity

We obtain the asymptotic representation for the nonlinear term. Define the norm

Lemma 6. The asymptotics is true for all , , where

Proof. In view of (37) we find for the new dependent variable where By Lemma 3 we have By Lemma 2 Therefore Also by the Leibnitz rule Then by Lemma 4 we get Hence we obtain In the same manner Hence the result of the lemma follows.

4. A Priori Estimates

We define where , , and is small. We have the local in time existence of solutions.

Theorem 7. Let the initial data Then there exists a time such that (1) has a unique solution in .

To get the desired results, we prove a priori estimates of solutions uniformly in time.

Lemma 8. Assume and the norm Then the estimate is true for all .

Proof. By continuity of the norm with respect to , arguing by the contradiction we can find the first time such that Consider a priori estimates of . To avoid the derivative loss in (1) we apply the operator and use the commutators , , where Also we represent with ; that is, . Then we get Define the high and short frequency projectors , where for and for , and also Then we get and the integral equation Hence applying the energy method to the first equation we find and by the integral equation Applying the estimate of Lemma 2 we have Hence Therefore And similarly from which it follows that By the identity , we obtain Next we estimate the norm Denote Applying the operator to (1) via the commutator , we get where Using the factorization formulas as in the derivation of (37) we find where we denote Then we get Next using identity (36) we find with Next using the relations and , we get , , and , where , and Therefore we obtain where By Lemma 2 we have for , and then by Lemma 4 we obtain Then we represent where We need to estimate the derivative . We have Since and also we find for the second summand Since , we obtain By Lemmas 4 and 5Therefore Thus we get Hence Then as the above using the projectors and we find We have and then we get Therefore Next we estimate In the domain we get by the Sobolev imbedding theorem if , so we need to estimate the function in the domain Next by (37) for , using Lemma 6, we get for all , . Multiplying this formula by we get in the domain Define the cut-off function , such that for and for , and define Thus we get for all The third term is estimated by To exclude the resonant term we make a change , where Then we get with Integrating by parts we obtain Thus we get the estimate in the domain Therefore we find the desired estimate . This is the desired contradiction. Lemma 8 is proved.

5. Proof of Theorem 1

The global existence of solution to Cauchy problem (1) satisfying a priori estimate follows by a standard continuation argument from Lemma 8 and local existence Theorem 7. We need only to prove asymptotic formula (20).

We need to compute the asymptotics of the function . As in the proof of Lemma 8 we get for any Therefore there exists a unique final state such thatfor all We writeWe study the asymptotics in time of the remainder term . We have By (185) we obtain for any Hence there exists a unique real-valued function such thatfor all Representation (186) and estimate (188) yield for all Thus we get the large time asymptotics Therefore we obtain the estimate with Using the factorization of we have This completes the proof of asymptotics (20). Theorem 1 is proved.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The work of Nakao Hayashi is partially supported by JSPS KAKENHI Grant nos. JP25220702 and JP15H03630. The work of Pavel I. Naumkin is partially supported by CONACYT and PAPIIT Project IN100616.