International Journal of Differential Equations
Volume 2017 (2017), Article ID 3879017, 21 pages
https://doi.org/10.1155/2017/3879017

Research Article

Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

Fernando Bernal-Vílchis,¹ Nakao Hayashi,² and Pavel I. Naumkin³

¹Instituto de Matemáticas, UNAM, Campus Morelia, AP 61-3 Xangari, 58089 Morelia, MICH, Mexico
²Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan
³Centro de Ciencias Matemáticas, UNAM, Campus Morelia, AP 61-3 Xangari, 58089 Morelia, MICH, Mexico

Correspondence should be addressed to Nakao Hayashi; nhayashi@math.sci.osaka-u.ac.jp

Received 19 July 2016; Accepted 30 August 2016; Published 23 January 2017

Academic Editor: Baoxiang Wang

Copyright © 2017 Fernando Bernal-Vílchis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 space

under the restriction Rewrite (1) as

Multiplying both sides of (8) by , integrating in space, using (7), we obtain

which 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 asymptotics

is 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 identity

with , , 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 identity

with , , 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 using

we 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 5

Therefore

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 that

for all We write

We 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 that

for 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.

References

N. Hayashi and P. I. Naumkin, “Large time asymptotics for the reduced Ostrovsky equation,” Communications in Mathematical Physics, vol. 335, no. 2, pp. 713–738, 2015.

L. A. Ostrovsky and Y. A. Stepanyants, “Nonlinear surface and internal waves in rotating fluids,” in Nonlinear Waves 3, Research Reports in Physics, pp. 106–128, Springer, Berlin, Germany, 1990.

F. Linares and A. Milanes, “Local and global well-posedness for the Ostrovsky equation,” Journal of Differential Equations, vol. 222, no. 2, pp. 325–340, 2006.

K. Tsugawa, “Well-posedness and weak rotation limit for the Ostrovsky equation,” Journal of Differential Equations, vol. 247, no. 12, pp. 3163–3180, 2009.

V. Varlamov and Y. Liu, “Cauchy problem for the Ostrovsky equation,” Discrete and Continuous Dynamical Systems Series A, vol. 10, no. 3, pp. 731–753, 2004.

G. Gui and Y. Liu, “On the Cauchy problem for the Ostrovsky equation with positive dispersion,” Communications in Partial Differential Equations, vol. 32, no. 10–12, pp. 1895–1916, 2007.

Z. Huo and Y. Jia, “Low-regularity solutions for the Ostrovsky equation,” Proceedings of the Edinburgh Mathematical Society, vol. 49, no. 1, pp. 87–100, 2006.

P. Isaza and J. Mejía, “Cauchy problem for the Ostrovsky equation in spaces of low regularity,” Journal of Differential Equations, vol. 230, no. 2, pp. 661–681, 2006.

J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation,” Geometric and Functional Analysis, vol. 3, no. 3, pp. 209–262, 1993.

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Sharp global well-posedness for KdV and modified KdV on and ,” Journal of the American Mathematical Society, vol. 16, no. 3, pp. 705–749, 2003.

T. Schäfer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D. Nonlinear Phenomena, vol. 196, no. 1-2, pp. 90–105, 2004.

A. Stefanov, Y. Shen, and P. G. Kevrekidis, “Well-posedness and small data scattering for the generalized Ostrovsky equation,” Journal of Differential Equations, vol. 249, no. 10, pp. 2600–2617, 2010.

Y. Liu, D. Pelinovsky, and A. Sakovich, “Wave breaking in the short-pulse equation,” Dynamics of Partial Differential Equations, vol. 6, no. 4, pp. 291–310, 2009.

Y. Liu, D. Pelinovsky, and A. Sakovich, “Wave breaking in the Ostrovsky-Hunter equation,” SIAM Journal on Mathematical Analysis, vol. 42, no. 5, pp. 1967–1985, 2010.

D. Pelinovsky and A. Sakovich, “Global well-posedness of the short-pulse and sine-Gordon equations in energy space,” Communications in Partial Differential Equations, vol. 35, no. 4, pp. 613–629, 2010.

J. P. Boyd, “Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves),” European Journal of Applied Mathematics, vol. 16, no. 1, pp. 65–81, 2005.

J. Hunter, “Numerical solutions of some nonlinear dispersive wave equations,” Lectures in Applied Mathematics, vol. 26, pp. 301–316, 1990.

R. Grimshaw and D. Pelinovsky, “Global existence of small-norm solutions in the reduced Ostrovsky equation,” Discrete and Continuous Dynamical Systems Series A, vol. 34, no. 2, pp. 557–566, 2014.

N. Hayashi, P. I. Naumkin, and T. Niizato, “Asymptotics of solutions to the generalized Ostrovsky equation,” Journal of Differential Equations, vol. 255, no. 8, pp. 2505–2520, 2013.

N. Hayashi, P. I. Naumkin, and T. Niizato, “Nonexistence of the usual scattering states for the generalized Ostrovsky-Hunter equation,” Journal of Mathematical Physics, vol. 55, no. 5, Article ID 053502, 2014.

M. V. Fedoryuk, Asymptotics: Integrals and Series, Mathematical Reference Library, Nauka, Moscow, Russia, 1987.

M. V. Fedoryuk, “Asymptotic methods in analysis,” in Analysis I, vol. 13 of Encyclopaedia of Mathematical Sciences, pp. 83–191, Springer, New York, NY, USA, 1989.

E. M. Stein and R. Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis, vol. 4 of Princeton Lectures in Analysis, Princeton University Press, Princeton, NJ, USA, 2011.