Hm convergence rates of solutions of GBBM

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Progress in Natural Science 19 (2009) 1053–1057 www.elsevier.com/locate/pnsc

Hm convergence rates of solutions of GBBM equations in multi-dimensional spaces Shaomei Fang a,*, Boling Guo b a

Department of Mathematics, South China Agricultural University, Guangzhou 510642, China b Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Received 16 October 2008; received in revised form 4 November 2008; accepted 5 November 2008

Abstract In this paper, we study the decay rates of the generalized Benjamin–Bona–Mahony equations in n-dimensional space. By using Fourier analysis for long wave and by applying the energy method for short wave, we obtain the Hm convergence rates of the solutions when the initial data are in the bounded subset of the phase space H m ðRn Þðn P 3Þ. The optimal decay rates are obtained in our results and are found to be the same as the Heat equation. Ó 2009 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. Keywords: Generalized Benjamin–Bona–Mahony equations; Fourier analysis; Energy method; Optimal convergence rates

1. Introduction The Benjamin–Bona–Mahony (BBM) equation and its counterpart, the Kortewegde Vries (KdV) equation, were both proposed as model equations for long waves in nonlinear dispersive media. The BBM equation was advocated and studied in [3] by Benjamin, Bona and Mahony. In [8], Guo proposed and studied a generalized BBM equation in multi-dimensions. The existence and uniqueness of solutions for GBBM equations have been proved by many authors (see [2,3,7,8]). The large time behaviors and decay rates of solutions to the initial value problems and the initial boundary value problems were also studied in [1,4,5,9]. But these studies do not give a clear picture of the decay rate in multi-dimensions. The goal of this paper is to give the optimal decay rates of solutions to Cauchy problems of the generalized BBM equations in multi-dimensional space.

*

Corresponding author. Tel.: +86 20 87344806. E-mail address: [email protected] (S. Fang).

We consider the system defined on Rn ðn P 3Þ ut Dut gDu þ ðb rÞu þ divðf ðuÞÞ ¼ 0;

ð1Þ

where g is a positive constant and b is a real constant vecs tor, f ðuÞ ¼ ðf1 ðuÞ; . . . ; fn ðuÞÞ . The initial data are given by ujt¼0 ¼ u0 ðxÞ:

ð2Þ

In fact, we have given an optimal decay rate in the phase space H 1 ðRn Þ for the case of the nonlinear functions fj ðuÞ ¼ u2 in [6]. The paper is a follow-up of [6]. We obtained the optimal decay rates in the phase space Hm(Rn) for the generalized nonlinear functions. As we know, it is difficult to get the optimal decay rates of solutions using only energy estimates. Some new techniques are needed. As in [6], we give the frequency of decomposition for solutions. Then, by using the Fourier analysis for long wave and by applying the energy method for short wave to solutions, the L2 convergence rates of the solution were obtained when the initial data are in the bounded subset of the phase space Hm(Rn). This is the main difference between our study and related results of previous

1002-0071/$ - see front matter Ó 2009 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. doi:10.1016/j.pnsc.2008.11.009

1054

S. Fang, B. Guo / Progress in Natural Science 19 (2009) 1053–1057

works. But how to obtain the Hm convergence rates of the solution is a new difficulty. We need more technology and more mature methods than those mentioned in [6]. Throughout this paper, we denote the generic constants as C. Wm,p(Rn), m 2 Z þ , p 2 ½1; 1; and we define the usual Sobolev space with the norm m X kf kW m;p :¼ k@ a f kLp : jaj¼0

m,2

m

In particular, W = H . As usual, Fourier transformation to the variable x 2 Rn is Z pffiffiffiffi fb ðn; tÞ ðf f Þðn; tÞ ¼ f ðx; tÞe 1xn dx; Rn

We will first recall some basic results for Fourier analysis. Lemma 2.1 (Bernstein inequality). For all 1 6 p 6 q < 1, we have kvðDÞDax f kLq ðRn Þ 6 C p;q ejajþnð1=p1=qÞ kf kLp ðRn Þ; where Cp,q is a constant depending only on p and q. Now we come back to considering (1). As in Ref. [6], applying the operator v(D) to (1), we have ðuL Þt DðuL Þt gDuL þ ðb rÞuL þ divðvðDÞf ðuÞÞ ¼ 0: ð5Þ Define U(u) = div u(u) with u(u) = (v(D)f(u)); we can then write (5) as

and the inverse Fourier transform to the variable n is Z pffiffiffiffi n 1 b f ðx; tÞ ðf f Þðx; tÞ ¼ ð2pÞ fb ðn; tÞe 1xn dn:

ðuL Þt DðuL Þt gDuL þ ðb rÞuL ¼ UðuÞ:

Rn

By using the above notations, we can now state the main result of this paper as follows. n 1 n Theorem 1.1. Given a ball B ¼ fv 2 Hmþ1 ðR Þ \ L ðR Þ : n maxðkvkL1 ; kvkH mþ1 Þ 6 Rg ðn P 3; m P 2 þ 1; if k P 3; P 0 if k ¼ 2Þ. If the initial data are u0 ðxÞ 2 B and fj ðuÞ ¼ uk ðj ¼ 1; ; n; k 2 Z þ Þ, then there exists a solution uðx; tÞ 2 L1 ð½0; 1; H mþ1 Þ to (1), defined globally in time. Moreover, for jaj 6 m þ 1

For the long wave of the solutions, we first consider the Cauchy problem of the linear part of (5) as follows: Et DEt gDE þ ðb rÞE ¼ 0; ð6Þ Ejt¼0 ¼ dðxÞ;

ð3Þ

where d(x) is the Dirac function. After making the Fourier transformation to the variable x 2 Rn in (6), we obtain the b following ordinary differential equation for E: ( p ffiffiffiffiffiffi ffi b þ jnj2 ð EÞ b þ gjnj2 E b þ 1ðb nÞ E b ¼ 0; ð EÞ t t ð7Þ b ¼ 1; Ej

Remark 1. If k ¼ 2, this theorem is a generalization of the result in [6].

ð@x1 ; . . . ; @xn Þ. By direct where n corresponds to Dx ¼ p1ffiffiffiffi 1 calculation, we have

k@ ax uð; tÞkL2 6 Cð1 þ tÞ

jaj

n4 2

:

Remark 2. The decay rate of solutions to Heat equations is n

jaj

also ð1 þ tÞ4 2 : The rest of the paper is arranged as follows. In Section 2, we will give the estimates for long wave of solutions by using Fourier analysis. In Section 3, the estimates for short wave of solutions obtained by using the energy method are given. Then, the time-asymptotic behavior of the solutions to (1) and (2) follows these estimates. 2. The estimates for long wave In this section, we will study the estimates for long wave of solutions, using Fourier analysis. Let 1; jnj < e; vðnÞ ¼ ð4Þ 0; jnj > 2e; be a smooth cut-off function, with e being sufficiently small. Set v(D) as a pseudodifferential operator with symbol v(n). Define uL ðx; tÞ ¼ vðDÞuðx; tÞ; uS ðx; tÞ ¼ ð1 vðDÞÞuðx; tÞ: Here, uL(x, t) and uS(x, t) represent long wave of solutions and short wave of solutions, respectively.

t¼0

b tÞ ¼ ekðnÞt ; Eðn;

ð8Þ

with 2

kðnÞ

gjnj

pffiffiffiffiffiffiffi 1ðb nÞ

1 þ jnj

ð9Þ

2

Some properties of the solutions Eðx; tÞ can be obtained as follows. Lemma 2.2 [6]. For any fixed e there exists a positive constant C, such that jaj

b L ðn; tÞj 6 Cð1 þ tÞ 2 ; jna E Z n b L ðn; tÞj2 dn 6 Cð1 þ tÞjaj2 : jna E

ð10Þ

Rn

The proof of Lemma 2.2 can be seen in [6]. By using this lemma, we get the following proposition about the estimates on uL. The solution of (5) can be formulated by the Duhamel principle as Z t EL ð; t sÞ Uðuð; sÞÞds: uL ðx; tÞ ¼ ðEL ðvðDÞu0 ÞÞðx; tÞ þ 0

ð11Þ


Since kDax ðEL ðvðDÞu0 ÞÞkL2 6 kDax EL kL2 kðvðDÞu0 ÞkL1 ; by using Lemma 2.2, one has kDax ðEL ðvðDÞu0 ÞÞkL2 6 Cð1 þ tÞ

jaj

2 n4

ku0 kL1 :

ð12Þ

For the second term of (11), first Z

t

kDax EL ð; t sÞ Uðuð; sÞÞkL2 ds ¼

Z

0

3. The estimates for short wave In this section, we will establish some L2 estimates for the short wave by using the energy method. By applying the operator 1 v(D) to (1), we obtain ðuS Þt DðuS Þt gDuS þ ðb rÞuS þ divðð1 vðDÞÞf ðuÞÞ ¼ 0;

t=2

ð18Þ

kDax EL ð; t sÞ Uðuð; sÞÞkL2 ds 0

þ

Z

n

t

kEL ð; t sÞ Dax Uðuð; sÞÞkL2 ds t=2

Z

t=2

kDax EL ð; t sÞ Uðuð; sÞÞkL1 ds

6C þC

Z

0 t

kEL ð;t sÞDax Uðuð; sÞÞkL1 ds:

by integrating its product with uS over R [0, t], we find 1 d 2 2 2 ðkuS ðtÞkL2 þ kruS ðtÞkL2 Þ þ gkruS ðtÞkL2 2 dt Z þ ððb rÞuS þ divðð1 vðDÞÞf ðuÞÞÞuS dx ¼ 0; ð19Þ Rn

t=2

Let jaj

IðtÞ ¼

þn

sup 06s6t;jaj6mþ1

kDa uð; sÞkL2 ð1 þ sÞ 2 4 :

ð13Þ

jaj þn sup kDa uð; tÞkL2 6 ð1 þ tÞ ð 2 4Þ IðtÞ;

jaj6mþ1

sup kDa uð; tÞkL1 6 ð1 þ tÞð

jaj n 2 þ4

Þ IðtÞ:

since Z ððb rÞuS ÞuS dx ¼ 0; Rn

and Z ðdivðf ðuÞÞÞu dx ¼ 0;

By (13) and Sobolev’s embedded theorem, we have

Rn

ð14Þ

jaj6m½n2

Then, by using the Bernstein inequality, we have kDax Uðuð; sÞÞkL1 6 CekDax uðuð; sÞÞkL1 X 6 Ce kDax 1 uð; sÞkL2 kDax 2 uð; sÞkL2 kRa3 ðuk2 ÞkL1 ; ja1 jþja2 jþja3 j¼jaj

one has Z ððb rÞuS þ divðð1 vðDÞÞf ðuÞÞÞuS dx Rn Z ¼ ðdivðvðDÞÞf ðuÞÞuL dx n ZR ððdivðð1 vðDÞÞf ðuÞÞÞÞuL dx n ZR ðdivðvðDÞÞf ðuÞÞuS dx ¼ R1 þ R2 þ R3 : Rn

where Ra3 ðu

1055

k2

Þ¼

Y Pk2 j¼1

Using the Bernstein inequality and (14), we get b Dx j u;

jR1 j 6 CkdivðvðDÞÞf ðuÞkL2 kuL kL2

bj ¼a3

6 Cen=2 kdivðvðDÞÞf ðuÞkL1 kuL kL2 2

and bj 6 minða1 ; a2 Þ: By (14), we have

6 Cen=2 kdiv ukL2 kukL2 kuk2 kL1

kDax Uðuð; sÞÞkL1 6 CekDax uðuð; sÞÞkL1

6 Cen=2 ð1 þ tÞ

6 Ceð1 þ sÞ

jaj

2 n4 k

I ðtÞ:

ð15Þ

Thus, by using Lemma 2.1 and (15), we get Z t kDax EL ð; t sÞ Uðuð; sÞÞkL2 ds 0 Z t=2 jaj n n ð1 þ t sÞ 2 4 ð1 þ sÞ 2 ds 6 Ce 0 ! Z t

þ

n

ð1 þ t sÞ 4 ð1 þ sÞ

jaj

2 n2

nðkþ1Þ 4

I kþ1 ðtÞ:

For R2, set 1; jnj < 2e; v1 ðnÞ ¼ 0; jnj > 3e; as a smooth cut-off function. It is easy to see that ð1 vðnÞÞvðnÞ 6 v1 ðnÞvðnÞ: Thus, jR2 j 6 Ckdivðv1 ðDÞÞf ðuÞkL2 kuL kL2 6 Cen=2 kdivðv1 ðDÞÞf ðuÞkL1 kuL kL2

ds I k ðtÞ

t=2

6 Ceð1 þ tÞ

12

2

jaj 2 n4

6 Cen=2 kdiv ukL2 kukL2 kuk2 kL1 I k ðtÞ:

ð16Þ

Summing up (11), (12), and (16), we get jaj 2 n4

kDa uL ðtÞkL2 6 Cð1 þ tÞ

ðku0 kL1 þ eI k ðtÞÞ:

6 Cen=2 ð1 þ tÞ

12

nðkþ1Þ 4

I kþ1 ðtÞ:

For R3, similar to R1, one has ð17Þ

1

jR3 j 6 Cen=2 ð1 þ tÞ2

nðkþ1Þ 4

I kþ1 ðtÞ:

1056


d 2 2 2 kruS kL2 þ kDuS kL2 þ gkDuS kL2 dt 6 2C g kdivð1 vðDÞÞðf ðuÞÞk2L2 :

In summary, by the above inequalities, we obtain Z ððb rÞuS þ divðð1 vðDÞÞf ðuÞÞÞuS dx n R

12

6 Cen=2 ð1 þ tÞ

nðkþ1Þ 4

I kþ1 ðtÞ:

ð20Þ

As (21), by using Plancherel’s theorem 2

u ðtÞk2L2 P e2 kuS ðtÞk2L2 : kruS ðtÞk2L2 ¼ knð1 vðnÞÞb

ð21Þ

Taking l 6 minf1; e2 g g4 ; we have

ð22Þ

kruS ðtÞk2L2 þ kDuS ðtÞk2L2 6 Celt ðkruS ð0Þk2L2 þ kDuS ð0Þk2L2 Þ Z t 2 þ 2C g elðtsÞ kdivð1 vðDÞÞðf ðuðsÞÞÞkL2 ds:

Then, taking l 6 minf1; e2 g g4 ; one has 2

2

2

lðkuS ðtÞkL2 þ kruS ðtÞkL2 Þ þ gkruS ðtÞkL2 P 0: Combining (19), (20), and (22) gives 1 d lt 2 2 e ðkuS ðtÞkL2 þ kruS ðtÞkL2 Þ 2 dt 6 Cen=2 elt ð1 þ tÞ

nðkþ1Þ 12 4

Since 2

I kþ1 ðtÞ:

ð26Þ

0

Thus

2

kDuS kL2 P e2 kruS kL2 :

By using Plancherel’s theorem

kuS ðtÞk2L2 þ kruS ðtÞk2L2 6 Celt ðku0 k2L2 þ kru0 k2L2 Þ Z t

1 nðkþ1Þ þ Cen=2 elðtsÞ ð1 þ sÞ2 4 ds I kþ1 ðtÞ: 0

1þn2

Setting hðtÞ ¼ elt ð1 þ tÞ ; we get n 2 1þn 2 ð1 þ tÞ2 kuS ðtÞkL2 þ ð1 þ tÞ 2 kruS ðtÞkL2 2 2 6 ChðtÞ ku0 kL2 þ kru0 kL2 þ en=2 I kþ1 ðtÞ:

ð23Þ

Here h(0) = 1, and it is easy to see that there exists t1 > 0 such that h(t) < 1, when t > t1. Thus, we can obtain from (17) and (23) that n n ð1 þ tÞ2 kuðtÞk2L2 þ ð1 þ tÞ1þ2 kruðtÞk2L2 6 C ku0 k2L2 þ kru0 k2L2 þ ku0 k2L1 þ en=2 I kþ1 ðtÞ þ e2 I 2k ðtÞ: ð24Þ 2 krus ðtÞkL2 ;

In order to obtain the estimates of inner product of (18) with DuS yields

taking the

ððuS Þt þ DðuS Þt þ gDuS ðb rÞuS divð1 vðDÞÞðf ðuÞÞ; DuS Þ ¼ 0:

ð25Þ

It is easy to see that ððuS Þt þ DðuS Þt þ gDuS ; DuS Þ 1 d 2 2 2 ¼ kruS k þ kDuS k þ gkDuS k : 2 dt Since ððb rÞuS ; DuS Þ ¼ 0; and

2

2

kdivð1 vðDÞÞðf ðuðsÞÞÞkL2 6 Ckdiv ukL2 kuk1 kL1 ;

ð27Þ

by using (24), (26), and (27), we obtain n n ð1 þ tÞ1þ2 kruS ðtÞk2L2 þ ð1 þ tÞ2þ2 kDuS ðtÞk2L2 2 2 6 C ðku0 kH 2 þ ku0 kL1 Þ þ en=2 I kþ1 ðtÞ þ e2 I 2k ðtÞ :

ð28Þ

Thus, (28) and (17) also give 1þn 2 2þn 2 ð1 þ tÞ 2 kruðtÞkL2 þ ð1 þ tÞ 2 kDuðtÞkL2 6 C ðku0 k2H 2 þ ku0 k2L1 Þ þ en=2 I kþ1 ðtÞ þ e2 I 2k ðtÞ :

ð29Þ

For a similar method, for any 0 6 h 6 m, we have nþh nþhþ1 2 2 k@ hþ1 uS ðtÞkL2 ð1 þ tÞ2 k@ h uðtÞkL2 þ ð1 þ tÞ2 2 2 6 ChðtÞ ðku0 kH hþ1 þ ku0 kL1 Þ þ en=2 I kþ1 ðtÞ þ e2 I 2k ðtÞ : ð30Þ l

P

a

Here, @ ¼ jaj¼l @ . Now we come back to prove Theorem 1.1 by using (30). In fact X nþjaj 2 ð1 þ tÞ2 k@ a ukL2 : ð31Þ I 2 ðtÞ 6 jaj6mþ1

By combining (30) and (31), we have I 2 ðtÞ 6 C ku0 k2H mþ1 þ ku0 k2L1 þ e2 ðI kþ1 ðtÞ þ I 2k ðtÞÞ : 2

ð32Þ

2

Let M 2 ¼ 4Cðku0 kH mþ1 þ ku0 kL1 Þ > 0. Since C is a constant independent of e and t, and the above inequality holds for 1 any e > 0, taking e2 ¼ ð2CðM 2k2 þ M k1 ÞÞ , we see that 1 2 2 ð33Þ I 2 ð0Þ 6 2C ku0 kH mþ1 þ ku0 kL1 ¼ M 2 : 2

g jðdivð1 vðDÞÞðf ðuÞÞ; DuS Þj 6 kDuS k2L2 þ C g kdivð1 vðDÞÞ 2 2 ðf ðuÞÞkL2 ;

We will prove that I2(t) 6 M2 for all t > 0. For this, we define

we have

If T < 1, then by the definition we find I2(t) = M2, then

T ¼ supft P 0 : I 2 ðtÞ 6 M 2 g:


1 3 M 2 6 M 2 þ e2 ðM 2k þ M kþ1 Þ 6 M 2 : ð34Þ 4 4 It then follows that M2 < 0; the contradiction tells us that T = 1 and therefore 2 2 ð35Þ I 2 ðtÞ 6 4C ku0 kH mþ1 þ ku0 kL1 : From (35) and the definition of I(t), we know that Theorem 1.1 is valid. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 10576013 and 10871075) and the Natural Science Foundation of Guangdong Province (Grant No. 05300889). References [1] Albert J. On the decay of solutions of the generalized Benjamin–Bona– Mahony equation. J Math Anal Appl 1989;141:527–37.

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[2] Avrin J, Goldstein JA. Global existence for the Benjamin–Bona– Mahony equations. Nonlinear Anal 1985;9:861–5. [3] Benjamin TB, Bona JL, Mahony J. Model equations for long waves in nonlinear dispersive systems. Philos Trans Roy Soc London 1972;272:47–78. [4] Biler P. Long time behavior of solutions of the generalized Benjamin– Bona–Mahony equation in two-space dimensions. Differential Integral Equations 1992;5:891–901. [5] Fang SM, Guo BL. Long time behavior for solution of initialboundary value problem for one class of system with multidimensional inhomogeneous GBBM equations. Appl Math Mech 2005;26(6):665–75. [6] Fang SM, Guo BL. The decay rates of solutions of generalized Benjamin–Bona–Mahony equations in multi-dimensions. Nonlinear Anal 2008;69(7):2230–5. [7] Goldstein JA, Wichnoski BJ. On the Benjamin–Bona–Mahony equation in higher dimensions. Nonlinear Anal 1980;4:861–5. [8] Guo BL. Initial boundary value problem for one class of system of multidimensional inhomogeneous GBBM equations. Chin Ann Math Ser B 1987;8(2):226–38. [9] Zhang LH. Decay of solutions of generalized Benjamin–Bona– Mahony equation in n-space dimensions. Nonlinear Anal 1995;25:1343–69.