Research article

Pointwise convergence problem of free Benjamin-Ono-Burgers equation

  • Received: 08 May 2021 Revised: 06 November 2021 Accepted: 11 November 2021 Published: 10 January 2022
  • MSC : 42B15, 42B25

  • In this paper, we show the almost everywhere pointwise convergence of free Benjamin-Ono-Burgers equation in Hs(R) with s>0 with the aid of the maximal function estimate.

    Citation: Fei Zuo, Junli Shen. Pointwise convergence problem of free Benjamin-Ono-Burgers equation[J]. AIMS Mathematics, 2022, 7(4): 5527-5533. doi: 10.3934/math.2022306

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  • In this paper, we show the almost everywhere pointwise convergence of free Benjamin-Ono-Burgers equation in Hs(R) with s>0 with the aid of the maximal function estimate.



    In this paper, we investigate the pointwise convergence problem of the free Benjamin-Ono-Burgers equation

    ut2xu+H2xu=0, (1.1)
    u(x,0)=f(x). (1.2)

    Carleson [4] initiated the pointwise converge problem, more precisely, Carleson showed pointwise convergence problem of the one dimensional Schrödinger equation in Hs(R) with s14. Dahlberg and Kenig [8] showed that the pointwise convergence of the Schrödinger equation does not hold for s<14 in any dimension. The pointwise convergence problem of Schrödinger equations in higher dimension has been investigated by many authors, for example, see [1,2,6,7,9,14,16,17,20,21,22,23,24,25]. The results of Bourgain [3] and Du et al. [11,12] showed that s>n2(n+1) is the necessary condition for the pointwise convergence problem of n dimensional Schrödinger equation. Moreover, Miao et al. [19] studied the the maximal inequality for 2D fractional order Schrödinger operators.

    The Benjamin-Ono-Burgers equation (1.1) was obtained by Ewdin and Roberts [13] in the study of intense magnetic flux tubes of the solar atmosphere. The dissipative effects ϵ2xu in that literature are due to weak thermal conduction, where ϵ is a measure of the importance of thermal conduction and is assumed small. Some people have studied the Cauchy problem for the Benjamin-Ono-Burgers equation [5,15,18]. In this paper, motivated by [10,25], we show the pointwise convergence of free Benjamin-Ono-Burgers equation in Hs(R) with s>0.

    We present some notations before stating the main results. We always assume that 0<ϵ<108. |E| denotes by the Lebesgue measure of set E.

    Fxf(ξ)=12πReixξf(x)dx,F1xf(ξ)=12πReixξf(x)dx,U1(t)u0=12πReixξitξ|ξ|tξ2Fxu0(ξ)dξ,fLqxLpt=(R(R|f(x,t)|pdt)qpdx)1q.

    Hs(R)={fS(R):fHs(R)=ξsFxfL2ξ(R)<}, where ξs=(1+ξ2)s2.

    The main result is as follows.

    Theorem 1.1. Let fHs(R) with s>0. Then, we have

    U1(t)f(x)f(x) (1.3)

    almost everywhere as t0.

    In this section, we present the proof of Lemma 2.1.

    Lemma 2.1. For t0, we have

    |Reixξitξ|ξ|tξ|ξ|dξ|ξ|ϵ|C|x|1+ϵ. (2.1)

    Proof. Let xξ=η. Then, we have

    Reixξitξ|ξ|tξ|ξ|dξ|ξ|ϵ=Cx1|x|ϵReiηitx1|x|1η|η|tx2η2dη|η|ϵ. (2.2)

    We define

    I:=Reiηtx1|x|1η|η|tx2η2dη|η|ϵ. (2.3)

    We consider x0,x<0, respectively. When x>0, we have

    I:=ReiηiAη|η|Aη2dη|η|ϵ=:I1+I2,. (2.4)

    where I1:=0eiηiAη2Aη2dηηϵ,I2:=0eiη+iAη2Aη2dη(η)ϵ. Here A=tx2. For 12A2, then, we have I1:=I11+I12. Here, I11:=40eiηiAη2Aη2dηηϵ,I12:=4eiηiAη2Aη2dηηϵ. Obviously, we have

    |I11||40eiηiAη2Aη2dηηϵ|40ηϵ=41ϵ1ϵ. (2.5)

    By using the integration by parts, we have

    I12:=4eiηiAη2Aη2dηηϵ=+41(12Aη)ηϵi+2Aη1+ϵdeiηiAη2Aη2=1(18A)4ϵi+2A41+ϵ+4eiηAη2(1(12Aη)ηϵi+2Aη1+ϵ)dη. (2.6)

    Thus, from (2.6), we have

    |(1u+iv)|2|u|u2+v2+2|v|u2+v2. (2.7)

    From (2.7), we have

    (1(12Aη)ηϵi+2Aη1+ϵ)2+3ϵη1+ϵ((12Aη)2+4A2η2)2+3ϵη1+ϵ3η1+ϵ. (2.8)

    Combining (2.5) with (2.8), we have

    |I12||4eiηiAη2Aη2dηηϵ||1(18A)4ϵi+2A41+ϵ|++4|(12Aη1+ϵ+ηϵi)|dη|18A|4ϵ(18A)242ϵ+4A242+2ϵ+4A242+2ϵ(18A)242ϵ+4A242+2ϵ+3+4η1ϵdη32+3ϵ6ϵ. (2.9)

    When 12A2, we have

    I1:=0eiηiAη2Aη2dηηϵ=10+14A1+1A14A+1A. (2.10)

    We have

    |10eiηiAη2Aη2dηηϵ|10ηϵdη11ϵ. (2.11)

    Since |12Aη|12, we have

    |14A1eiηiAη2Aη2(1(12Aη)ηϵi+2Aη1+ϵ)||14A12+3ϵη1+ϵ((12Aη)2+4A2η2)dη|1214A11η1+ϵdη12ϵ. (2.12)

    Since 4A2η214, we have

    |1A14AeiηiAη2Aη2(1(12Aη)ηϵi+2Aη1+ϵ)||1A14A2+3ϵη1+ϵ((12Aη)2+4A2η2)dη|1214A11η1+ϵdη12ϵ. (2.13)

    Since Aη1, we have

    |1AeiηiAη2Aη2(1(12Aη)ηϵi+2Aη1+ϵ)||1A2+3ϵη1+ϵ((12Aη)2+4A2η2)dη|1A1η1+ϵdη1ϵ. (2.14)

    For I2, let y=η, we have

    I2:=0eiy+iAy2Ay2dyyϵ. (2.15)

    Thus, I2 can be proved similarly to I1. When x<0, we have

    I:=Reiη+Aη|η|dη|η|ϵ=:I3+I4, (2.16)

    where I3:=0eiη+Aη2dηηϵ,I4:=0eiηAη2dη(η)ϵ. Thus, I3,I4 can be proved similarly to I1.

    This completes the proof of Lemma 2.1.

    In this section, we establish the maximal function estimate.

    Lemma 3.1. For fHϵ(R), we have

    U1(t)fL1xLtCf˙Hϵ(R). (3.1)

    Proof. To prove Lemma 3.1, it suffices to prove

    |11U1(t(x))f(x)dx|C(ϵ)f˙Hϵ(R) (3.2)

    for all measurable functions t:RR. By using the Fubini's theorem and the Cauchy-Schwarz inequality as well as Lemma 2.1, we have

    11Reixξit(x)ξ|ξ|t(x)ξ2Fxf(ξ)dξdxCRFxf(ξ)[11Reixξit(x)ξ|ξ|t(x)ξ2dx]dξC|R|Fxf(ξ)|2|ξ|ϵdξ|12R|11eixξit(x)ξ|ξ|t(x)ξ2dx|2dξ|ξ|ϵCf˙Hϵ(R)1111Rei(xy)ξit(x)ξ|ξ|(t(x)+t(y))ξ2dξ|ξ|ϵdxdyCf˙Hϵ(R)1111|xy|1+ϵdxdyCϵf˙Hϵ(R). (3.3)

    This completes the proof of Lemma 3.1.

    In this section, we apply Lemma 2.1 and the density theorem to prove Theorem 1.1.

    If f is rapidly decreasing function, ϵ>0, then, we have

    |U1(t)ff||U1(t)fU2(t)f|+|U2(t)ff|C|t|R|ξ|2|Fxf(ξ)|dξ0 (4.1)

    as t0. Here, U2(t)=12Reixξeit|ξ|ξFxf(ξ)dξ.

    When fHs(R)(sϵ), by using the density theorem which can be seen in Lemma 2.2 of [10], there exists a rapidly decreasing function g such that f=g+h, where hHs(R)<ϵ(sϵ). Thus, we have

    limt0|U1(t)ff|limt0|U1(t)gg|+limt0|U1(t)hh|. (4.2)

    We define Eα={xR:limt0|U1(t)ff|>α}. Obviously, EαE1αE2α,

    E1α={xR:limt0|U1(t)gg|>α2},E2α={xR:limt0|U1(t)hh|>α2}. (4.3)

    Obviously,

    EαE1αE2α. (4.4)

    From (4.1), we have

    |E1α|=0. (4.5)

    Obviously,

    E2αE21αE22α, (4.6)

    where

    E21α={xR:supt>0|U1(t)h|>α4},E22α={xR:|h|>α4}. (4.7)

    Thus, from Lemma 3.1, we have

    |E21α|=E21αdxCE21αsupt>0|U1(t)h|αdxCα1hL1xLtCα1f˙Hϵ(R). (4.8)

    Obviously, we have

    |E22α|E22αdxCE22α|h|2α2dxCα2h2L2. (4.9)

    From (4.5), (4.8) and (4.9), we have

    |Eα||E1α|+|E2α||E1α|+|E21α|+|E22α|Cϵα+Cϵ2α2. (4.10)

    Thus, for any α>0, from (4.10) we have

    |Eα|=0. (4.11)

    Thus, we have

    U1(t)ff0 (4.12)

    almost everywhere as t goes to zero.

    This completes the proof of Theorem 1.1.

    The authors declare there are no conflicts of interest.



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