线性阻尼Boussinesq方程解的Phragmén-Lindelof二择一性.pdf

应用数学MATHEMATICA APPLICATA2024,37(2):466-475线性阻尼Boussinesq方程解的Phragm en-Lindel of二择一性石金诚(广州华商学院数据科学学院,广东 广州 511300)摘要:研究一类含有双调和算子的双曲方程解的空间性态.利用能量方法,构造一个能量表达式,推导出该能量表达式所满足的二阶微分不等式,通过求解该不等式,得到解的Phragm en-Lindel of型的二择一结果.该结果表明,Saint-Venant原则对于线性阻尼Boussinesq方程同样适用.关键词:Phragm en-Lindel of二择一;双曲方程;Saint-Venant原则;双调和方程中图分类号:O175.29AMS(2010)主题分类：35G31;35B53文献标识码:A文章编号:1001-9847(2024)02-0466-101.引言Saint-Venant原理在数学和力学领域具广泛的应用,但在研究抛物方程的Saint-Venant原理时,需要假设在无限远端方程的解趋于零,而经典的Phragm en-Lindel of 二择一定理,无需添加在无限远端趋于零的限制.经典的Phragm en-Lindel of定理指出:调和方程的解从有限的一端到无穷远处必须随距离呈指数增长或指数衰减.许多学者致力于抛物型方程的空间行为的研究13,但对双曲型方程的研究较少.Horgan和Knowles4和Horgan56指出研究弹性波传播的双曲型方程组解的二择一性较少.Knops和Payne7首次研究了双曲型方程的Saint-Venant原理.近年来,对双曲型和拟双曲型方程的端部衰减效应的研究获得了新的进展810.双调和方程在应用数学和力学的研究中有着重要的应用,Knowles11、Roseman12、Flavin13、Flavin和Knops14,Horgan15,Payne和Scaefer16,以及Varlamov17等人研究了双调和方程在R2半无限长条带上解的空间性质.最近文18等也取得了一些新的结果.本文采取以下符号约定,用逗号表示求偏导,用,i表示对xi求偏导,如:u,i表示为uxi,重复指标表示求和,u,u,=2,=1(2uxx)2.2.准备知识定义一个半无限长条带为R=(x1,x2)|0 x1+,0 x2 h,(2.1)其中h是已知的大于零的常数.Rz为如下的区域Rz=(x1,x2)|0 z x1 0,(2.4)其中u(x,t)为自由面的高度,为取决于流体深度和长波的特征速度的常数.方程(2.4)是描述波在色散介质中运动的非线性演化的方程.考虑到在实际过程中,粘度也起着重要作用,因此需要进一步研究如下方程u,tt 2bu,txx u,xxxx+u,xx(u2),xx=0,x R1,t 0,(2.5)与经典的Boussinesq方程(2.4)不同的是左边第二项考虑了耗散.方程(2.5)是阻尼Boussinesq方程17,此时,b为大于零的常数,是常数.本文考虑具有如下初边值问题的线性情况:u,tt u u,t+2u=0,(2.6)u(x1,0,t)=u,2(x1,0,t),(2.7)u(x1,h,t)=u,2(x1,h,t)=0,(2.8)u(0,x2,t)=g1(x2,t),u,1(0,x2,t)=g2(x2,t),(2.9)u(x1,x2,0)=u,t(x1,x2,0)=0,(2.10)其中表示调和算子,2表示双调和算子.可微函数gi(x2,t),i=1,2是给定的函数并满足如下的相容性条件:g1(0,t)=g1(h,t)=g2(0,t)=g2(h,t)=g1(0,t),(2.11)其中g1表示关于x2的偏导数.3.一些引理引理3.1设u是方程(2.6)满足条件(2.7)-(2.11)的经典解,定义函数1(z,t)如下:1(z,t)=32t0Lzexp()u2,dx2d+Lzexp(t)u2,1dx2+2t0Lzexp()u2,1dx2d t0Lzexp()u,u,dx2d+2t0Lzexp()u,1u,1dx2d 2t0Lzexp()u,1u,1dx2d 2t0Lzexp()u,u,1dx2d.(3.1)则1(z,t)也可表示为1(z,t)=2t0z0Lexp()(z )u2,dAd+12z0Lexp(t)(z )u2,tdA+2t0z0Lexp()(z )u,u,dAd+12z0Lexp(t)(z )u,u,dA+t0z0Lexp()(z )u,u,dAd+2t0z0Lexp()(z )u,u,dAd468应用数学2024+12z0Lexp(t)(z )uudA t0z0Lexp()u,u,1dAd 2t0z0Lexp()u,u,1dAd+2t0z0Lexp()u,u,1dAd+2z0Lexp(t)u,tu,1dA zt0L0exp()u,u,1dx2d zt0L0exp()u,u,1dx2d+zt0L0exp()uu,1dx2d zt0L0exp()u,u,1dx2d 1(0,t).(3.2)证将方程(2.6)两边乘以exp()(z )u,并积分,由散度定理和初边值条件(2.7)-(2.11),可得0=t0z0Lexp()(z )u,(u,u,u,+u,)dAd=2t0z0Lexp()(z )u2,dAd+12z0Lexp(t)(z )u2,tdA+t0z0Lexp()(z )u,u,dAd t0z0Lexp()u,u,1dAd zt0L0exp()u,u,1dx2d+t0z0Lexp()(z )u,u,dAdt0z0Lexp()u,1u,dAd zt0L0exp()u2,dx2d+t0z0Lexp()(z )u,u,dAd.(3.3)下面处理项t0z0Lexp()(z )u,u,dAd由散度定理,可知t0z0Lexp()(z )u,u,dAd=2t0z0Lexp()(z )u,u,dAd+12z0Lexp(t)(z )u,u,dA 2t0z0Lexp()u,u,1dAd+zt0L0exp()u,u,1dx2d zt0L0exp()u,u,1dx2d.(3.4)对于式(3.4)右边第3项,由散度定理,可得 2t0z0Lexp()u,u,1dAd=2t0z0Lexp()u,u,1dAd 2t0Lzexp()u,1u,1dx2d+2t0L0exp()u,1u,1dx2d.(3.5)第 2 期石金诚：线性阻尼Boussinesq方程解的Phragm en-Lindel of二择一性469将式(2.6)代入到式(3.5)右边第1项,并由散度定理,可得2t0z0Lexp()u,u,1dAd=2t0z0Lexp()(u,u,+u,)u,1dAd=t0Lzexp()u2,dx2d+t0L0exp()u2,dx2d+2t0z0Lexp()u,u,1dAd+2z0Lexp(t)u,1u,tdA+2t0z0Lexp()u,1u,dAd+2t0z0Lexp()u,u,1dAd 2t0Lexp()u2,1dx2d+2t0L0exp()u2,1dx2d.(3.6)对于式(3.6)右边第5项,由散度定理,可得2t0z0Lexp()u,1u,dAd=2t0z0Lexp()u,1u,dAd+2t0Lzexp()u,1u,1dx2d 2t0L0exp()u,1u,1dx2d=t0Lzexp()u,u,dx2d t0L0exp()u,u,dx2d 2t0Lzexp()u,1u,1dx2d+2t0L0exp()u,1u,1dx2d+2t0Lzexp()u,1u,1dx2d 2t0L0exp()u,1u,1dx2d.(3.7)联合式(3.1)、(3.3)-(3.7),可得式(3.2).证毕.引理3.2设u是方程(2.6)满足条件(2.7)-(2.11)的经典解,定义函数2(z,t)如下：2(z,t)=12t0Lzexp()u2,dx2d+12t0Lzexp()u,u,dx2dt0Lzexp()u,1u,1dx2d+12t0Lzexp()u2,2dx2d,(3.8)则2(z,t)也可表示为2(z,t)=t0z0Lexp()(z )u2,1dAd z0Lexp(t)u,1u,tdA+z0Lexp(t)(z )u,1u,1tdA z0Lexp(t)(z )u,11u,tdA+t0z0Lexp()(z )u,1u,1dAd+t0z0Lexp()(z )u,1u,1dAd470应用数学2024+t0z0Lexp()(z )u,12u,2dAd zt0L0exp()u,1u,dx2d+zt0L0exp()u,12u,2dx2d+2(0,t).(3.9)证将方程(2.6)两边乘以exp()(z )u,11并积分,可得0=t0z0Lexp()(z )u,11(u,u,u,+u,)dAd=t0z0Lexp()(z )u,11u,dAd t0z0Lexp()(z )u,11u,dAdt0z0Lexp()(z )u,11u,dAd+t0z0Lexp()(z )u,11u,dAd.(3.10)对于式(3.10)右边第1项,由散度定理,可得t0z0Lexp()(z )u,11u,dAd=t0z0Lexp()(z )u2,1dAd 12t0Lzexp()u2,dx2d+12t0L0exp()u2,dx2d+zt0L0exp()u,1u,dx2d+z0Lexp(t)(z )u,11u,tdA.(3.11)对于式(3.10)右边第2项,由散度定理,可得t0z0Lexp()(z )u,11u,dAd=t0z0Lexp()(z )u,11u,dAd t0z0Lexp()u,11u,1dAd+zt0L0exp()u,11u,1dx2d=t0z0Lexp()(z )u,1u,1dAd+12t0Lzexp()u,u,dx2d12t0L0exp()u,u,dx2d zt0L0exp()u,1u,dx2d12t0Lzexp()u2,1dx2d+12t0L0exp()u2,1dx2d+zt0L0exp()u,11u,1dx2d.(3.12)对于式(3.10)右边第3项,由散度定理,可得t0z0Lexp()(z )u,11u,dAd=12z0Lexp(t)(z )u,1u,1dA zt0L0exp()u,12u,2dx2d第 2 期石金诚：线性阻尼Boussinesq方程解的Phragm en-Lindel of二择一性471t0z0Lexp()u,12u,2dAd.(3.13)对于式(3.10)右边第4 项,由散度定理,可得t0z0Lexp()(z )u,11u,dAd=t0z0Lexp()(z )u,1u,1dAd+12t0Lexp()u,1u,1dx2d12t0L0exp()u,1u,1dx2d 12t0Lzexp()u,1u,1dx2d+12t0L0exp()u,1u,1dx2d+zt0L0exp()u,11u,1dx2d+12t0Lzexp()u2,11dx2d 12t0L0exp()u2,11dx2d12t0Lzexp()u2,12dx2d+12t0L0exp()u2,12dx2d zt0L0exp()u,11u,1dx2d.(3.14)联合式(3.8)、(3.10)-(3.14),可得式(3.9).证毕.4.Phragm en-Lindel of 二择一结果本节在二阶微分方程的基础上,推导出Phragm en-Lindel of 二择一结果.现定义一个新的函数(z,t),(z,t)=1(z,t)+k2(z,t),(4.1)其中k为任意大于零的常数.引理4.1对于表达式(z,t),有如下估计|(z,t)|m2(z,t)z2,(4.2)其中m为可计算的常数.证将式(3.2)对z求二阶偏导,可得21(z,t)z2=2t0Lzexp()u2,dx2d+12Lzexp(t)u2,tdx2+2t0Lzexp()u,u,dx2d+12Lzexp(t)u,u,dx2+t0Lzexp()u,u,dx2d+2t0Lzexp()u,u,dx2d+12Lzexp(t)u,u,dx2+2t0Lzexp()u2,1dx2d+12Lzexp(t)u2,1dx2+t0Lzexp()u,u,11dx2dLzexp(t)u,1u,1dx2 2t0Lzexp()u,u,11dx2d t0Lzexp()u,1u,1dx2d+Lzexp(t)u2,1dx2472应用数学2024+t0Lzexp()u2,1dx2d.(4.3)将式(3.9)对z求二阶偏导,可得22(z,t)z2=t0Lzexp()u2,1dx2d Lzexp(t)u,11u,tdx2+t0Lzexp()u,1u,1dx2d+t0Lzexp()u,1u,1dx2d+t0Lzexp()u,112u,2dx2d+12Lzexp(t)u2,12dx2+2t0Lzexp()u2,12dx2d.(4.4)联合式(3.1)、(3.8)和(4.1),可得(z,t)=32t0Lzexp()u2,dx2d+Lzexp(t)u2,1dx2+2t0Lzexp()u2,1dx2d t0Lzexp()u,u,dx2d+2t0Lzexp()u,1u,1dx2d 2t0Lzexp()u,1u,1dx2d 2t0Lzexp()u,u,dx2d k2t0Lzexp()u2,dx2d+k2t0Lzexp()u,u,dx2d kt0Lzexp()u,1u,1dx2d+k2t0Lzexp()u2,2dx2d.(4.5)联合式(4.3)和式(4.4),并由Schwarz不等式,可得2(z,t)z24t0Lzexp()u2,dx2d+4t0Lzexp()u,u,dx2d+12t0Lzexp()u,u,dx2d+4t0Lzexp()u,u,dx2d+k2t0Lzexp()u,1u,1dx2d+k2t0Lzexp()u,1u,1dx2d+k4Lzexp(t)u2,12dx2+k4t0Lzexp()u2,12dx2d=F(z,t),(4.6)其中 4，k 4.联合式(4.5)和式(4.6),可得|(z,t)|m2(z,t)z2,其中m为可计算的常数.证毕.引理4.2设u是方程(2.6)满足条件(2.7)-(2.11)的经典解,若存在z0 0,使得(z0,t)z 0,则有如下的不等式成立limzeazG(z,t)n1(t),(4.7)其中a=1m,n1(t)=1b(z1,t)z+a(z1,t)eaz1,b为大于1 的常数,G(z,t)为式(4.17)中定义的函数.第 2 期石金诚：线性阻尼Boussinesq方程解的Phragm en-Lindel of二择一性473证由于2(z,t)z2 0(z 0)和(z0,t)z 0,可知当z z0时,有(z,t)z 0和(z,t)(z0,t)+(z0,t)z(z z0)成立,因此当z 时,(z,t)0,故存在z1 z0,使得(z1,t)z0和(z1,t)0成立.由式(4.2),可得2(z,t)z2 a2(z,t)0.(4.8)式(4.8)可以变形成下列两个不等式:z(eaz(z,t)z+a(z,t)0,(4.9)z(eaz(z,t)z a(z,t)0.(4.10)对式(4.9)和(4.10)从z1到z积分(z z1),可得(z,t)z+a(z,t)(z1,t)z+a(z1,t)ea(zz1),(4.11)(z,t)z a(z,t)(z1,t)z a(z1,t)ea(zz1).(4.12)联合式(4.11)和(4.12),可得(z,t)z(z1,t)zea(zz1)+ea(zz1)2+(z1,t)ea(zz1)ea(zz1)2.(4.13)仿照式(4.6)的推导过程,可得2(z,t)z2 bF(z,t),(4.14)其中b是大于1 可计算的常数.将式(4.14)从z1到z积分,可得(z,t)z bzz1F(z,t)d+(z1,t)z.(4.15)联合式(4.13)和(4.15),可得bzz1F(z,t)d(z1,t)zea(zz1)+ea(zz1)2 1+a(z1,t)ea(zz1)ea(zz1)2.(4.16)设G(z,t)=zz1F(z,t)d.(4.17)联合式(4.6)和(4.17),可得G(z,t)=4t0zz1Lexp()u2,dAd+4t0zz1Lexp()u,u,dAd+12t0zz1Lexp()u,u,dAd+4t0zz1Lexp()u,u,dAd+k2t0zz1Lexp()u,1u,1dAd+k2t0zz1Lexp()u,1u,1dAd+k4zz1Lexp(t)u2,12dA+k4t0zz1Lexp(t)u2,12dAd.(4.18)联合式(4.16)-(4.18),可得limzeazG(z,t)n1(t).(4.19)证毕.474应用数学2024引理4.3设u是方程(2.6)满足条件(2.7)-(2.11)的经典解,若对任意z 0,都有(z,t)z 0,则有如下的不等式成立:H(z,t)eazn2(t),(4.20)其中n2(t)=z(0,t)+a(0,t),H(z,t)为式(4.23)中定义的函数.证假设存在一点z0 0,使得(z0,t)z0时,(z,t)(z0,t)0.由式(4.2),可得(z,t)z(z0,t)z=2(,t)z2(z z0)1m(,t)(z z0),(4.21)其中z0 0,与引理4.3中的条件(z,t)z 0 矛盾.故对对任意z0,有(z,t)0.式(4.8)从0 到z积分,可得(z,t)z+a(z,t)eazn2(t).(4.22)由式(4.22)中当z 时,(z,t)0,(z,t)z 0,可知(z,t)z=+z2(,t)2d+zF(,t)d=H(z,t).(4.23)联合式(4.6)和(4.23),可得H(z,t)=4t0zLexp()u2,dAd+4t0zLexp()u,u,dAd+12t0zLexp()u,u,dAd+4t0zLexp()u,u,dAd+k2t0zLexp()u,1u,1dAd+k2t0zLexp()u,1u,1dAd+k4zLexp(t)u2,12dA+k4t0zLexp()u2,12dAd.(4.24)联合式(4.22)和(4.23),可得H(z,t)ean2(t).证毕.由引理4.2和引理4.3,可得如下定理.定理4.1设u是定义在半无限条R上的初边值问题(2.6)-(2.11)的经典解,当空间变量z趋于无穷时,解的能量表达式或者满足limzeazG(z,t)n1(t),(4.25)或者满足H(z,t)eazn2(t),(4.26)式(4.25)与(4.26)有且只有一个成立.定理4.1表明要么能量函数G(z,t)呈指数式增长,要么能量函数H(z,t)呈指数式衰减.参考文献：1 EDELSTEIN W S.A spatial decay estimate for the heat equationJ.Zeitschrift F ur AngewandteMathematik Und Physik Zamp,1969,20(6):900-905.2 LIN Changhao,PAYNE L E.A Phragmn-Lindel of Alternative for a class of quasi-linear second orderparabolic problemsJ.Differential and Integral Equations,1995,8(3):539-551.3 LIN Changhao,PAYNE L E.Spatial decay bounds in time dependent pipe flow of an incompressibleviscous fluidJ.Siam Journal on Applied Mathematics,2004,66(2):458-474.第 2 期石金诚：线性阻尼Boussinesq方程解的Phragm en-Lindel of二择一性4754 HORGAN C O.Recent developments concerning Saint-Venants principle:an updateJ.AppliedMechanics Reviews,1989,42(11):295-302.5 HORGAN C O.Recent development concerning Saint-Venants principle:A second updateJ.Ap-plied Mechanics Reviews,1996,49(10):101-111.6 HORGAN C O,KNOWLES J K.Recent developments concerning Saint-Venants principleJ.Ad-vances in Applied Mechanics,1996,23(10):179-269.7 KNOPS R J,PAYNE L E.A Saint-Venants principle for nonlinear elasticityJ.Archive for RationalMechanicsAnalysis,1983,81:1-12.8 BORRELLI A,PATRIA M C.Energy bounds in dynamical problems for a semi-infinite magnetoelastic beamJ.Zeitschrift fr Angewandte Mathematik Und Physik Zamp,1996,47(6):880-893.9 DIAZ J I,QUINTALLINA R.Spatial and continuous dependence estimates in linear viscoelasticityJ.Journal of Mathematical Analysis and Applications,2002,273(1):1-16.10 QUINTANILLA R.A spatial decay estimate for the hyperbolic heat equationJ.Society for Industrialand Applied Mathematics,1998,27:78-9111 KNOWLES J K.On Saint-Venants principle in the two dimensional linear theory of elasticityJ.Archive for Rational Mechanics Analysis,1966,21(1):1-22.12 ROSEMAN J J.The principle of Saint-Venant in linear and non-linear plane elasticityJ.Archive forRational Mechanics Analysis,1967,26(2):142-162.13 FLAVIN J N.On knowlesversion of saint-venants principle in two-dimensional elastostaticsJ.Archive for Rational Mechanics and Analysis,1973,53:366-375.14 FLAVIN J N,KNOPS R J.Some convexity considerations for a two-dimensional traction problemJ.Journal of Applied Mathematics and Physics,1988,39:166-176.15 HORGAN C O.Decay estimates for the biharmonic equation with application to Saint-Venants,sprinciple in plane elasticity and stokes flowsJ.Quarterly of Applied Mathematics,1989,42:147-157.16 PAYNE L E,SCHAEFER P W.Some Phragmn-Lindel of type results for the biharmonic equationJ.Zeitschrift fr angewandte Mathematik und Physik,1994,45(3):414-432.17 VARLAMOV V.Existence and uniqueness of a solution to the Cauchy problem for the dampedBoussinesq equationJ.Mathematical Methods in the Applied Sciences,1996,19(8):639-649.18 李远飞.在一个半无穷柱体上的非标准Stokes流体方程的二择一问题J.应用数学和力学,2020,41(4):406-419.Phragmn-Lindel of Alternative for Solutions of LinearlyDamped Boussinesq EquationSHI Jincheng(School of Date Science,Guangzhou Huashang College,Guangzhou 511300,China)Abstract:The spatial properties of a class of biharmonic hyperbolic equations were studied.Basedon energy methods,an energy expression was constructed,and a second order differential inequality wasderived for the energy expression.The Phragmn-Lindel of alternative results were obtained by solvingthis inequality.These results showed that the Saint-Venant Principle was also valid for the linear dampedBoussinesq equation.Key words:Phragmn-Lindel of alternative;Hyperbolic equation;Saint-Venant principle;Bihar-monic equation