一类退化拟线性抛物方程解的唯一性和存在性
摘要
渗流是自然界中一种普遍存在的自然现象，它指的是液体在多孔介质中的运动，例如水在土壤中的流动就是一种渗流现象，渗流的研究对地下水资源的开发，石油天然气的开采，特别对农业生产都有重要的意义，同时，在研究土壤的盐碱化和改良，肥料的合理施用，工业废水处理和地下水资源的保护等问题时，进一步涉及到渗流中溶质迁移和热量传递过程，都必须考虑渗流中溶质和热量输送的动力学问题。
渗流现象的研究起源于1956年H. Darcy[1]的著名实验。在以后的几十年中，众多数学家建立了大量关于渗流现象的数学模型，并在数值计算以及理论的定性研究上都取得了巨大的进展。本文所研究的问题来源于一种不可压流体在均匀，各项同性的多孔介质中流动。首先由连续性方程有
(?)θ/(?)t+div(?)=0，(1)其中θ为介质的孔隙率，(?)表示为渗流速度，Darcy定律给出
(?)=-κ(θ)▽Φ，(2)
一类退化拟线性抛物方程解的唯一性和存在性
其中无(的为液导系数，中为总位势.在假设忽略吸附的作用，化学作
用，渗透效应的条件下，中可以写成
中=重十之
(3)
其中第一项平是由毛细管作用产生的吸力而引起的静力学位势，第二
项是重力位势，z为沿重力方向的坐标变量.
联合(1)，(2)，(3)得到
div(无(0)甲中)+
口无(8)
口之
(4)
一一
即一决
在许多介质中，宙可以看成是夕的函数，即平二中(因，则我们
可以得到如下形式的方程
幻此
△A(0)斗一d ivB(8).
二
即一次
而由实验表明，液导系数峨哟一般是非负的，即A(、)是非减函数，
时方程(5)就是典型的渗流方程.
另一方面，若0依赖于寸，即口=砚重)，方程(4)可化为
口8(平)
口t
div(无(重)V垂)+
口K(中)
口x
在一维情形下经过适当的变换，
口C(。)
口亡
则得到如下形式的方程
口2牡
口了2
(6)
如果不考虑重力的作用(例如x的方向为水平的情形)，方程(6)的形式
为
口C(。)口2二
a艺口xZ’
(7)
第35页
舍魂
走军硕士学位论文
其中C(司一般也为单调不减函数，这类方程常用于带有饱和区和非饱
和区的渗流问题的研究.
对于方程(5)，(6)，(7)，在数学上我们所感兴趣的主要是带有退化
的情形.一般来讲，方程(5)是典型的抛物一双曲方程，退化发生在使
A’(s)=0的地方，而方程(6)，(7)是椭圆一抛物方程，退化发生在使
已(s)=0的地方.
关于退化抛物型方程(5)弱解理论的研究可以追溯到1958年.
Oleinik，Kalashinkov和周毓麟!2」发表了关于方程
口。口2动(二，t，二)
口t口了2
CauChy问题的研究.
在这篇文章中，他们要求《t，、，司对、全0有定义且
叻(t，x，二
必(‘，x，o
>O，叻。(t，x，?，)>O，当。>O时，
=功。(t，x，O)=O
、、，，矛了、龟.尸/
由于方程具有退化性质，一般来说是不存在古典解的，因而必须考虑方
程的弱解.他们给出了第一边值和第二边值问题弱解的定义，利用抛物
正则化方法证明了弱解的存在性，同时也给出了唯一性的证明以及扰动
有限传播的条件.
Gilding和Peletier{3」于1976年研究了方程
口视口2视“‘口u，，
而一万万.+丽
1
的Cauchy问题，其中饥>1，n>0，并且证明了当“全烈m+l)时
乙
弱解唯一，当“0全O连续，有界且。邵LipSChitz连续时弱解存在.这
第36页
一类退化拟线性抛物方程解的唯一性和存在性
个结果随后被Gilding 14}推广到更一般的方程
口廿口/山八_口廿
而一丽\a(司司+乙叫厉囚
其中a(。)，b(。)连续，且a(二)>O(。>O)，a(O)=0.他证明了当
62(二)=O(a(u))(。一、O+)
时弱解的唯一性.后来，陈亚浙教授!5}去掉了(s)中由a(u)控制b(司
的条件，集中对a(动加条件，证明了弱解的唯一性.而方程(s)研究
中的一个实质性的进展是由赵俊宁教授{6}得到的，他不要求a(司与
试司之间有任何的关系，只假设a(司全。，但集合F={、，a(、)=0}
不含内点，而且唯一性是在有界可测函数类中证明的.这方面的发展历
史可详见综述文章【7」及所附的文献表.
1959年，周毓麟教授!s]研究了方程
口2锐.，、口u_口:‘、口社、一
下万-下刃=A(x，t，七)下二一，+万(x，t，祝，:二一)一万一+c(x，t，祝)+户’(x，t)
口X‘、“‘dt、“口X‘口T
的混合边值问题，他利用差分方法证明了弱解的存在性，并研究了解的
有界性.
1982年，Van Duyn和Peletier【9{，!10{发表了关于方程(7)的
第一边值问题的研究结果，包括弱解的存在唯一性，解的性质及饱和与
非饱和区域交界面的连续性.另外，在1987年，他们提出了方程(7)
的自由边界问题(参见【n〕)，证明了自由边界的连续性.在他们的研究
中，C(。)一般都有如下的性质
当二<0时，C(哟严格递增(对应于非饱和情形);
当二全0时，C(司=1(对应于饱和情形).
Filtration is a kind of common phenomenon in nature, which indicates the movement of liquid in porous media. For example, the water flowing among the soil is a kind of filtration. The research of nitration is very important to the exploitation of ground water resources and the discovery of petroleum or gas, especially to the agriculture. At the same time, when we investigate the problem about the saline-alkali soil and melioration, the using fertilizer intelligently, the industrial waste water disposal and the protection of the ground water resource, which are involved the solute movement and the heat transfer, we must consider the dynamics of the solute in the fiitration and the heat transportation.
The experimental research of filtration phenomenon originated from the famous experiment of H. Dary's [1] in 1956. In many years after that, a lot of mathematical models were established, and researches on numerical computation and the theoretical qualitative analysis have been achieved a great deal progress. In this paper,
we are interested in a flow with a homogeneous, isotropic and rigid porous medium filled with a fluid. Firstly, by the continuity equa-
tion, we have
39
+ divu = 0, (1)
ot
where v denotes the macroscopic velocity of the fluid, 9 the volumetric moistrue content. The Darcy's law yields
(2)
where k(9] denotes the hydraulic conductivity and the total potential. If we ignore the absorption and chemical, osmotic and thermal effects could be expressed as
(3)
where the first term is the hydrostatic potential due to capillary suction and z the gravitational potential. Here z is a variable which direction accords with gravitation.
Combining (1), (2), (3), we obtain
For many medium, could be a function of 9, i.e.\I = {9}. Then we have the following equation of the form
an
. (5)
ot
And the experimentation yields that the hydraulic conductivity
is not negative, i.e. A(s] is a non-decreasing function. This is a kind
of typical filtration equation.
On the other hand, if 9 depends on , i.e. 0 = ), the equation (4) is induced to
.
In one dimensional case, we could get the following equation by some proper transform
dC(u) _d_ dB(u]
If the effect of gravitation is ignored (e.g. the direction of x is horizontal), the equation (6) has the form of
dC(u)
dt dxr
where C(u) is in general a non-decreasing function. This equation is applied to the research of filtration with saturated and unsaturated region.
For the equations (5), (6). (7) of above, we are interested in the degenerate case. Generally, equation (5) is the typical parabolic-
4
hyperbolic mixed equation, which degenerates when A'(s] = 0. While equations (6) and (7) are elliptic-parabolic mixed types, which degenerates when C'(s] = 0.
The theory on the solutions of the degenerate equations (5) could ascend to 1958. In this year. Oleinik, Kalashinkov and Zhou Yulin [2] studied the Cauchy problem of the equations with the
following form
du d(f)(x,t,u}
' Where they required (, x,u) is defined for u 0 and with the
following properties
> 0, 0u(t,x-,u) > 0, when u > 0, ,0 = 0.
Due to the degeneracy, classical solutions may not exist. They put forward the definitions of the generalized solutions of the first boundary value condition and the second boundary value condition. Using the method of parabolic regularitzation, they proved the existence of generalized solutions. And they also proved the uniqueness of solutions and obtained the conditions for solutions to have the properties of finite propagation of disturbances.
After that. Gilding and Peletier [3] considered the Cauchy problem for the equation
du
dt dx2 dx and proved that it admits at most one generatlized solution whenever n |(m + 1), and it admits a generalized solution if is
nongenative, bounded and continuous with u[" lying Lipschitz continuous. Soon
渗流现象的研究起源于1956年H. Darcy[1]的著名实验。在以后的几十年中，众多数学家建立了大量关于渗流现象的数学模型，并在数值计算以及理论的定性研究上都取得了巨大的进展。本文所研究的问题来源于一种不可压流体在均匀，各项同性的多孔介质中流动。首先由连续性方程有
(?)θ/(?)t+div(?)=0，(1)其中θ为介质的孔隙率，(?)表示为渗流速度，Darcy定律给出
(?)=-κ(θ)▽Φ，(2)
一类退化拟线性抛物方程解的唯一性和存在性
其中无(的为液导系数，中为总位势.在假设忽略吸附的作用，化学作
用，渗透效应的条件下，中可以写成
中=重十之
(3)
其中第一项平是由毛细管作用产生的吸力而引起的静力学位势，第二
项是重力位势，z为沿重力方向的坐标变量.
联合(1)，(2)，(3)得到
div(无(0)甲中)+
口无(8)
口之
(4)
一一
即一决
在许多介质中，宙可以看成是夕的函数，即平二中(因，则我们
可以得到如下形式的方程
幻此
△A(0)斗一d ivB(8).
二
即一次
而由实验表明，液导系数峨哟一般是非负的，即A(、)是非减函数，
时方程(5)就是典型的渗流方程.
另一方面，若0依赖于寸，即口=砚重)，方程(4)可化为
口8(平)
口t
div(无(重)V垂)+
口K(中)
口x
在一维情形下经过适当的变换，
口C(。)
口亡
则得到如下形式的方程
口2牡
口了2
(6)
如果不考虑重力的作用(例如x的方向为水平的情形)，方程(6)的形式
为
口C(。)口2二
a艺口xZ’
(7)
第35页
舍魂
走军硕士学位论文
其中C(司一般也为单调不减函数，这类方程常用于带有饱和区和非饱
和区的渗流问题的研究.
对于方程(5)，(6)，(7)，在数学上我们所感兴趣的主要是带有退化
的情形.一般来讲，方程(5)是典型的抛物一双曲方程，退化发生在使
A’(s)=0的地方，而方程(6)，(7)是椭圆一抛物方程，退化发生在使
已(s)=0的地方.
关于退化抛物型方程(5)弱解理论的研究可以追溯到1958年.
Oleinik，Kalashinkov和周毓麟!2」发表了关于方程
口。口2动(二，t，二)
口t口了2
CauChy问题的研究.
在这篇文章中，他们要求《t，、，司对、全0有定义且
叻(t，x，二
必(‘，x，o
>O，叻。(t，x，?，)>O，当。>O时，
=功。(t，x，O)=O
、、，，矛了、龟.尸/
由于方程具有退化性质，一般来说是不存在古典解的，因而必须考虑方
程的弱解.他们给出了第一边值和第二边值问题弱解的定义，利用抛物
正则化方法证明了弱解的存在性，同时也给出了唯一性的证明以及扰动
有限传播的条件.
Gilding和Peletier{3」于1976年研究了方程
口视口2视“‘口u，，
而一万万.+丽
1
的Cauchy问题，其中饥>1，n>0，并且证明了当“全烈m+l)时
乙
弱解唯一，当“0全O连续，有界且。邵LipSChitz连续时弱解存在.这
第36页
一类退化拟线性抛物方程解的唯一性和存在性
个结果随后被Gilding 14}推广到更一般的方程
口廿口/山八_口廿
而一丽\a(司司+乙叫厉囚
其中a(。)，b(。)连续，且a(二)>O(。>O)，a(O)=0.他证明了当
62(二)=O(a(u))(。一、O+)
时弱解的唯一性.后来，陈亚浙教授!5}去掉了(s)中由a(u)控制b(司
的条件，集中对a(动加条件，证明了弱解的唯一性.而方程(s)研究
中的一个实质性的进展是由赵俊宁教授{6}得到的，他不要求a(司与
试司之间有任何的关系，只假设a(司全。，但集合F={、，a(、)=0}
不含内点，而且唯一性是在有界可测函数类中证明的.这方面的发展历
史可详见综述文章【7」及所附的文献表.
1959年，周毓麟教授!s]研究了方程
口2锐.，、口u_口:‘、口社、一
下万-下刃=A(x，t，七)下二一，+万(x，t，祝，:二一)一万一+c(x，t，祝)+户’(x，t)
口X‘、“‘dt、“口X‘口T
的混合边值问题，他利用差分方法证明了弱解的存在性，并研究了解的
有界性.
1982年，Van Duyn和Peletier【9{，!10{发表了关于方程(7)的
第一边值问题的研究结果，包括弱解的存在唯一性，解的性质及饱和与
非饱和区域交界面的连续性.另外，在1987年，他们提出了方程(7)
的自由边界问题(参见【n〕)，证明了自由边界的连续性.在他们的研究
中，C(。)一般都有如下的性质
当二<0时，C(哟严格递增(对应于非饱和情形);
当二全0时，C(司=1(对应于饱和情形).
Filtration is a kind of common phenomenon in nature, which indicates the movement of liquid in porous media. For example, the water flowing among the soil is a kind of filtration. The research of nitration is very important to the exploitation of ground water resources and the discovery of petroleum or gas, especially to the agriculture. At the same time, when we investigate the problem about the saline-alkali soil and melioration, the using fertilizer intelligently, the industrial waste water disposal and the protection of the ground water resource, which are involved the solute movement and the heat transfer, we must consider the dynamics of the solute in the fiitration and the heat transportation.
The experimental research of filtration phenomenon originated from the famous experiment of H. Dary's [1] in 1956. In many years after that, a lot of mathematical models were established, and researches on numerical computation and the theoretical qualitative analysis have been achieved a great deal progress. In this paper,
we are interested in a flow with a homogeneous, isotropic and rigid porous medium filled with a fluid. Firstly, by the continuity equa-
tion, we have
39
+ divu = 0, (1)
ot
where v denotes the macroscopic velocity of the fluid, 9 the volumetric moistrue content. The Darcy's law yields
(2)
where k(9] denotes the hydraulic conductivity and the total potential. If we ignore the absorption and chemical, osmotic and thermal effects could be expressed as
(3)
where the first term is the hydrostatic potential due to capillary suction and z the gravitational potential. Here z is a variable which direction accords with gravitation.
Combining (1), (2), (3), we obtain
For many medium, could be a function of 9, i.e.\I = {9}. Then we have the following equation of the form
an
. (5)
ot
And the experimentation yields that the hydraulic conductivity
is not negative, i.e. A(s] is a non-decreasing function. This is a kind
of typical filtration equation.
On the other hand, if 9 depends on , i.e. 0 = ), the equation (4) is induced to
.
In one dimensional case, we could get the following equation by some proper transform
dC(u) _d_ dB(u]
If the effect of gravitation is ignored (e.g. the direction of x is horizontal), the equation (6) has the form of
dC(u)
dt dxr
where C(u) is in general a non-decreasing function. This equation is applied to the research of filtration with saturated and unsaturated region.
For the equations (5), (6). (7) of above, we are interested in the degenerate case. Generally, equation (5) is the typical parabolic-
4
hyperbolic mixed equation, which degenerates when A'(s] = 0. While equations (6) and (7) are elliptic-parabolic mixed types, which degenerates when C'(s] = 0.
The theory on the solutions of the degenerate equations (5) could ascend to 1958. In this year. Oleinik, Kalashinkov and Zhou Yulin [2] studied the Cauchy problem of the equations with the
following form
du d(f)(x,t,u}
' Where they required (, x,u) is defined for u 0 and with the
following properties
> 0, 0u(t,x-,u) > 0, when u > 0, ,0 = 0.
Due to the degeneracy, classical solutions may not exist. They put forward the definitions of the generalized solutions of the first boundary value condition and the second boundary value condition. Using the method of parabolic regularitzation, they proved the existence of generalized solutions. And they also proved the uniqueness of solutions and obtained the conditions for solutions to have the properties of finite propagation of disturbances.
After that. Gilding and Peletier [3] considered the Cauchy problem for the equation
du
dt dx2 dx and proved that it admits at most one generatlized solution whenever n |(m + 1), and it admits a generalized solution if is
nongenative, bounded and continuous with u[" lying Lipschitz continuous. Soon
引文
[1] 孙正讷等，多孔介质流体力学，中国建筑工业出版社，北京，1985．
[2] Oleinik O.A., Kalashnikov A.S.. Chzou Yui-lin, The Cauchy problem and boundary problems for equations of the type of nonstationary filtration, IZV, Akd. Nauk, SSSR Ser, Mat., 22(1958), 667-704(Russion).
[3] Gilding B.H., Peletier L.A., The Cauchy problem for au equation in the theory of infiltration. Arch. Rat. Mech., 61(1976),127-140.
[4] Gilding B.H., A nonlinear degenerate parabolic equation, Annali della Scuola Norm. Sup di Pisa, 4(3)(1977): 393-432.
[5] Chen Yazhe, Uniqueness of weak solutions of quasilinear degenerate parabolic equations, The Proc. of the 1982 Changchun Sym. on Diff. Geom. and Diff. Eqs., 317-332.
[6] Zhao Junning, Uniqueness of solutions of the first boundary value problem for quasilinear degenerate parabolic equation, Northeastern Math. J., 1(1)(1985), 153-165.
[7] 伍卓群，拟线性退缩抛物方程．数学进展，16(2)(1987)，121-158．
[8] 李德元，周毓麟论文集，科学出版社，1992．
[9] Van Duyn C.J., Peletier L.A., Nonstationary filration in partially saturated porous media, Arch. Rat. Mech. Anal 78(1982),173-198.
[10] Van Duyn, C.J., Nonstationary filtration in partially Saturated porous media:Continuity of the free boundary, Arch. Rat. Mech. Aual., 79:3(1982), 261-265.
[11] Van Duyn, C.J., Hulshof, J., An elliptic-parabolic problem with a non-local boundary condition, Arch Rat. Mech. Anal., 99:1(1987), 61-73.
[12] Comparini E., A One-Dimensional Bingham Flow. J. Math. Anai. Appl. 169(1992), 127-139.
[13] Roberto Gianni, Riccardo Ricci, An elliptic-parabolic problem in Bingham fluid motion. Reridiconti, dell'istituto di matematica, 28,(2002)247-261.
[14] Zhao Junning, Applications of the theory of compensated compactness to quasilinear degenerate parabolic equations and quasi linear degenerate elliptic equations. Northeastern, Math. J. 2(1)(1986), 33-48
[15] 伍卓群，赵俊宁，尹景学，李辉来，非线性扩散方程，吉林大学出版社，长春，1996．
[16] 伍卓群，尹景学，王春朋，椭圆与抛物型方程引论，科学出版社，北京，2003．
[17] Kersner R., Nonlinear heat conduction with absorption: space localization and extinction in finite time, SIAM J. Appl. Math., 43(1983), 1274-1285.
[18] Jingxue Yin, Chunpeng Wang, Young measure solutions of a
class of forward-backward diffusion equations, J. Math. Anal. Appl. 279(2003)659-683.
[19] Friedman A．，抛物型偏微分方程，科学出版社，北京，1984．
[20] Zhou Yulin, Initial value problems for nonlinear degenerate systems of filtration type, Chin. Ann. Math., 5B(4)(1984),633652.
[21] K. H. Karlsen, N. H. Risebro, J. D. Towers, On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient, (2002).
[22] 陈亚浙，二阶抛物型偏微分方程，北京大学出版社，北京，2003．
[23] H. Holden, N. H. Riserbro, A. Tveito. Maximum principles for a class of conservation laws.SIAM J Appl. Math.,55(3)，651-661,1995.
[2] Oleinik O.A., Kalashnikov A.S.. Chzou Yui-lin, The Cauchy problem and boundary problems for equations of the type of nonstationary filtration, IZV, Akd. Nauk, SSSR Ser, Mat., 22(1958), 667-704(Russion).
[3] Gilding B.H., Peletier L.A., The Cauchy problem for au equation in the theory of infiltration. Arch. Rat. Mech., 61(1976),127-140.
[4] Gilding B.H., A nonlinear degenerate parabolic equation, Annali della Scuola Norm. Sup di Pisa, 4(3)(1977): 393-432.
[5] Chen Yazhe, Uniqueness of weak solutions of quasilinear degenerate parabolic equations, The Proc. of the 1982 Changchun Sym. on Diff. Geom. and Diff. Eqs., 317-332.
[6] Zhao Junning, Uniqueness of solutions of the first boundary value problem for quasilinear degenerate parabolic equation, Northeastern Math. J., 1(1)(1985), 153-165.
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