在這篇論文中，我們考慮兩種類型的Regularized Buckley-Leverett方程（縮寫成RBL方程）。第一種類型的RBL方程是拋物線型的偏微分方程，而第二類的RBL方程為具有耗散和色散的偏微分方程。在第2節，我們將推導出這兩種型號的偏微分方程。在第3節，我們將使用固定點定理證明這兩個RBL方程的柯西問題的古典解的局部存在及唯一性。

In this thesis, we consider two types of regularized Buckley-Leverett equations (RBL equations for short). The first type of RBL equations are the scalar partial differential equations of parabolic type, while the second type of RBL equations are the scalar partial differential equations consist of both the dissipative and dispersive terms. In Section 2 we will derive these two models of PDEs. In Section 3 we will use the fixed point theorem to show the local existence and uniqueness of classical solutions to the Cauchy problem of these two RBL equations.

[1] C.J.Amick, J.L.Bona and M.E.Schonbek, Decay of
solutions of somenon-linear wave equations,
J.Diff.Eq.81 (1989), 1-49.
[2] T.B.Benjamin, J.L.Bona, J.J.Mahony, Model Equations
for Long Waves in Nonlinear Dispersive Systems,
Philos.Royal Soc.London Series A, 272,(1972), 47-78.
[3] B.Boczar-Karakiewicz, J.L.Bona and D.Cohen,
Interaction of shallowwater waves and bottom
topography, In Dynamical problems in continuum
physics, IMA Series in Mathematics and Its
Aplications,4,Springer-Verlag (1987), 131-176.
[4] B.Boczar-Karakiewicz, J.L.Bona and B.Pelchat,
Interaction of internalwaves with the sea bed on
continental shelves, Continental Shelf Res.11
(1991), 1181-1197.
[5] J.L.Bona, H.Chen, S.Sun and B.Zhang, Comparison of
quarter-plane and two-point boundary value problems:
the BBM-equation, Discrete Contin.Dyn.Sys.13
(2005), 921-940.
[6] J.L.Bona and L.Luo, Initial-boundary value problems
for model equations for the propagation of long waves,
in: G.Gerrayra, G.Goldstein and F.Neubrander (Ed.),
Lecture Notes in Pure and Appl.Math.,168, Dekker,
New York, (1995), 65-94.
[7] J.L.Bona, W.G.Pritchard and L.R.Scott, An evaluation
of a model equation for water waves, Philos.Royal
Soc.London Series A 302 (1981), 457-510.
[8] J.L.Bona, S.Sun and B.-Y.Zhang, Forced
oscillations of a damped Korteweg-de Vries equation in
a quarter plane, Commun.Contemp.Math.5 (2003),
369-400.
[9] J.L.Bona and J.Wu, Temporal growth and eventual
periodicity for dispersive wave equations in a quarter
plane, to appear in Discrete Contin.Dyn.Sys.
[10] J.M.Hong, J.Wu and J.-M. Yuan, Explicit solution
representation for the BBM equation in a quarter plane
and the eventual periodicity.
[11] E.H.Lieb and M.Loss, Analysis (second edition),
American Mathematical Society, Providence, RI (2001).
[12] J.Shen, J.Wu and J.-M. Yuan, Eventual periodicity for
the KdV equation on a half-line, Physica D 227 (2007),
105-119.
[13] C.J.Van Duijn, A.Mikelic, and I.S.Pop.Effective
equations for twophase flow with trapping on the micro
scale. SIAM Journal on Applied Mathematics,
62(5):1531-1568, 2002.
[14] S.Hassanizadeh and W.Gray.Mechanics and
thermodynamics of multiphase flow in porous media
including interphase boundaries. Adv. Water Resour.,
13:169-V186, 1990.
[15] S.Hassanizadeh and W.Gray.Thermodynamic basis of
capillary pressure in porous media. Water Resour.
Res., 29:3389-3405, 1993.
[16] A.Corey.The interrelation between gas and oil
relative permeabilities.Producer’s Monthly,
19(1):38-41, 1954.
[17] C.J.Van Duijn, A.Mikelic, and I.Pop.Effective
Buckley-Leverett equations by homogenization. Progress
in industrial mathematics at ECMI, pages 42-52, 2000.
[18] Y.Wang, Central schemes for the modified Buckley-
Leverett equation, Ph.D. thesis, Ohio State Univ.,
2010.