一类六阶抛物方程的若干问题
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摘要
在研究油-水-表面活性剂三相系统动力学性质时得到了如下模型方程(1)是一个典型的高阶抛物方程,其中γ>0,这里u与油和水的局部浓度差成正比,f(u)=F'(u),而F(u)和a(u)由如下六次和二次多项式给出
在过去的几年里,六阶抛物方程得到了广泛的关注,很多学者研究了六阶抛物方程,得到了一些结果,例如解的存在性,惟一性和正则性[4]-[8].然而,据我们所知,对于方程(1)的研究结果却很少,Pawlow和Za j aczkowski[9]考虑了问题(1)-(3)当γ1=1,α2>0的初边值问题,证明了初边值问题存在惟一的全局光滑解,并且解连续依赖初值G.Schimperna等[10]研究了带有粘性项△ut的方程
其中f(r)=F'(r),F(r)具如下的对数位势他们研究了在参数γ趋于0时,六阶方程解的相应性质,惟一性和正则性也在文章中得到证明G.schimperna[42]等考虑了如下带有奇异扩散项的高阶抛物方程,其中m(r)=M'(r),M(r)=(1-r)log(1-r)+(1+r)log(1+r),r∈[一1,1]及。(r)=2/1-γ2,作者证明了对于任意时间T,问题都存在一个定义在(0,T)上的能量型弱解.以上研究的都是具有常数迁移率的油水表面活性剂模型.实际问题中迁移率可能是浓度相关的函数,Liu[11]研究了具有非常数迁移率的方程并且在二维情况下证明了古典解的存在性.
本文的目的是研究具常数迁移率的油-水-表面活性剂方程(1)的解的某些性质.
在第二章中,我们首先研究其中Ω是Rn(n≤3)上的具有光滑边界的有界区域,并且γ>0,这里f(u)=F'(u),F(u)和α(u)在(2),(3)中给定,对方程(1)附加如下边值条件和初值条件
值得注意的是,在油水模型方程推导的过程中,若不加表面活性剂,α(u)为正,而加入表面活性剂之后,α(u)系统的最小值出现在u=0时,但是随着表面活性剂的增加,则两亲分子的浓度α(u)在微乳液相中可以变为负数([1]),所以从物理化学及数学的角度,在我们的研究过程中是不要求α2,α0∈R的符号,我们首先考虑存在性,目的是在γ1,α2和α0不限制为正数的情况下,得到解的存在性.由于四阶扩散项和二阶扩散项的非线性,使得我们在做先验估计时遇到了一定的困难,我们主要利用了方程特有的能量泛函,F(u),f(u)及其导数所具备的结构,并利用了Nirenberg不等式等一系列不等式来处理困难.在研究解的爆破性质的时候,不仅要考虑α(u)和f(u)所带来的困难,还要考虑u2|(?)Vu|2及|(?)u|2.我们除利用方程的能量泛函外还要引入如下的函数w,而w为下述方程的惟一解借助w的性质,我们就证明了古典解不必整体存在,也就是说如果γ1>0,α2>0或α2<0且γ充分大时,问题有整体古典解,而在其他情况下,解有限时间爆破.
然后,我们考虑了油-水-表面活性剂系统的吸引子的存在性.
方程的动力学性质在高阶抛物方程的研究中尤为重要,例如,整体解的渐近行为,整体吸引子的存在性等.在过去的一段时间内,很多学者对非线性耗散动力系统的吸引子表现出了浓厚的兴趣,经典的结果是由Temam[79]和Hale[80]给出的,在这之后又得到了大量的结果,例如[12]-[15],[81]-[85].G.Schimperna做了大量的关于Cahn-Hilliard方程吸引子的工作,作者在三维情形下,考虑了其带有非常数迁移率的标准和粘性问题的吸引子的存在性[43],又考虑了带有惯性项的粘性方程[70],并给出了整体吸引的存在性,而在[45]中给出了一类具有依赖化学势迁移率的Cahn-Hilliard方程的整体吸引子的存在性.在此之后,吸引子又进一步得到了发展,L.Song等[49]首先研究了二阶半线性方程其中(d>0且9(χ,u)的形式为作者证明了上述方程的初边值问题在Hk(k≥0)空间吸引子的存在性,在此基础上又研究了[48]如下四阶方程的初边值问题其中9(u)具有多项式形式作者给出了上述问题的Hk(k≥0)吸引子,紧接着又在[46],[47]中给出另外两个四阶抛物方程在Hk(k≥0)空间上吸引子的存在性.
据我们所知,对于六阶抛物方程的整体吸引子的研究还不是很多,只有Korzec等[44],在一维和二维情形下,证明了六阶方程的整体吸引子的存在性.我们利用能量泛函的方法考虑了油水模型的整体吸引子的存在性.我们利用对线性半群的正则性估计,结合迭代技术和整体吸引子的经典存在性理论来证明(1)-(3)在Hk(k≥0)空间上存在整体吸引子.相比于L.Song等研究的四阶方程,本节的困难主要是由于非线性项△(α(u)△2u+(a'(u))/2｜▽u｜2)和△f(u)存在,我们需要作出更高阶的估计,这些估计是复杂的和很难得到的.我们也利用半群的相关知识调整迭代过程.
本章最后,我们考虑带有粘性项的方程
证明方法与标准方程的证明类似,由于粘性项δ△ut的存在,使得证明解的先验估计时,出现了一些差别,另外在证明解的吸引子时,迭代过程中与使用嵌入定理有所不同,也得到了类似于标准方程的结果.
第三章中,我们考虑最优控制问题.
从二十世纪50年代起,最优控制问题引起了学术界的高度重视.大部分的控制理论及应用都是由ODE来表示,但是随着科技应用的发展,同样需要解决由PDE表示的最优控制问题.目前,很多学者对抛物方程的最优控制问题做了大量的研究,首先关于Burgers方程得到了许多有价值的结果[29],[71]-[77]Armaou和Christofides[30]研究了Kuramto-Sivashing方程的反馈控制问题.除此之外,很多学者研究了高阶抛物方程的最优控制问题.Yong和Zheng[27]在具有光滑边界的有界区域上研究了Cahn-Hilliard方程的回馈稳定和最优控制问题Tian等[23]研究了粘性抛物方程的最优控制问题,例如粘性Camassa-Holm方程给出了最优解的存在惟一性,Tian等[25,26]利用类似的方法处理了粘性Degasperis-Procesi方程,粘性Dullin-Gottualld-Holm方程,同样得到了最优解的存在惟一性.最近,对于Cahn-Hilliard方程,同样得到了一些结果,参见[28],[20],[21]Wang [50]考虑了半线性抛物方程的最优控制的必要条件.
综上所述,上述的问题都是研究四阶及以下的方程的最优控制问题,还没有对油-水-表面活性剂模型的最优控制问题的研究,所以我们首先利用Galerkin方法给出油-水-表面活性剂模型的最优控制问题的弱解的存在性,与四阶方程相比较,由于非线性项△(α(u)△u+(a'(u))/2｜▽u｜2)的存在,我们在做估计时,需要利用方程的能量泛函并调整所使用的不等式,来达到证明弱解存在性的目的.在这些估计的基础上,利用Lions关于最优解的经典理论来证明了最优解的存在性,同样是由于非线性项的干扰,使得我们在研究的过程中需要结合非线性项的特点,选择变量收敛的空间及收敛的方式.我们也给出了最优控制条件.本章最后,我们考虑了具有粘性项的油-水-表面活性剂模型的最优控制问题,在证明的过程中,我们利用变换在形式上改变了最优控制问题,再利用前面相同的方法之后,得到了具有粘性项问题的最优控制解的存在性.
第四章,我们研究方程的周期解问题.
从十九世纪到现在,扩散问题得到了广泛的研究,特别是周期问题引起了学术界极大的兴趣.据我们所知,对于二阶方程的周期问题已经得到了很多结果,例如[51,52,53,54].注意到高阶扩散方程可以用来描述生物种群、人口的扩散和迁徙等周期模型[55,56],有一些数值结果用来解释此类现象[67,68,69].周期解对于高阶抛物方程尤为重要,过去一段时间,很多学者研究了高阶抛物方程的周期解.包括空间周期问题[57,58,59,60,61],周期边值问题[62,63,64,65,66].Yin等[35,36,37]考虑了一类Cahn-Hilliard型方程并在一维情形下证明时间周期解的存在性Wang等[34]利用Galerkin方法和Leray-Schguder不动点定理,在一维和二维情形下,证明了广义Ginzburg-Landau模型方程的广义周期解和古典周期解的存在惟一性.在[32]中作者利用类似的方法研究了Camassa-Holm方程证明了时间周期解的存在唯一性.
然而,还有一些物理模型需要在二维或高维空间上考虑周期解,例如油膜在固体表面的扩散.所以无论从数学本身还是物理背景都需要研究高维情形,因此我们要在高维空间研究油-水-表面活性剂模型的周期解.为达到这个目的,我们首先引入算子G,在得到了算子的紧性及解的一些必要估计后,我们将在合适的泛函空间上得到算子的不动点(其中σ=1),即为问题的解.相比于四阶方程,由于四阶项和二阶扩散项的非线性项的存在,得到uσ和(?)uσ的Holder连续性还不足以证明主要定理,我们还需要△uσ的Holder连续性,为解决这个困难,我们主要利用Schauder型先验估计的方法来处理,先验估计将由调整Campanato空间得到.
In this thesis,we consider the equation which describes dynamics of phase transitions in ternary oil-water-surfactant systems,where λ>0and u is the scalar order parameter which is pro-portional to the local difference between oil and water concentrations.Let f(u)=F'(u),F(u)and.(u)are approximated,respectiVely,by a sixth and a second order polynomial
It is worth noting that sixth order parabolic equations have attracted mathematicians' attention.During the past years,many authors have paid much attention to the sixth order parabolic equation,such as the existence, uniqueness and regularity of the solutions[4]一[8]. However,as far as we know,there are few investigations concerned equation(1). Pawlow and Zajaczkowski[9]proved that the initial-boundary-value problem(1)一(3) admits a unique global smooth solution which depends continuously on the initial datum when λ1and α2>0. G.Schimperna et.al.[10]studied the equation with viscous term△ut where f(r)=F'(r),F(u)is the logarithmic potential They investigated the behavior of the solutions to the sixth order system as the parameter λ tends to0. G.Schimperna et. al.[42]considered a class of higher order equations characterized by a singular diffusion term where m(r)=M'(r),M(r)=(1-r)log(1-r)+(1+r)log(1+r),r∈[-1,1] and α(r)=2/(1-r2),and authors proved that for any final time T,the problem admits a unique energy type weak solution,defined over(0,T). Liu[11] studied the equation with nonconstant mobiliy and he proved the existence of classical solutions for two dimensions.
The aim of the thesis is to consider some properties of solutions for the oil-water-surfactant equation(1).
In the section2,we first consider the eq uation where Q is a bounded domain in Rn(n≤3) with smooth boundary and λ>0.The equation is supplemented by the boundary value conditions and the initial value condition u(x,0)=u0(x).
It is worth noticing that,in the absence of amphiphile the α(u)is a positive constant.When amphiphile is added to the system a minimum of α(u)develops at u=0.With increasing amphiphile strength or amphiphile concentration α(u)becomes negative at the microemulsion phase.So,from the background of physical chemistry and mathematics,the purpose of this paper is devoted to the investigation of properties of solutions with γ1,α2,α0are not restricted to be positive.
Firstly,the difficulties of the discussion on the existence and unique-ness of the solution for the equation(1)are to deal with u2△u,u｜▽u｜2and u6from the nonlinear term α(u)and f(u).To overcome these difficulties, we use the energy functional of the equation,the structure of the F(u), f(u)and the derivatives of f(u)and some inequalities. When we study the blowup of the solution,we need to consider the sign of α2and α0and deal with u2｜▽u｜2and｜▽u｜2,we also use the energy functional and the properties of ω,which is the unique solution of the following problem: We show that the solutions might not be classical globally.In other words, in some cases,the classical solutions exist globally,while in some other cases,such solutions blow up at a finite time.
In section3,we give the existence of attractors of the equation(1).
The dynamic properties of the higher order parabolic equation,such as the global asymptotical behaviors of solutions and existence of global attractors,are important for the study of higher order parabolic system. During the past years,many authors have paid much attention to the attractors. There are many studies on the existence of global attractors for general nonlinear dissipative dynamical systems.For the classical re-gults we refer the reader to Temam[79]and Hale[80]. Many authors are interested in the existence of global attractors for general nonlinear dissipative dynamical systems such as[12]一[15],[81]一[85]. G.Schimperna [43]in a three-dimensional spatial setting,gave the existence of attractors for both the viscous and the standard Cahn-Hilliard equation with a non-constant mobility coefficient.G.Schimperna[70]considered the Viscous Cahn-Hilliard equation characterized by the presence of an inertial term which is a differential model describing nonisothermal fast phase separa-tion processes taking place in a three-dimensional bounded domain,and the author proved that the corresponding dynamical system is dissipative and possesses a global attractor.G.Schimperna et.al.[45]studied the ex-istence of global attractors for a family of Cahn-Hilliard equations with a mobility depending on the chemical potential.L.song et.al.[49]considered the semilinear parabolic equation whered>0and g(x,u)is Authors proved the existence of the global attractors inHk(k≥0)apace. L.Song et.al.[48]also considered the fourth order parabolic equation where g(u)is a polynomial given by g(s)=(?)aksk,and the authors[46],[47] proved some parabolic equations possesses a global attractor in Hk(k≥0),which attracts any bounded subset of Hk(k≥0)in the Hk-norm.
To the best of our knowledge,there are few studies on the existence of the global attractors of the higher order parabolic equations. As Ko-rzec et.al.[44]established the the existence of global attractor for a six order equation describing the evolution of a growing crystalline surface in one di-mension and two dimensions.So,in this chapter,We discuss the existence of global attractor.We will use the regularity estimates for the linear semi-groups,combining with the iteration technique and the classical existence theorem of global attractors,to prove that the problem possesses a global attractor in Hk(k≥0)space.The difficulties are the higher order of u,▽u,△u and the emerge of▽△u,△4u caused by the△(a(u)△2u+(a'(u))｜▽u｜2) and△f(u),we should make some adjustment on the space,the embedding theorems and iteratiom.
In the section4,we also consider the problem with viscous terms δ△ut We obtained the similarly result.
In chapter3,we consider the optimal control of the oil-water-surfactant model.
From1950s,the optimal control of distributed parameter system had become much more active in academic ficld.Modern optimal control the-ories and applied models are both represented by ODE.With the develop-ment and application of technology,it is necessary to solve the problem of optimal control theories for PDE.There are many profound results on the optimal control problems for PDE,such as the optimal control for Burgers equation[29],([71]一[77]) Armaou and Christofides[30]studied the feedback control of Kuramto-Sivashing equation Besides,many papers have already been published to study the control problems of nonlinear parabolic equations.Yong and Zheng[27]studied the feedback stabilization and optimal control of the Cahn-Hilliard equa-tion in a bounded domain with smooth boundary.Tian et al [23,25,26] also studied the optimal control problems for parabolic equations,such as viscous Camassa-Holm equation,viscous Degasperis-Procesi equation,vis-cous Dullin-Gottualld-Holm equation and so on.Recently,Zhao and Liu [28]considered the optimal control problem for1D viscous Cahn-Hilliard equation.M.Kubo[20]considered the Cahn-Hilliard equation with time-dependent constraint. M.Hintermuller and D.Wegner[21]studied the distributed optimal control for the Cahn-Hilliard system.In their papers, the optimal control under boundary condition was given and the existence of optimal solution to the equation was proved.Wang[50]was concerned about necessary conditions for optimal control problems governed by some semi-linear parabolic differential equations.
Wc consider the standard oil-water-surfactant equation and the equa-tion with viscous term.We first employ Galerkin method to give the existence of the weak solution of equation. Because of nonlinear term△(a(u)△u+(a'(u))｜▽u｜2),we need to use the energy functional and make the adjustment on the inequalities to prove the existence of the weak so一lution of the equation. In section2,we prove the existence of optimal solution basing on Lions'theory. In section3,the optimality conditions are showed and the optimality system is derived. In section4,we used similarly methods and proved the existence of optimal solution.
In chapter4,we study the time periodic solution for the oil-water-surfact ant system.
From the early19th century,diffusion equations have been widely in-vestigated,specially,the periodic problems have been paid much attention. As far as we know,the researches on second order periodic diffhsion equa-tions are extensive,and many profound results have been obtained,such as[51,52,53,54],noticing that the higher diffusion equations can be used to describe models with periodic factors,for example biological groups, the diffusion and migration of population[55,56]and so on.so,the time periodic solutions are important for the higher-order parabolic equation.
During the past years,many authors have paid much attention to the time periodic solutions of higher order parabolic equations.We can find many corresponding numerical results which provide the refcerences to explain certain problems[67,68,69].Furthermore,some authors paid at-tention to periodic problems,including spatial periodic problems[57,58,59,60,61],periodic boundary problems[62,63,64,65,66].Yin et.a1.[35],[36],[37]considered time periodic problems that gave the existence of time periodic solutions for the Cahn-Hilliard type equation in one dimension. Using the Galerkin method and the Leray—Schauder fixed point theorem, wang et.al.[34]proved the existence and uniqueness of time-periodic gen-eralized solutions and time-periodic classical solutions to the generalized Ginzburg-Landau model equation in1D and2D cases. Y. Fu et.al.[32] employed the similar methods to consider the Camassa-Holm equation and proved the existence of the time-periodic solution. However, many physical phenomena, such as the diffusion of oil film over a solid surface, need to be discussed in two-dimension or multi-dimension case. So we should study the multi-dimensional case considered not only from mathe-matics itself but also from physical background. As far as we know, there are few investigations concerned with the time periodic solutions of such kind of equations.
In this chapter, we prove the existence of time periodic solutions of the problem in two space dimensions. For this purpose, we first introduce an operator G by considering a linear sixth-order equation with a param-eter σ∈[0,1]. After proving the compactness of the operator and some necessary estimates of the solutions, we then obtain a fixed point of the operator in a suitable functional space with σ=1, which is the desired solution of the problem.
Compare to the fourth order parabolic equation, we should not only obtain the Holder continuity of uσ and Vtσ, but also the Holder continuity of Δuσ, because of the nonlinearity of the fourth order term and the second order diffusive factors. The main method that we use is based on the Schauder type priori estimates, which here will be obtained by means of a modified Campanato space.
在过去的几年里,六阶抛物方程得到了广泛的关注,很多学者研究了六阶抛物方程,得到了一些结果,例如解的存在性,惟一性和正则性[4]-[8].然而,据我们所知,对于方程(1)的研究结果却很少,Pawlow和Za j aczkowski[9]考虑了问题(1)-(3)当γ1=1,α2>0的初边值问题,证明了初边值问题存在惟一的全局光滑解,并且解连续依赖初值G.Schimperna等[10]研究了带有粘性项△ut的方程
其中f(r)=F'(r),F(r)具如下的对数位势他们研究了在参数γ趋于0时,六阶方程解的相应性质,惟一性和正则性也在文章中得到证明G.schimperna[42]等考虑了如下带有奇异扩散项的高阶抛物方程,其中m(r)=M'(r),M(r)=(1-r)log(1-r)+(1+r)log(1+r),r∈[一1,1]及。(r)=2/1-γ2,作者证明了对于任意时间T,问题都存在一个定义在(0,T)上的能量型弱解.以上研究的都是具有常数迁移率的油水表面活性剂模型.实际问题中迁移率可能是浓度相关的函数,Liu[11]研究了具有非常数迁移率的方程并且在二维情况下证明了古典解的存在性.
本文的目的是研究具常数迁移率的油-水-表面活性剂方程(1)的解的某些性质.
在第二章中,我们首先研究其中Ω是Rn(n≤3)上的具有光滑边界的有界区域,并且γ>0,这里f(u)=F'(u),F(u)和α(u)在(2),(3)中给定,对方程(1)附加如下边值条件和初值条件
值得注意的是,在油水模型方程推导的过程中,若不加表面活性剂,α(u)为正,而加入表面活性剂之后,α(u)系统的最小值出现在u=0时,但是随着表面活性剂的增加,则两亲分子的浓度α(u)在微乳液相中可以变为负数([1]),所以从物理化学及数学的角度,在我们的研究过程中是不要求α2,α0∈R的符号,我们首先考虑存在性,目的是在γ1,α2和α0不限制为正数的情况下,得到解的存在性.由于四阶扩散项和二阶扩散项的非线性,使得我们在做先验估计时遇到了一定的困难,我们主要利用了方程特有的能量泛函,F(u),f(u)及其导数所具备的结构,并利用了Nirenberg不等式等一系列不等式来处理困难.在研究解的爆破性质的时候,不仅要考虑α(u)和f(u)所带来的困难,还要考虑u2|(?)Vu|2及|(?)u|2.我们除利用方程的能量泛函外还要引入如下的函数w,而w为下述方程的惟一解借助w的性质,我们就证明了古典解不必整体存在,也就是说如果γ1>0,α2>0或α2<0且γ充分大时,问题有整体古典解,而在其他情况下,解有限时间爆破.
然后,我们考虑了油-水-表面活性剂系统的吸引子的存在性.
方程的动力学性质在高阶抛物方程的研究中尤为重要,例如,整体解的渐近行为,整体吸引子的存在性等.在过去的一段时间内,很多学者对非线性耗散动力系统的吸引子表现出了浓厚的兴趣,经典的结果是由Temam[79]和Hale[80]给出的,在这之后又得到了大量的结果,例如[12]-[15],[81]-[85].G.Schimperna做了大量的关于Cahn-Hilliard方程吸引子的工作,作者在三维情形下,考虑了其带有非常数迁移率的标准和粘性问题的吸引子的存在性[43],又考虑了带有惯性项的粘性方程[70],并给出了整体吸引的存在性,而在[45]中给出了一类具有依赖化学势迁移率的Cahn-Hilliard方程的整体吸引子的存在性.在此之后,吸引子又进一步得到了发展,L.Song等[49]首先研究了二阶半线性方程其中(d>0且9(χ,u)的形式为作者证明了上述方程的初边值问题在Hk(k≥0)空间吸引子的存在性,在此基础上又研究了[48]如下四阶方程的初边值问题其中9(u)具有多项式形式作者给出了上述问题的Hk(k≥0)吸引子,紧接着又在[46],[47]中给出另外两个四阶抛物方程在Hk(k≥0)空间上吸引子的存在性.
据我们所知,对于六阶抛物方程的整体吸引子的研究还不是很多,只有Korzec等[44],在一维和二维情形下,证明了六阶方程的整体吸引子的存在性.我们利用能量泛函的方法考虑了油水模型的整体吸引子的存在性.我们利用对线性半群的正则性估计,结合迭代技术和整体吸引子的经典存在性理论来证明(1)-(3)在Hk(k≥0)空间上存在整体吸引子.相比于L.Song等研究的四阶方程,本节的困难主要是由于非线性项△(α(u)△2u+(a'(u))/2｜▽u｜2)和△f(u)存在,我们需要作出更高阶的估计,这些估计是复杂的和很难得到的.我们也利用半群的相关知识调整迭代过程.
本章最后,我们考虑带有粘性项的方程
证明方法与标准方程的证明类似,由于粘性项δ△ut的存在,使得证明解的先验估计时,出现了一些差别,另外在证明解的吸引子时,迭代过程中与使用嵌入定理有所不同,也得到了类似于标准方程的结果.
第三章中,我们考虑最优控制问题.
从二十世纪50年代起,最优控制问题引起了学术界的高度重视.大部分的控制理论及应用都是由ODE来表示,但是随着科技应用的发展,同样需要解决由PDE表示的最优控制问题.目前,很多学者对抛物方程的最优控制问题做了大量的研究,首先关于Burgers方程得到了许多有价值的结果[29],[71]-[77]Armaou和Christofides[30]研究了Kuramto-Sivashing方程的反馈控制问题.除此之外,很多学者研究了高阶抛物方程的最优控制问题.Yong和Zheng[27]在具有光滑边界的有界区域上研究了Cahn-Hilliard方程的回馈稳定和最优控制问题Tian等[23]研究了粘性抛物方程的最优控制问题,例如粘性Camassa-Holm方程给出了最优解的存在惟一性,Tian等[25,26]利用类似的方法处理了粘性Degasperis-Procesi方程,粘性Dullin-Gottualld-Holm方程,同样得到了最优解的存在惟一性.最近,对于Cahn-Hilliard方程,同样得到了一些结果,参见[28],[20],[21]Wang [50]考虑了半线性抛物方程的最优控制的必要条件.
综上所述,上述的问题都是研究四阶及以下的方程的最优控制问题,还没有对油-水-表面活性剂模型的最优控制问题的研究,所以我们首先利用Galerkin方法给出油-水-表面活性剂模型的最优控制问题的弱解的存在性,与四阶方程相比较,由于非线性项△(α(u)△u+(a'(u))/2｜▽u｜2)的存在,我们在做估计时,需要利用方程的能量泛函并调整所使用的不等式,来达到证明弱解存在性的目的.在这些估计的基础上,利用Lions关于最优解的经典理论来证明了最优解的存在性,同样是由于非线性项的干扰,使得我们在研究的过程中需要结合非线性项的特点,选择变量收敛的空间及收敛的方式.我们也给出了最优控制条件.本章最后,我们考虑了具有粘性项的油-水-表面活性剂模型的最优控制问题,在证明的过程中,我们利用变换在形式上改变了最优控制问题,再利用前面相同的方法之后,得到了具有粘性项问题的最优控制解的存在性.
第四章,我们研究方程的周期解问题.
从十九世纪到现在,扩散问题得到了广泛的研究,特别是周期问题引起了学术界极大的兴趣.据我们所知,对于二阶方程的周期问题已经得到了很多结果,例如[51,52,53,54].注意到高阶扩散方程可以用来描述生物种群、人口的扩散和迁徙等周期模型[55,56],有一些数值结果用来解释此类现象[67,68,69].周期解对于高阶抛物方程尤为重要,过去一段时间,很多学者研究了高阶抛物方程的周期解.包括空间周期问题[57,58,59,60,61],周期边值问题[62,63,64,65,66].Yin等[35,36,37]考虑了一类Cahn-Hilliard型方程并在一维情形下证明时间周期解的存在性Wang等[34]利用Galerkin方法和Leray-Schguder不动点定理,在一维和二维情形下,证明了广义Ginzburg-Landau模型方程的广义周期解和古典周期解的存在惟一性.在[32]中作者利用类似的方法研究了Camassa-Holm方程证明了时间周期解的存在唯一性.
然而,还有一些物理模型需要在二维或高维空间上考虑周期解,例如油膜在固体表面的扩散.所以无论从数学本身还是物理背景都需要研究高维情形,因此我们要在高维空间研究油-水-表面活性剂模型的周期解.为达到这个目的,我们首先引入算子G,在得到了算子的紧性及解的一些必要估计后,我们将在合适的泛函空间上得到算子的不动点(其中σ=1),即为问题的解.相比于四阶方程,由于四阶项和二阶扩散项的非线性项的存在,得到uσ和(?)uσ的Holder连续性还不足以证明主要定理,我们还需要△uσ的Holder连续性,为解决这个困难,我们主要利用Schauder型先验估计的方法来处理,先验估计将由调整Campanato空间得到.
In this thesis,we consider the equation which describes dynamics of phase transitions in ternary oil-water-surfactant systems,where λ>0and u is the scalar order parameter which is pro-portional to the local difference between oil and water concentrations.Let f(u)=F'(u),F(u)and.(u)are approximated,respectiVely,by a sixth and a second order polynomial
It is worth noting that sixth order parabolic equations have attracted mathematicians' attention.During the past years,many authors have paid much attention to the sixth order parabolic equation,such as the existence, uniqueness and regularity of the solutions[4]一[8]. However,as far as we know,there are few investigations concerned equation(1). Pawlow and Zajaczkowski[9]proved that the initial-boundary-value problem(1)一(3) admits a unique global smooth solution which depends continuously on the initial datum when λ1and α2>0. G.Schimperna et.al.[10]studied the equation with viscous term△ut where f(r)=F'(r),F(u)is the logarithmic potential They investigated the behavior of the solutions to the sixth order system as the parameter λ tends to0. G.Schimperna et. al.[42]considered a class of higher order equations characterized by a singular diffusion term where m(r)=M'(r),M(r)=(1-r)log(1-r)+(1+r)log(1+r),r∈[-1,1] and α(r)=2/(1-r2),and authors proved that for any final time T,the problem admits a unique energy type weak solution,defined over(0,T). Liu[11] studied the equation with nonconstant mobiliy and he proved the existence of classical solutions for two dimensions.
The aim of the thesis is to consider some properties of solutions for the oil-water-surfactant equation(1).
In the section2,we first consider the eq uation where Q is a bounded domain in Rn(n≤3) with smooth boundary and λ>0.The equation is supplemented by the boundary value conditions and the initial value condition u(x,0)=u0(x).
It is worth noticing that,in the absence of amphiphile the α(u)is a positive constant.When amphiphile is added to the system a minimum of α(u)develops at u=0.With increasing amphiphile strength or amphiphile concentration α(u)becomes negative at the microemulsion phase.So,from the background of physical chemistry and mathematics,the purpose of this paper is devoted to the investigation of properties of solutions with γ1,α2,α0are not restricted to be positive.
Firstly,the difficulties of the discussion on the existence and unique-ness of the solution for the equation(1)are to deal with u2△u,u｜▽u｜2and u6from the nonlinear term α(u)and f(u).To overcome these difficulties, we use the energy functional of the equation,the structure of the F(u), f(u)and the derivatives of f(u)and some inequalities. When we study the blowup of the solution,we need to consider the sign of α2and α0and deal with u2｜▽u｜2and｜▽u｜2,we also use the energy functional and the properties of ω,which is the unique solution of the following problem: We show that the solutions might not be classical globally.In other words, in some cases,the classical solutions exist globally,while in some other cases,such solutions blow up at a finite time.
In section3,we give the existence of attractors of the equation(1).
The dynamic properties of the higher order parabolic equation,such as the global asymptotical behaviors of solutions and existence of global attractors,are important for the study of higher order parabolic system. During the past years,many authors have paid much attention to the attractors. There are many studies on the existence of global attractors for general nonlinear dissipative dynamical systems.For the classical re-gults we refer the reader to Temam[79]and Hale[80]. Many authors are interested in the existence of global attractors for general nonlinear dissipative dynamical systems such as[12]一[15],[81]一[85]. G.Schimperna [43]in a three-dimensional spatial setting,gave the existence of attractors for both the viscous and the standard Cahn-Hilliard equation with a non-constant mobility coefficient.G.Schimperna[70]considered the Viscous Cahn-Hilliard equation characterized by the presence of an inertial term which is a differential model describing nonisothermal fast phase separa-tion processes taking place in a three-dimensional bounded domain,and the author proved that the corresponding dynamical system is dissipative and possesses a global attractor.G.Schimperna et.al.[45]studied the ex-istence of global attractors for a family of Cahn-Hilliard equations with a mobility depending on the chemical potential.L.song et.al.[49]considered the semilinear parabolic equation whered>0and g(x,u)is Authors proved the existence of the global attractors inHk(k≥0)apace. L.Song et.al.[48]also considered the fourth order parabolic equation where g(u)is a polynomial given by g(s)=(?)aksk,and the authors[46],[47] proved some parabolic equations possesses a global attractor in Hk(k≥0),which attracts any bounded subset of Hk(k≥0)in the Hk-norm.
To the best of our knowledge,there are few studies on the existence of the global attractors of the higher order parabolic equations. As Ko-rzec et.al.[44]established the the existence of global attractor for a six order equation describing the evolution of a growing crystalline surface in one di-mension and two dimensions.So,in this chapter,We discuss the existence of global attractor.We will use the regularity estimates for the linear semi-groups,combining with the iteration technique and the classical existence theorem of global attractors,to prove that the problem possesses a global attractor in Hk(k≥0)space.The difficulties are the higher order of u,▽u,△u and the emerge of▽△u,△4u caused by the△(a(u)△2u+(a'(u))｜▽u｜2) and△f(u),we should make some adjustment on the space,the embedding theorems and iteratiom.
In the section4,we also consider the problem with viscous terms δ△ut We obtained the similarly result.
In chapter3,we consider the optimal control of the oil-water-surfactant model.
From1950s,the optimal control of distributed parameter system had become much more active in academic ficld.Modern optimal control the-ories and applied models are both represented by ODE.With the develop-ment and application of technology,it is necessary to solve the problem of optimal control theories for PDE.There are many profound results on the optimal control problems for PDE,such as the optimal control for Burgers equation[29],([71]一[77]) Armaou and Christofides[30]studied the feedback control of Kuramto-Sivashing equation Besides,many papers have already been published to study the control problems of nonlinear parabolic equations.Yong and Zheng[27]studied the feedback stabilization and optimal control of the Cahn-Hilliard equa-tion in a bounded domain with smooth boundary.Tian et al [23,25,26] also studied the optimal control problems for parabolic equations,such as viscous Camassa-Holm equation,viscous Degasperis-Procesi equation,vis-cous Dullin-Gottualld-Holm equation and so on.Recently,Zhao and Liu [28]considered the optimal control problem for1D viscous Cahn-Hilliard equation.M.Kubo[20]considered the Cahn-Hilliard equation with time-dependent constraint. M.Hintermuller and D.Wegner[21]studied the distributed optimal control for the Cahn-Hilliard system.In their papers, the optimal control under boundary condition was given and the existence of optimal solution to the equation was proved.Wang[50]was concerned about necessary conditions for optimal control problems governed by some semi-linear parabolic differential equations.
Wc consider the standard oil-water-surfactant equation and the equa-tion with viscous term.We first employ Galerkin method to give the existence of the weak solution of equation. Because of nonlinear term△(a(u)△u+(a'(u))｜▽u｜2),we need to use the energy functional and make the adjustment on the inequalities to prove the existence of the weak so一lution of the equation. In section2,we prove the existence of optimal solution basing on Lions'theory. In section3,the optimality conditions are showed and the optimality system is derived. In section4,we used similarly methods and proved the existence of optimal solution.
In chapter4,we study the time periodic solution for the oil-water-surfact ant system.
From the early19th century,diffusion equations have been widely in-vestigated,specially,the periodic problems have been paid much attention. As far as we know,the researches on second order periodic diffhsion equa-tions are extensive,and many profound results have been obtained,such as[51,52,53,54],noticing that the higher diffusion equations can be used to describe models with periodic factors,for example biological groups, the diffusion and migration of population[55,56]and so on.so,the time periodic solutions are important for the higher-order parabolic equation.
During the past years,many authors have paid much attention to the time periodic solutions of higher order parabolic equations.We can find many corresponding numerical results which provide the refcerences to explain certain problems[67,68,69].Furthermore,some authors paid at-tention to periodic problems,including spatial periodic problems[57,58,59,60,61],periodic boundary problems[62,63,64,65,66].Yin et.a1.[35],[36],[37]considered time periodic problems that gave the existence of time periodic solutions for the Cahn-Hilliard type equation in one dimension. Using the Galerkin method and the Leray—Schauder fixed point theorem, wang et.al.[34]proved the existence and uniqueness of time-periodic gen-eralized solutions and time-periodic classical solutions to the generalized Ginzburg-Landau model equation in1D and2D cases. Y. Fu et.al.[32] employed the similar methods to consider the Camassa-Holm equation and proved the existence of the time-periodic solution. However, many physical phenomena, such as the diffusion of oil film over a solid surface, need to be discussed in two-dimension or multi-dimension case. So we should study the multi-dimensional case considered not only from mathe-matics itself but also from physical background. As far as we know, there are few investigations concerned with the time periodic solutions of such kind of equations.
In this chapter, we prove the existence of time periodic solutions of the problem in two space dimensions. For this purpose, we first introduce an operator G by considering a linear sixth-order equation with a param-eter σ∈[0,1]. After proving the compactness of the operator and some necessary estimates of the solutions, we then obtain a fixed point of the operator in a suitable functional space with σ=1, which is the desired solution of the problem.
Compare to the fourth order parabolic equation, we should not only obtain the Holder continuity of uσ and Vtσ, but also the Holder continuity of Δuσ, because of the nonlinearity of the fourth order term and the second order diffusive factors. The main method that we use is based on the Schauder type priori estimates, which here will be obtained by means of a modified Campanato space.
引文
[1] G. Gompper, J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys.Rev. E,50(1994),1325-1335.
[2] G. Gompper, M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaus-sian interface uctuations, Phys. Rev. E,47(1993),4289-4300.
[3] G. Gompper, M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. MonteCarlo simulations, Phys. Rev. E,47(1993),4301-4312.
[4] J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation:I. Blow-up similarity solutions, Nonlinearity,20(2007),1799-1841.
[5] J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation:II. Global similarity patterns, Nonlinearity,20(2007),1843-1881.
[6] Z. Li, C. Liu, On the nonlinear instability of travelingwaves for a sixth order parabolicequation, Abstr. Appl. Anal.,2012, Article ID739156,17pages,2012.
[7] C. Liu, Qualitative properties for a sixth-order thin film equation, Math. Model. Anal.,15(4)(2010),457-471.
[8] C. Liu, Y. Tian, Weak solutions for a sixth-order thin film equation, Rocky Mountain J.Math.,41(5)(2011),1547-1565.
[9] I. Pawlow, W. Zaja czkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal.,10(2011),1823-1847.
[10] G. Schimperna, I. Pawlow, On a Class of Cahn-Hilliard Models with Nonlinear Difusion,SIAM J. Math. Anal.,45(1)(2013),31-63.
[11] C. Liu, Regularity of solutions for a sixth order nonlinear parabolic equation in two spacedimensions, Annales Polonici Mathematici,107(3)(2013),271-291.
[12] B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class ofpattern formation equations, Comm. Partial Diferential Equations,14(2)1989,245-297.
[13] T. Dlotko, Global attractor for the Cahn-Hilliard equation in H2and H3, J. DiferentialEquations,113(1994),381-393.
[14] D. Li, C. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlin-earity, J. Diferential Equations,149(2)(1998),191-210.
[15] H. Wu, S. M. Zheng, Global attractor for the1-D thin film equation, Asymp. Anal.,51(2007),101-111.
[16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Pysics, Springer,New York,1988.
[17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Diferential Equa-tions, in: Appl. Math. Sci., vol.44, Springer-Verlag,1983.
[18] L. Song, Y. Zhang and T. Ma, Global attractor of the Cahn-Hilliard euqation in Hkspaces,J. Math. Anal. Appl.,355(2009),53-62.
[19] L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equationin Hkspace, Nonlinear Anal.,72(2010),183-191.
[20] M. Kubo, The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Anal.,75(2012),5672-5685.
[21] M. Hintermu¨ller, D. Wegner, Distributed optimal control of the Cahn-Hilliard systemincluding the case of a double-obstacle homogeneous free energy density, SIAM J. ControlOptim.,50(2012),388-418.
[22] J. L. Lions, Optimal control of systems governed by partial diferential equations, Springer,Berlin,1971.
[23] C. Shen, L. Tian and A. Gao, Optimal control of the viscous Dullin-Gottwalld-Holmequation, Nonlinear Anal. RWA,11(2010),480-491.
[24] J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density,and pressure, SIAM J. Math. Anal.,21(5)(1990),1093-1117.
[25] L. Tian, C. Shen, Optimal control of the viscous Degasperis-Procesi equation, J. Math.Phys.,48(11)(2007),113513,
[26] L. Tian, C. Shen and D. Ding, Optimal control of the viscous Camassa-Holm equation,Nonlinear Anal. RWA,10(1)(2009),519–530.
[27] J. Yong, S. Zheng, Feedback stabilization and optimal control for the Cahn-Hilliard equa-tion, Nonlinear Anal. TMA,17(1991),431-444.
[28] X. Zhao, C. Liu, Optimal control problem for viscous Cahn-Hilliard equation, NonlinearAnal.,74(2011),6348-6357.
[29] K. Kunisch, S. Volkwein, Control of the Burgers equation by a reduced-order approachusing proper orthogonal decomposition, J. Optim. Theory Appl.,102(2)(1999),345-371.
[30] A. Armaou, P. D. Christofides, Feedback control of Kuramto-Sivashing equation, PhysicaD,137(2000),49-61.
[31] A. Wouk, A Course of Applied Functional Analysis, Wiley-Interscience, New York,1979.
[32] Y. Fu, B. Guo, Time periodic solution of the viscous Camassa-Holm equation, J. Math.Anal. Appl.,313(2006),311-321.
[33] M. Giaquinta, M. Struwe, On the partial regularity for weak solutions to nonlinearparabolic systems, Math. Z.,179(1982),437-451.
[34] Y. P. Wang, Y. L. Zhang, Time-periodic solutions to a nonlinear parabolic type equationof higher order, Acta Mathematica Applicatae Sinica, English Series,24(2008),129-140.
[35] L. Yin, Y. Li, R. Huang and J. Yin, Time periodic solutions for a Cahn-Hilliard typeequation, Math. Comput. Modelling,48(2008),11-18.
[36] J. Yin, Y. Li and R. Huang, The Cahn-Hilliard type equations with periodic potentialsand sources, Appl. Math. Comput.,211(2009),211-221.
[37] Y. Li, J. Yin, The viscous Cahn-Hilliard equation with periodic potentials and sources, J.Fixed Point Theory Appl.,9(1)(2011),63-84.
[38] G. Gompper, M. Schick, Correlation between structural and interfacial properties of am-phiphilic systems, Phys. Rev. Lett.,65(1990),1116-1119.
[39] G. Gompper, M. Hennes, Layering, dewetting, and first-order wetting in ternary am-phiphilic systems, J. Chem. Phys.,102(2871)(1995); doi:10.1063/1.468665.
[40] G. Gompper, M. Schick, Self-assembling amphiphilic system, in Phase Transitions andCritical Phenomena”(C. Domb and J. Lebowitz eds.), vol.16, pages1-176, London,1994,Academic Press.
[41] G. Gompper, S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures, Phys.Rev. A,46(1992),4836-4851.
[42] G. Schimperna, I. Pawlow, A Cahn-Hilliard equation with singular difusion, J. DiferentialEquations,254(2)(2013),779-803.
[43] G. Schimperna, Global attractors for Cahn-Hilliard equations with nonconstant mobility,Nonlinearity,20(10)(2007),2365-2387.
[44] M. D. Korzec, P. Nayar and P. Rybka, Global attractors of sixth order PDEs describingthe faceting of growing surfaces, In press.
[45] M. Grasselli, A. Miranville, R. Rossi and G. Schimperna, Analysis of the Cahn-Hilliardequation with a chemical potential dependent mobility, Comm. Partial Diferential Equa-tions,36(7)(2011),1193-1238.
[46] L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equationin Hkspaces, Nonlinear Anal.,72(1)(2010),183-191.
[47] L. Song, Y. Zhang and T. Ma, Global attractor of the Cahn-Hilliard equation in Hkspaces,J. Math. Anal. Appl.,355(1)(2009),53-62.
[48] Y. Zhang, L. Song and T. Ma, The existence of global attractors for2D Navier-Stokesequations in Hkspaces, Acta Math. Sin.(Engl. Ser.),25(1)(2009),51-58.
[49] L. Song, Y. He and Y.Zhang, The existence of global attractors for semilinear parabolicequation in Hkspaces, Nonlinear Anal.,68(11)(2008),3541-3549.
[50] G. Wang, Pontryagin’s Maximum principle of optimal control governed by some non-wellposed semi-linear parabolic diferential equations, Nonlinear Anal.,53(2003),601-618.
[51] D. W. Bange, Periodic solutions of a quasilinear parabolic diferential equation, J. Difer-ential Equations,17(1975),61-72
[52] P. Fife, Solutions of parabolic boundary problems existing for all time, Arch. RationalMech. Anal.,16(1964),155-186.
[53] M. Nakao, On boundedness, periodicity, and almost periodicity of solutions of some non-linear parabolic equations, J. Diferential Equations,19(1975),371-385.
[54] T. I. Seidman, Periodic solutions of a nonlinear parabolic equations, J. Diferential Equa-tions,19(1975),242-257.
[55] D. S. Cohen, J. D. Murray, A generalized difusion model for growth and dispersal in apopulation, J. Math. Biology,12(1981),237-249.
[56] J. D. Murray, Mathematical Biology, Second edition, Springer-Verlag, Belin,(1993).
[57] A. Bonfoh, A. Miranville, On Cahn-Hilliard-Gurtin equations, Nonlinear Anal.,47(5)(2001),3455-3466.
[58] R. Choksi, P. Sternberg, Periodic phase separation: the periodic Cahn-Hilliard and isoperi-metric problems, Interfaces Free Bound.,8(3)(2006),371-392.
[59] Y. He, Y. Liu and T. Tang, On large time-stepping methods for the cahn-Hilliard equation,Appl. Numer. Math.,57(5-7)(2007),616-628.
[60] A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl.Math.,(4)(2003),165-185.
[61] X. Ye, X. Cheng, The Fourier spectral method for the Cahn-Hilliard equation, Appl. Math.Comput.,171(1)(2005),345-357.
[62] S. M. Choo, S. K. Chung and Y. J. Lee, Error estimates of nonstandard finite diferenceschemes for generalized Cahn-Hilliard and Kuramoto-Sivashinsky equations, J. KoreanMath. Soc.,42(6)(2005),1121-1136.
[63] M. I. M. Copetti, Numerical analysis of spectral approximations to the cahn-hilliard equa-tion, Comput. Appl. Math.,19(3)(2000),307-321.
[64] A. Eden, V. K. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett.,20(4)(2007),455-461.
[65] A. Miranville, Some models of Cahn-Hilliard equations in nonisotropic media, M2ANMath. model Numer. Anal.,34(3)(2000),539-554.
[66] X. Ye, The Fourier collocation method for the Cahn-Hilliard equation, Comput. Math.Appl.,44(1-2)(2002),213-229.
[67] S. M. Choo, S. K. Chung, Conservative nonlinear diference scheme for Cahn-Hilliardequation, Comput. Math. Appl.,36(7)(1998),31-39.
[68] J. Wei, M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H.Poincar′e Anal. Non Lin′eaire,15(4)(1998),459-492.
[69] H. Wu, S. Zheng, Convergence to equilibrium for the cahn-Hilliard equation with dynamicboundary conditions, J. Diferential Equations,204(2004),511-531.
[70]M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations,239(1)(2007),38-60.
[71]K. Kunisch, S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl.,102(2)(1999),345-371.
[72]M. Hinze, S Volkwein, Analysis of instantaneous control for the Burgers equation, Nonlin-ear Anal.,50(1)(2002),1-26.
[73]S. Volkwein, Application of augmented Lagrangian-SQP method to optimal control prob-lems for the stationary Burgers equation, Comput. Optim. Appl.,16(2000),57-81.
[74]S. Volkwein, Distributed control problems for the Burgers equation, Comput. Optim. Appl.,16(2000),57-81.
[75]R. Vedantham, Optimal control of the viscous Burgers equation using an equivalent index method, J. Global Optim.,18(2000),255-263.
[76]H. M. Park, M. W. Lee, Y. D. Jang, An efficient computational method of boundary optimal control problems for the Burgers equation, Comput. Methods Appl. Mech. Engrg.,166(1998),289-308.
[77]J. Yong, Existence theory of optimal controls for distributed parameter systems, Kodai. Math. J.,15(1992),193-220.
[78]王柔怀,关于一般抛物边值问题的Schauder型理论,吉林大学自然科学学报,2(1964),35-64.
[79]R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,2nd edn, in:Applied Mathematical Sciences, vol.68, Springer-Verlag, New York,1997.
[80]J. K. Hale, Asymptotic Behaviour of Dissipative Systems, AMS, Providence RI,1988.
[81]S. Lu, H. Wu and C. K. Zhong, Attractors for nonautonomous2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst.,13(3)(2005),701-719.
[82]Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J.,51(6)(2002),1541-1559.
[83] C. K. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-difusion equation,J. Diferential Equations,223(2)(2006),367-399.
[84] C. K. Zhong, C. Sun and M. Niu, On the existence of global attractor for a class of infinitedimensional nonlinear dissipative dynamical systems, Chinese Ann. Math.,26B(3)(2005),1-8.
[85] J. W. Cholewa, T. Dlotko, Global attractor of the Cahn-Hilliard system, Bull. Austral.Math. Soc.,49(2)(1994),277-292.
[2] G. Gompper, M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaus-sian interface uctuations, Phys. Rev. E,47(1993),4289-4300.
[3] G. Gompper, M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. MonteCarlo simulations, Phys. Rev. E,47(1993),4301-4312.
[4] J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation:I. Blow-up similarity solutions, Nonlinearity,20(2007),1799-1841.
[5] J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation:II. Global similarity patterns, Nonlinearity,20(2007),1843-1881.
[6] Z. Li, C. Liu, On the nonlinear instability of travelingwaves for a sixth order parabolicequation, Abstr. Appl. Anal.,2012, Article ID739156,17pages,2012.
[7] C. Liu, Qualitative properties for a sixth-order thin film equation, Math. Model. Anal.,15(4)(2010),457-471.
[8] C. Liu, Y. Tian, Weak solutions for a sixth-order thin film equation, Rocky Mountain J.Math.,41(5)(2011),1547-1565.
[9] I. Pawlow, W. Zaja czkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal.,10(2011),1823-1847.
[10] G. Schimperna, I. Pawlow, On a Class of Cahn-Hilliard Models with Nonlinear Difusion,SIAM J. Math. Anal.,45(1)(2013),31-63.
[11] C. Liu, Regularity of solutions for a sixth order nonlinear parabolic equation in two spacedimensions, Annales Polonici Mathematici,107(3)(2013),271-291.
[12] B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class ofpattern formation equations, Comm. Partial Diferential Equations,14(2)1989,245-297.
[13] T. Dlotko, Global attractor for the Cahn-Hilliard equation in H2and H3, J. DiferentialEquations,113(1994),381-393.
[14] D. Li, C. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlin-earity, J. Diferential Equations,149(2)(1998),191-210.
[15] H. Wu, S. M. Zheng, Global attractor for the1-D thin film equation, Asymp. Anal.,51(2007),101-111.
[16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Pysics, Springer,New York,1988.
[17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Diferential Equa-tions, in: Appl. Math. Sci., vol.44, Springer-Verlag,1983.
[18] L. Song, Y. Zhang and T. Ma, Global attractor of the Cahn-Hilliard euqation in Hkspaces,J. Math. Anal. Appl.,355(2009),53-62.
[19] L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equationin Hkspace, Nonlinear Anal.,72(2010),183-191.
[20] M. Kubo, The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Anal.,75(2012),5672-5685.
[21] M. Hintermu¨ller, D. Wegner, Distributed optimal control of the Cahn-Hilliard systemincluding the case of a double-obstacle homogeneous free energy density, SIAM J. ControlOptim.,50(2012),388-418.
[22] J. L. Lions, Optimal control of systems governed by partial diferential equations, Springer,Berlin,1971.
[23] C. Shen, L. Tian and A. Gao, Optimal control of the viscous Dullin-Gottwalld-Holmequation, Nonlinear Anal. RWA,11(2010),480-491.
[24] J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density,and pressure, SIAM J. Math. Anal.,21(5)(1990),1093-1117.
[25] L. Tian, C. Shen, Optimal control of the viscous Degasperis-Procesi equation, J. Math.Phys.,48(11)(2007),113513,
[26] L. Tian, C. Shen and D. Ding, Optimal control of the viscous Camassa-Holm equation,Nonlinear Anal. RWA,10(1)(2009),519–530.
[27] J. Yong, S. Zheng, Feedback stabilization and optimal control for the Cahn-Hilliard equa-tion, Nonlinear Anal. TMA,17(1991),431-444.
[28] X. Zhao, C. Liu, Optimal control problem for viscous Cahn-Hilliard equation, NonlinearAnal.,74(2011),6348-6357.
[29] K. Kunisch, S. Volkwein, Control of the Burgers equation by a reduced-order approachusing proper orthogonal decomposition, J. Optim. Theory Appl.,102(2)(1999),345-371.
[30] A. Armaou, P. D. Christofides, Feedback control of Kuramto-Sivashing equation, PhysicaD,137(2000),49-61.
[31] A. Wouk, A Course of Applied Functional Analysis, Wiley-Interscience, New York,1979.
[32] Y. Fu, B. Guo, Time periodic solution of the viscous Camassa-Holm equation, J. Math.Anal. Appl.,313(2006),311-321.
[33] M. Giaquinta, M. Struwe, On the partial regularity for weak solutions to nonlinearparabolic systems, Math. Z.,179(1982),437-451.
[34] Y. P. Wang, Y. L. Zhang, Time-periodic solutions to a nonlinear parabolic type equationof higher order, Acta Mathematica Applicatae Sinica, English Series,24(2008),129-140.
[35] L. Yin, Y. Li, R. Huang and J. Yin, Time periodic solutions for a Cahn-Hilliard typeequation, Math. Comput. Modelling,48(2008),11-18.
[36] J. Yin, Y. Li and R. Huang, The Cahn-Hilliard type equations with periodic potentialsand sources, Appl. Math. Comput.,211(2009),211-221.
[37] Y. Li, J. Yin, The viscous Cahn-Hilliard equation with periodic potentials and sources, J.Fixed Point Theory Appl.,9(1)(2011),63-84.
[38] G. Gompper, M. Schick, Correlation between structural and interfacial properties of am-phiphilic systems, Phys. Rev. Lett.,65(1990),1116-1119.
[39] G. Gompper, M. Hennes, Layering, dewetting, and first-order wetting in ternary am-phiphilic systems, J. Chem. Phys.,102(2871)(1995); doi:10.1063/1.468665.
[40] G. Gompper, M. Schick, Self-assembling amphiphilic system, in Phase Transitions andCritical Phenomena”(C. Domb and J. Lebowitz eds.), vol.16, pages1-176, London,1994,Academic Press.
[41] G. Gompper, S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures, Phys.Rev. A,46(1992),4836-4851.
[42] G. Schimperna, I. Pawlow, A Cahn-Hilliard equation with singular difusion, J. DiferentialEquations,254(2)(2013),779-803.
[43] G. Schimperna, Global attractors for Cahn-Hilliard equations with nonconstant mobility,Nonlinearity,20(10)(2007),2365-2387.
[44] M. D. Korzec, P. Nayar and P. Rybka, Global attractors of sixth order PDEs describingthe faceting of growing surfaces, In press.
[45] M. Grasselli, A. Miranville, R. Rossi and G. Schimperna, Analysis of the Cahn-Hilliardequation with a chemical potential dependent mobility, Comm. Partial Diferential Equa-tions,36(7)(2011),1193-1238.
[46] L. Song, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equationin Hkspaces, Nonlinear Anal.,72(1)(2010),183-191.
[47] L. Song, Y. Zhang and T. Ma, Global attractor of the Cahn-Hilliard equation in Hkspaces,J. Math. Anal. Appl.,355(1)(2009),53-62.
[48] Y. Zhang, L. Song and T. Ma, The existence of global attractors for2D Navier-Stokesequations in Hkspaces, Acta Math. Sin.(Engl. Ser.),25(1)(2009),51-58.
[49] L. Song, Y. He and Y.Zhang, The existence of global attractors for semilinear parabolicequation in Hkspaces, Nonlinear Anal.,68(11)(2008),3541-3549.
[50] G. Wang, Pontryagin’s Maximum principle of optimal control governed by some non-wellposed semi-linear parabolic diferential equations, Nonlinear Anal.,53(2003),601-618.
[51] D. W. Bange, Periodic solutions of a quasilinear parabolic diferential equation, J. Difer-ential Equations,17(1975),61-72
[52] P. Fife, Solutions of parabolic boundary problems existing for all time, Arch. RationalMech. Anal.,16(1964),155-186.
[53] M. Nakao, On boundedness, periodicity, and almost periodicity of solutions of some non-linear parabolic equations, J. Diferential Equations,19(1975),371-385.
[54] T. I. Seidman, Periodic solutions of a nonlinear parabolic equations, J. Diferential Equa-tions,19(1975),242-257.
[55] D. S. Cohen, J. D. Murray, A generalized difusion model for growth and dispersal in apopulation, J. Math. Biology,12(1981),237-249.
[56] J. D. Murray, Mathematical Biology, Second edition, Springer-Verlag, Belin,(1993).
[57] A. Bonfoh, A. Miranville, On Cahn-Hilliard-Gurtin equations, Nonlinear Anal.,47(5)(2001),3455-3466.
[58] R. Choksi, P. Sternberg, Periodic phase separation: the periodic Cahn-Hilliard and isoperi-metric problems, Interfaces Free Bound.,8(3)(2006),371-392.
[59] Y. He, Y. Liu and T. Tang, On large time-stepping methods for the cahn-Hilliard equation,Appl. Numer. Math.,57(5-7)(2007),616-628.
[60] A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl.Math.,(4)(2003),165-185.
[61] X. Ye, X. Cheng, The Fourier spectral method for the Cahn-Hilliard equation, Appl. Math.Comput.,171(1)(2005),345-357.
[62] S. M. Choo, S. K. Chung and Y. J. Lee, Error estimates of nonstandard finite diferenceschemes for generalized Cahn-Hilliard and Kuramoto-Sivashinsky equations, J. KoreanMath. Soc.,42(6)(2005),1121-1136.
[63] M. I. M. Copetti, Numerical analysis of spectral approximations to the cahn-hilliard equa-tion, Comput. Appl. Math.,19(3)(2000),307-321.
[64] A. Eden, V. K. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett.,20(4)(2007),455-461.
[65] A. Miranville, Some models of Cahn-Hilliard equations in nonisotropic media, M2ANMath. model Numer. Anal.,34(3)(2000),539-554.
[66] X. Ye, The Fourier collocation method for the Cahn-Hilliard equation, Comput. Math.Appl.,44(1-2)(2002),213-229.
[67] S. M. Choo, S. K. Chung, Conservative nonlinear diference scheme for Cahn-Hilliardequation, Comput. Math. Appl.,36(7)(1998),31-39.
[68] J. Wei, M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H.Poincar′e Anal. Non Lin′eaire,15(4)(1998),459-492.
[69] H. Wu, S. Zheng, Convergence to equilibrium for the cahn-Hilliard equation with dynamicboundary conditions, J. Diferential Equations,204(2004),511-531.
[70]M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations,239(1)(2007),38-60.
[71]K. Kunisch, S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl.,102(2)(1999),345-371.
[72]M. Hinze, S Volkwein, Analysis of instantaneous control for the Burgers equation, Nonlin-ear Anal.,50(1)(2002),1-26.
[73]S. Volkwein, Application of augmented Lagrangian-SQP method to optimal control prob-lems for the stationary Burgers equation, Comput. Optim. Appl.,16(2000),57-81.
[74]S. Volkwein, Distributed control problems for the Burgers equation, Comput. Optim. Appl.,16(2000),57-81.
[75]R. Vedantham, Optimal control of the viscous Burgers equation using an equivalent index method, J. Global Optim.,18(2000),255-263.
[76]H. M. Park, M. W. Lee, Y. D. Jang, An efficient computational method of boundary optimal control problems for the Burgers equation, Comput. Methods Appl. Mech. Engrg.,166(1998),289-308.
[77]J. Yong, Existence theory of optimal controls for distributed parameter systems, Kodai. Math. J.,15(1992),193-220.
[78]王柔怀,关于一般抛物边值问题的Schauder型理论,吉林大学自然科学学报,2(1964),35-64.
[79]R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,2nd edn, in:Applied Mathematical Sciences, vol.68, Springer-Verlag, New York,1997.
[80]J. K. Hale, Asymptotic Behaviour of Dissipative Systems, AMS, Providence RI,1988.
[81]S. Lu, H. Wu and C. K. Zhong, Attractors for nonautonomous2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst.,13(3)(2005),701-719.
[82]Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J.,51(6)(2002),1541-1559.
[83] C. K. Zhong, M. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-difusion equation,J. Diferential Equations,223(2)(2006),367-399.
[84] C. K. Zhong, C. Sun and M. Niu, On the existence of global attractor for a class of infinitedimensional nonlinear dissipative dynamical systems, Chinese Ann. Math.,26B(3)(2005),1-8.
[85] J. W. Cholewa, T. Dlotko, Global attractor of the Cahn-Hilliard system, Bull. Austral.Math. Soc.,49(2)(1994),277-292.