Some Problems On A Class Of Sixth-order Parabolic Equations

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 polynomialIt 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.