Some Problems On The Cahn-Hilliard Type Equations

Diffusion phenomena appear widely in the nature,for example the well-known heat diffusion.It is Fourier who first investigated this phenomenon in mathematics and gave the following heat equation in his celebrated memoir "Théorie analytique de la chaleur"(1810-1822).The heat equation is a typical second diffusion equation.From then on,there have been a tremendous amount of papers on diffusion equations.Thereinto,the fourth order diffusion equations,as an important type of diffusion equations,have been an in-teresting problem naturally.Cahn-Hilliard equation is a typical class of nonlinear fourth order diffusion equations.It was propounded by Cahn and Hilliard in 1958 as a mathematical model describing the diffusion phenomena in phase transition. Later,such equations are suggested as mathematical models of physical problems in many fields such as competition and exclusion of biological groups[16],mov-ing process of river basin[30]and diffusion of oil film over a solid surface[53]. Along with the development of the research,more physical terms and physical laws are included in the mathematical modelling and more generalized equations,such as viscous Cahn-Hilliard equation and Cahn-Hilliard equation with gradient dependent energy,are propounded.In this monograph,we call these generalized Cahn-Hilliard equation as Cahn-Hilliard type equation.It was Elliott and Zheng[25]who first study the Cahn-Hilliard equations in mathematics.They considered the following so-called standard Cahn-Hilliard equa-tion with constant mobility Basing on global energy estimates,they proved the global existence and uniqueness of classical solution of the initial boundary problem with spatial dimension N≤3 under the conditions that the coefficient of the leading term of the interior chemical potential is a positive constant or the leading term of the interior chemical potential is negative constant but the initial energy E(u₀)is sufficiently small.They also discussed the blow-up property of classical solutions when the coefficient of the leading term of the interior chemical potential is a negative constant but initial energy E(u₀)is sufficiently large.Later,there are many remarkable results about the Cahn-Hilliard equation,for example the asymptotic behavior of solutions[13,42, 54,65],perturbation of solutions[14,52],and stability of solutions[8,12],and the properties of the solutions for the Can-Hilliard equations with dynamic boundary conditions[38,45,46,47,56,57,48]which appear recently,etc.This monograph includes two chapters.In the first chapter,we consider the viscous Cahn-Hilliard equations.In the second chapter,we investigate the Cahn-Hilliard equations with gradient dependent energy.The viscous Chan-Hilliard equation is propounded by Novick-Cohen[41]in 1988.It is used to describe the dynamic behavior of the phase transition in alloy, for example[4,24].Later,many mathematicians considered this equation and have done many remarkable works,for example[3,4,15,20,23,24,28,37,49],etc. In 1996,Gurtin[29]considered the balance of the microforce in the mathematical modelling and gave the a generalized viscous Cahn-Hilliard equation.In the first chapter,we consider the following initial boundary value problem of the viscous Cahn-Hilliard equation with concentration dependent mobility and non-constant viscosity coefficient whereμ=φ(u)-kΔu+β(u)u_t,v denotes the unit exterior normal to the boundary (?)Ω,φ(u)is the interior chemical potential with a typical formIn the second section of this chapter,we consider a typical case of the initial boundary value problem(1)-(3),i.e.the case ofβ(u)=0.In this case,the equa-tion(1)becomes the standard Cahn-Hilliard equation with concentration dependent mobility.We know from the reference[25]that the sign of the coefficientγ₂ of the leading term of the interior chemical potential is important to the properties of the solutions.Exactly speaking,ifγ₂＞0 and the initial datum is suitably small,then the solution exists globally in time,while ifγ₂＜0 and the initial datum is suitably large,the solution blows up in finite time.In this section,we point out that such a result will not be valid for the equation with non-constant mobility.In fact,we show that if the mobility function m(s)satisfy the condition that(?)|s|^pm(s)＜+∞, where p is a constant lager than or equal to 4,then the solution does not blow up in finite time despite of the sign ofγ₂.That is to say,for the Cahn-Hilliard equa-tion with concentration dependent mobility,besides the sign of the coefficient of the leading term of the interior chemical potential,the mobility will also play a role to the blow-up of the solutions.In the third section of this chapter,we consider another typical case of the initial boundary value problem(1)-(3),i.e.both of the viscosity coefficientβ(u)and the mobility m(u)are constants.Ke and Yin[33]considered the case ofγ₂＞0 and proved the existence of the classical solutions.While,as far as we know,there are no results about the case ofγ₂＜0 for higher spatial dimensions before the present work.We first do some uniform a apriori estimates on the local solutions,then we prove the global in time existence of the weak solutions under the condition that the initial datum is suitably small.Then,we remove the restrict of the smallness of the initial datum and prove that the nontrivial solutions will blow-up in finite time. Furthermore,we find that the time of blow-up T^* is dependent on the viscosity coefficient.T^* will be large if the value of the viscosity coefficient is large.which, to a certain extent,reflects exactly the physical reality,we would like to mention here that in the proof of the blow-up of the nontrivial solutions,we construct a new Lyapunov functional which is different from that in the previous references where H′(u)=φ(u).Such a Lyapunov functional plays an important role in the proof of the main result.In the fourth section of this chapter,we consider the viscous Cahn-Hilliard equation with constant mobility and concentration dependent viscosity coefficient. Our main interest is to study the effect of the concentration dependent viscosity coefficient to the property of the blow-up of the solutions.Under two different structural conditions on the viscosity coefficient,we prove that the solutions blow up in finite time and tend to infinity when the time tend to infinity respectively. In the second chapter,we consider the following initial boundary value problem of a type of Cahn-Hilliard equation with gradient dependent energy whereΩis a bounded domain in R^N with smooth boundary,v denotes the unit exterior normal to the boundary(?)Ω,μ=K▽Δu-(?)(â–½u),K is a positive constant.In the second section of this chapter,we consider the initial boundary value problem(4)-(6)with constant mobility,i.e.m(u)equals to a positive constant.We employ Galerkin method to prove the existence and uniqueness of the weak solution to the initial boundary value problem(4)-(6)in any finite spatial dimensions.We also obtain a result that the weak solution blows up in finite time under certain condition.Furthermore,we show the regularity of the weak solution in two spatial dimensions.In the proof of the regularity of the solution,one of the most pivotal step is to establish the H(o|¨)lder norm estimate ofâ–½u.It is obvious that the Schauder's estimates are certain kind of pointwise estimates and in many cases it is quite diffi-cult to derive pointwise estimates directly from the differential equation considered. However,to derive integral estimates is relatively easy.In fact,the Campanato spaces can be used to describe the integral characteristic of the H(o|¨)lder continuous functions.The idea to overcome this difficulty is to establish the a apriori estimate on the Campanato norm ofâ–½u by the energy method,then use the property that the Campanato spaces can be embedded into the spaces of H(o|¨)lder continuous func-tions to obtain the desired estimate.After obtaining the H(o|¨)lder norm estimate ofâ–½u,we can use the classical parabolic theory to complete the proof of the regularity of the weak solution to the initial boundary value problem(4)-(6). In the third section of this chapter,we consider the initial boundary value problem(4)-(6)with concentration dependent mobility.In the present case,the main difficulty is that we can not find a corresponding Lyapunov functional,which is quit different from the case of constant mobility.Thus,we have to overcome more difficulties to do the energy type estimates.In the first subsection,we consider the one spatial dimension case.After establishing the necessary a apriori estimates,we first use the Sobolev embedding theorem and the equation(4)itself to obtain the H(o|¨)lder norm estimate of the solution.Then,we employ a result from the reference [61]to obtain the Schauder type estimate.Finally,we use the Leray-Schauder fixed point theorem to prove the existence of the classical solution to the initial boundary value problem(4)-(6)and use the Holmgren's method to prove the uniqueness of the solution.In the second subsection,we consider the initial boundary value problem (4)-(6)with constant mobility and small initial datum.The spatial dimension considered here is N≤3.In order to use the Sobolev embedding theorem to get the H(o|¨)lder norm estimate of the solution,we have to obtain some a apriori estimates of higher order terms than that in one spatial dimension case.After obtaining the necessary a apriori estimates by the energy method,we can obtain the existence and uniqueness of the classical solution to the problem(4)-(6)using the similar method to that in the second section.In the third subsection,we remove the restriction of the smallness of the initial datum to prove the existence and uniqueness of the classical solution to the problem(4)-(6).The main difficulty is the H(o|¨)lder norm estimate of the solution.Since we do not have the smallness of the initial datum any more,we can not use the normal energy method and the Sobolev embedding theorem to get the H(o|¨)lder norm estimate of the solution.The idea to overcome this difficulty is to use the Morrey theorem.That is to say,we first establish the L^p estimates of the solution and then obtain the boundedness of the solution in a suitable Morrey space.Then,the desired H(o|¨)lder norm estimate of the solution can be obtained by the embedding from the Morrey space.Finally,the main results can be obtained by a similar method to that in the first subsection.