Properties Of Solutions For Some Higher Order Differential Equations (Systems)

In this thesis, we mainly study the model of oil-water-surfactant mixtures with con-centration dependent mobility, the Non-Newtonian Navier-Stokes equations coupled with the model of oil-water-surfactant mixtures, the model of oil-water-surfactant mixtures with inertial term. During the past years, many authors have paid much attention to the model of oil-water-surfactant mixtures with constant mobility. Pawlow and Za-jaczkowski [6] proved that the initial-boundary-value problem for the model of oil-water-surfactant mixtures with constant mobility, admits a unique global smooth solution which depends continuously on the initial datum. They [9] applying the approach based on the Backlund transformation and the Leray-Schauder fixed point theorem, generalize the existence result of [6] by imposing weaker assumptions on the data. Miranville [10] s-tudied the asymptotic behavior. They proved the existence of (finite-dimensional) global attractors and of exponential attractors. G. Schimperna etc [8] considered the model oil-water-surfactant mixtures with viscous term and logarithmic potential. They investigated the behavior of the solutions to the sixth order system as the parameter γ tends to 0. The uniqueness and regularization properties of the solutions have been discussed. Liu and Wang [12] studied the optimal control problem. Liu and Wang [5] proved the existence of time-periodic solutions in two space dimensions. However, as far as we know, there are few investigations concerned the model of oil-water-surfactant mixtures with concentra- tion dependent mobility. We will study the model of oil-water-surfactant mixtures with concentration dependent mobility in this thesis.In Chapter 2, we study the global existence of weak solutions for the model of oil-water-surfactant mixtures in Ω × (0, T), where Ω ∈ R3 is a bounded domain, m(u)=|u|n, n> 1 is mobility, k>0, a(u)=a1u2+a2, and a1>0,02 are constants ([64]). From the physical consideration, we prefer to consider a typical case of the volumetric free energy F(u), that is F1(u)=f(u), in the following form ([6,64]) The equation (13) is supplemented by the boundary value conditions and the initial value conditionThe method of Backhund transformation used in [9] seems not applicable to the present situation.Study on Pawlow and Zajaczkowski with constant mobility of Backlund transform method is not applicable to this equation. Our method is based on Galerkin approximation and Simon’s compactness results. The main difficulties for treating the problem are caused by the degeneracy of the principal part, nonlinearity of the fourth order term and the lack of maximum principle.In Chapter 3, we study the Non-Newtonian Navier-Stokes equations coupled with the model of oil-water-surfactant mixtures in Ω×(0,T), where Ω is a bounded domain in Rn, n≤3, with smooth boundary. The system (14)-(17) is supplemented by the initial and boundary conditions The function f(φ) stands for the derivative of a potential F(φ), with F(φ) approximated by a sixth order polynomial, r)(u) is the kinematic viscosity coefficient e(u) is the symmetric deformations velocity tensor whose components are given Here u denotes the mean velocity of the fluid mixture, φ is the concentration difference of the two components, μ, is the chemical potential, p denotes the pressure of the fluid mixture and g the external body force, κ,δ and λi(i=1,2) are positive constants.During the past years, many authors have paid much attention to the incompressible isothermal Navier-Stokes equation to coupled with a convective Cahn-Hilliard equation, such as the existence and uniqueness of solutions [16], the existence of an exponential attractors [17], the instability of certain stationary solutions [18]. Zhou and Fan [30] studied the vanishing viscosity limit for the 2D Cahn-Hilliard-Navier-Stokes system in a bounded domain with a slip boundary condition.Recently, Pierluigi, Sergio and Maurizio [27] considered the nonlocal Cahn-Hilliard-Navier-Stokes system where a(x)=(?)Ω J(x-y)dy, J*φ=(?)Ω J(x-y)φ(y)dy. They establish the global existence of a weak solution. Frigeri and Grasselli [21] establish the existence of the Global and trajectory attractor.Abels and Feireisl [15] considered the compressible isothermal Navier-Stokes system coupled with the Cahn-Hilliard equation, with S=2λ(φ)D(u)+v(φ)div(uI),D(u)=1/2(▽u+▽uT)-1/3div(uI) and the pressure P=ρ2af/ap(ρ,φ).I is unit matrix with free energy and total energy Our main purpose is to establish the global existence of weak solutions. The main diffi-culties for treating the problem are caused by the non-Newtonian (P-Laplace) term and the nonlinearity fourth order term. Our method is based on Galerkin approximation, energy estimates, Korn inequality and Simon’s compactness results.In Chapter 4, We consider the global existence and decay rate of the classical solu-tions to the Cauchy problem for the model of oil-water-surfactant mixtures with inertial term The function f(u) stands for the derivative of a potential F(u), respectively, where k>0,71>0,a2>0 are constants. In what follows, without loss of generality, let k=1.The energy method seems not applicable to the present situation. Our method is based on the long wave-short wave decomposition, Green’s function method and energy estimates.In Chapter 5, we investigate the sixth order nonlinear parabolic equation where Q=(0,1), k>0, D=a/ax. Prom the physical consideration, we prefer to consider a typical case of the potential F(u), that is F’(u)=-A(u)=u-u3, in the following form [24] namely, the well-known double well potential.The equation (19) is supplemented by the boundary conditions and the initial value condition The equation (19) arises naturally as a continuum model for the formation of quantum dots and their faceting, see [54]. Here u(x,t) denotes the surface slope, and v is propor-tional to the deposition rate. The high order derivatives are the result of the additional regularization energy which is required to form an edge between two plane surfaces with different orientations.The main difficulties for treating the problem 19 are caused by the nonlinearity of both the fourth order diffusive and the convective factors. The method used for treating Cahn-Hilliard equation seems not applicable to the present situation. We shall use the regularity estimates for the linear semigroups, combining with the iteration technique and the classical existence theorem of global attractors, to prove that the problem possesses a global attractor in the Hk (k≥0) space.