Graduate School of Fundamental Science and Engineering Waseda University
博 士 論 文 概 要
Doctoral Thesis Synopsis
論 文 題 目
Thesis Theme
Dynamical Motion of Newtonian and Non- Newtonian Fluid Flows with Free Surfaces
申 請 者 (Applicant Name)
Sri Maryani
スリ マルヤニ
Department of Pure and Applied Mathematics, Research on Partial Differential Equations
December, 2015
The thesis is primarily concerned with the well-posedness of the Oldroyd- B model in the compressible barotropic fluid flow case and the existence of R bounded solution operators of the two phase problem for the Stokes resolvent equations.
In 1965, J. G. Oldroyd introduced the Oldroyd-B model in the incompressible viscous fluid case to describe the flow of viscoelastic fluids (Proc. Roy. Soc.
London 245 (1958), 278–297). Several recent studies investigating this model have been carried out on some polymer application using the numerical analysis.
Despite this, the mathematical investigation has not been developed yet in the compressible viscous fluid case. This reason is an important motivational factor to study the local and global well-posedness in this thesis.
On the other hand, the two phase problem for the Navier-Stokes equations appears in a lot of problems in fluid engineering, for example, problem of de- struction of the fluid machine by the cavitation. The benchmark is formulated mathematically by the two phase problem for the viscous fluid flows with sharp interface. In 1996, Denisova (St. Petersburg Math. J. 7 (1996)) studied this problem in L2 setting. In spite of better regularity of solution and less com- patibility condition, the same problem should be treated in theLp-Lq maximal regularity class which motivation to consider the R bounded solution operator of the resolvent problem for the two phase problem of the Stokes equations in this thesis. The R is the main step to prove that is the local and the global well-posedness for the two phase problems in the maximalLp-Lqregularity class.
This thesis is composed of four themed chapters. First chapter is introduc- tion including physical motivation and the notation. Chapter Two begins by laying out the proof of local well-posedness of the Oldroyd-B model. The third chapter of this thesis examined the global well-posedness of the Oldroyd-B type.
Chapter 4 analyzed theR-bounded solution operator of two phase problem for the Stokes resolvent equations
From now on, more detailed explanation of main parts is given. In Chapters 2 and 3, the free boundary problem of the Oldroyd-B model is treated, which is formulated as follows:
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∂tρ+ div (ρu) = 0 in Ωt,
ρ(∂tu+u·∇u)−DivT(u, P(ρ)) =βDivτ in Ωt,
∂tτ+u·∇τ+γτ =δD(u) +gα(∇u,τ) in Ωt, (T(u, P(ρ)) +βτ)nt=−P(ρ∗)nt on Γt, (ρ,u,τ)|t=0 = (ρ∗+θ0,u0,τ0) in Ω, Ωt|t=0 =Ω, Γt|t=0 =Γ.
(1)
for 0 < t < T. Let Ωt and Γt be the evolution of reference body Ω and its boundaryΓ, respectively. The problem is to determine the regionΩt ⊂RN, the
1
density fieldρ=ρ(x, t), τ =τ(x, t) be the elastic part of the stress tensor and the velocity fieldu= (u1(x, t), . . . , uN(x, t)).
Here, ρ∗ is a positive constant describing the mass density of the reference domain Ω,T(u, P(ρ)) the stress tensor of the form
T(u,ρ) =S(u)−P(ρ)I with S(u) =µD(u) + (ν−µ)divuI,
D(u) the doubled deformation tensor whose (i, j) components are Dij(u) =
∂iuj +∂jui (∂i = ∂/∂xj), I the N ×N identity matrix, µ, ν, β, γ and δ are positive constants (µ and ν are the first and second viscosity coefficients, respectively),nt is the unit outer normal toΓt, P(ρ) aC∞ function defined for ρ>0 which satisfies thatP′(ρ)>0 forρ>0. Moreover, the functiongα(∇u,τ) has a form
gα(∇u,τ) =W(u)τ−τW(u) +α(τD(u) +D(u)τ),
where αis a constant with −1≤ α≤1 and W(u) the doubled antisymmetric part of the gradient ∇u whose (i, j) components are Wij(u) = ∂iuj −∂jui. Finally, for any matrix field K whose components are Kij, the quantity DivK is an N vector whosei-th component is!N
j=1∂jKij, and also for any vector of functionsu= (u1, . . . , uN), divu=!N
j=1∂juj, andu·∇uis anN vector whose i-th component is !N
j=1uj∂jui.
Aside from the dynamical system (1), a further kinematic condition forΓtis satisfied, which gives
Γt ={x∈RN |x=x(ξ, t) (ξ∈Γ)}, wherex=x(ξ, t) is the solution to the Cauchy problem:
dx
dt =u(x, t) (t >0), x|t=0=ξ∈Ω.
In Chapter 2, the local well-posedness for the equations in Lagrange coordi- nates is proved. The local well-posedness means the existence of solutions in a finite time interval for any initial data. The chapter 2 consists of four sections.
Since the domain is unknown, problem (1) is transformed to the equations de- scribed in the Lagrange coordinate in Sect.1. After this transformation, the linearized problem is also explained. And then, main theorems concerning the local well-posedness for the nonlinear problem as well as the maximalLp-Lq reg- ularity for linearized equations are stated. In Sect. 2, it is proved the existence ofRbounded solution operators of the resolvent problem for the linearized equa- tions. In Sect. 3, the maximalLp-Lq regularity theorem is proved by combining the results in Sect. 2 with the Weis operator valued Fourier multiplier theorem.
In Sect. 4, the local well-posedness is proved by the Banach fixed point theorem based on the results obtained in Sect. 3.
In Chapter 3, the global well-posedness for small initial data is proved, which means the unique existence of solutions defined on the whole time interval (0,∞).
The main ingredient is the exponential decay property of solutions to the lin- earized equations. The chapter 3 consists of four sections. The first section is devoted to stating the main results about the global well-posedness for the nonlinear problem. In Sect. 2, it is proved the main result concerning the decay properties of solutions to the linearized equations by using results obtained in Sect. 3. In Sect. 3, the exponential stability of semi-group associated with lin- earized equations is proved by showing that the resolvent set contains the whole non-negative real half plan provided that the underlying space is orthogonal to the rigid motion. In Sect. 4, the global well-posedness is proved by a bootstrap argument based on the decay theorem obtained in Sect. 2.
In Chapter 4, the resolvent problem for the two phase problem of the Stokes equations is discussed. The equations is formulated as follows: Let Ω be a domain in RN, N ≥2, with two boundariesΓ± which satisfying Γ−∩Γ+ = ∅. Assume that some hyper-surfaceΓdividesΩinto two sub-domain ofΩ, that is, there are connected subsets Ω± ofΩ such thatΩ\Γ=Ω+∪Ω−. And we also suppose that Γ∩Γ+ =∅, Γ∩Γ− = ∅. Ω±’s boundaries consist of two part, Γ andΓ±, respectively. Let ˙Ω=Ω+∪Ω−. NowΩ+is occupied by one of the fluids with the viscosity coefficient µ+ and the density ρ+, whilst Ω− is occupied by another fluid with the viscosity coefficient µ− and the densityρ−. Hereρ± and µ± are positive constants. Ω± are occupied by different incompressible viscous fluids. The system of equations describing problem are figured by
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λu−ρ−1DivT(u,θ) =f, divu=g in ˙Ω, [[T(u,θ)n]] = [[h]] inΓ, [[u]] = 0 onΓ, T(u,θ)n+=k onΓ+,
u= 0 inΓ−.
(2)
The chapter 4 consists of four sections. In the first section, after it is stated the main result concerning the existence ofRbounded solution operators associated with (2), the reduced Stokes equations is given and the equivalence between the Stokes equations and reduced Stokes equations are proved. The main reason why the reduced Stokes equations are introduced is that the divergence condition is not stable under the localization procedure. In Sect. 2, the model problem in the whole space with planer interface is studied. In Sect. 3, the perturbed half- space problem is studied. In Sect. 4, the main result is proved by constructing the parametric with the help of the partition of unity and the results in Sect. 3.
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No.1
早稲田大学 博士（理学） 学位申請 研究業績書
(List of research achievements for application of doctorate (Dr. of Science), Waseda University)
氏 名 Sri Maryani 印( seal)
(As of December, 2015
） 種 類 別(By Type)
題名、 発表・発行掲載誌名、 発表・発行年月、 連名者（申請者含む）
(theme, journal name, date & year of publication, name of authors inc. yourself)
Thesis S. Maryani, Global Well-Posedness for Free Boundary Problem of The Oldroyd-B Model Fluid Flow, Mathematical Methods in the Applied Sciences, in press.
