Graduate School of Fundamental Science and Engineering Waseda University
博 士 論 文 概 要
Doctoral Thesis Synopsis
論 文 題 目
Maximal 𝐿 𝑝 − 𝐿 𝑞 regularity theorem for some initial-boundary value problems appearing in
mathematical physics
申 請 者 (Applicant Name)
Suma Inna
スマ イナ
Department of Pure and Applied Mathematics, Research on Partial Differential Equations
December, 2018
No. 1
This research discusses about The maximal 𝐿𝑝− 𝐿𝑞 regularity theorem for some initial-boundary value problem. Maximal regularity is a powerful tool in the treatment of nonlinear PDEs, as it provides a priori estimates that give local or global existence results. By using maximal regularity, it is possible to solve quasilinear and fully nonlinear PDEs by linearization techniques combined with the contraction mapping principle. In this doctoral thesis, we are concerned with the maximal 𝐿𝑝− 𝐿𝑞 regularity theorem for some initial-boundary value problem arising in mathematical physics such as thermoelasticity and fluid dynamics. In particular, we discuss the maximal 𝐿𝑝− 𝐿𝑞 regularity for dealing with thermoelastic plates equation and Stokes equation.
Thermoelatic plate equations
We consider the following linearized system
𝑢𝑡𝑡+ 𝜇(𝑥) ∆2𝑢𝑡 + ∆𝜃 = 𝑓1 In (0, T) × Ω
𝜃𝑡− ∆𝜃 − ∆𝑢𝑡 = 𝑓2 In (0, T) × Ω} (1) with initial condition
𝑢|𝑡=0 = 𝑢0 in Ω, 𝑢𝑡|𝑡=0 = 𝑢1 in Ω, 𝜃|𝑡=0 = 𝜃0 in Ω (2) and Dirichlet boundary condition
𝑢|Γ= 0, 𝐷𝜈𝑢|Γ = 0, 𝜃|Γ = 0. (3) Here, we assume that 𝜇(𝑥) is a uniformly continuous function such that
𝐴1 ≤ 𝜇(𝑥) ≤ A2 (4) with positive constants 𝐴1 and 𝐴2 are satisfying 𝐴1≤ 𝐴2. Setting 𝑣 = 𝑢𝑡, we transform the problem (1) to (3) to the following first order system of equations:
𝑈𝑡 − 𝐴𝜇(𝐷)𝑈 = 𝐹 in Ω × (0, 𝑇), 𝐵(𝐷)𝑈|Γ= 0, 𝑈|{ 𝑡=0}= 𝑈0, (5) With
𝑈 = (𝑢, 𝑣, 𝜃 )⊤, 𝐹 = (𝑓, 𝑔, ℎ)⊤ and 𝑈0 = (𝑢0, 𝑣0, 𝜃0)⊤.
To prove the maximal 𝐿𝑝− 𝐿𝑞 regularity for the problem (5), we use an 𝑅-bounded solution operator for corresponding resolvent problem obtained by the Laplace transform with respect to time variable and Weis' operator valued Fourier multiplier theorem. To prove the existence of 𝑅 bounded solution operator, we analyse the problem in the whole space, in the half space and in a bent half space. In the whole space case, we assume that 𝜇 is a positive number satisfying 𝐴1 ≤ 𝜇 ≤ 𝐴2.
In addition, we analysis of a perturbed problem in the whole space. Let 𝜇(𝑥)be a real valued continuous function satisfying the assumption (4). Let 𝑥0 be any point in Ω and let 𝑀1 be any number in (0, 1). Let 𝑑0 > 0 be a small positive number such that
|𝜇(𝑥) − 𝜇(𝑥0)| ≤ 𝑀1
for any 𝑥 ∈ 𝐵𝑑0(𝑥0)(𝑥0). We may assume that 𝐵𝑑0(𝑥0) ⊂ Ω. Let 𝜑(𝑥) be a function in 𝐶0∞(𝑹𝑁)$ which equals one in 𝐵𝑑0
2
(𝑥0) and zero outside of 𝐵𝑑0(𝑥0)(𝑥0). Let
𝜈(𝑥) = 𝜑(𝑥)𝜇(𝑥) + (1 − 𝜑(𝑥))𝜇(𝑥0).
Then, we consider the resolvent problem:
𝜆𝑈 − 𝐴𝜈(𝐷)𝑈 = 𝐹 in 𝑹𝑁.
No. 2
Next, we consider the problem in the half space case. In this case, we assume that 𝜇is a positive number satisfying 𝐴1≤ µ ≤ 𝐴2. Then, we consider the problem in a bent half space by changing variable: let
Ω+ = Φ(𝑹+𝑁) = { 𝑦 ∈ 𝑹𝑁 ∣∣ 𝑦 = Φ(𝑥), 𝑥 ∈ 𝑹+𝑁 } and
Γ+ = Φ(𝑹0𝑁) = { 𝑦 ∈ 𝑹𝑁 ∣∣ 𝑦 = Φ(𝑥), 𝑥 ∈ 𝑹0𝑁}, where Φ be a diffeomorphism of class 𝐻∞4 on 𝑹𝑁.
Let 𝜇 be a real valued function satisfying the condition (4). Let 𝑦0 be any point on Γ+ and we assume that there exists a positive number 𝑑0 such that
|𝜇(𝑦) − 𝜇(𝑦0)| ≤ 𝑀1 for any 𝑦 ∈ 𝐵𝑑0(𝑥0)(𝑦0) ∩ Ω+. Let
𝜈(𝑦) = 𝜑(𝑦)𝜇(𝑦) + (1 − 𝜑(𝑦))𝜇(𝑦0).
Then, we consider the following resolvent problem in a bent half space:
𝜆 𝑈 − 𝐴𝜈(𝐷)𝑈 = 𝐹 in Ω+, 𝑢 = 𝜕𝜈+𝑢 = 𝜃 = 0 on Ω+.
Finally, we construct a parametrix in Ω to prove the existence of the 𝑅-bounded solution operator of the thermoelastic plate equation. By using the 𝑅-bounded solution operator and Weis' operator value Fourier multiplier theorem, the maximal 𝐿𝑝− 𝐿𝑞 regularity is derived as shown in the following result.
Let 𝑇 > 0. Let 1 < 𝑝, 𝑞 < ∞ . Assume that Ω is a uniform 𝐶4-domain in 𝑹𝑁. Let 𝐷𝑞,𝑝 (Ω) = ( 𝐇𝑞0(Ω), 𝐇𝑞2 (Ω))
1−1 𝑝,𝑝,
where (⋅,⋅)𝜃,𝑝 denotes the real interpolation functor. Then, there exists a number 𝜆1 such that for any initial data 𝑈0 = (𝑢0, 𝑣0, 𝜃0)⊤ ∈ 𝐷𝑞,𝑝 (Ω) and right-hand side 𝐹 = (𝑓, 𝑔, ℎ)⊤ ∈ 𝐿𝑝( (0, 𝑇), 𝐇𝑞0(Ω)), the problem (1) – (3) admits a unique solution 𝑈 = (𝑢, 𝑣, 𝜃 )⊤ with
𝑈 ∈ 𝐿𝑝((0, 𝑇), 𝐇𝑞2(Ω)) ∩ 𝐇𝑝1((0, 𝑇), 𝐇𝑞0(Ω)) possessing the estimate:
‖𝑈‖𝐿𝑝((0,𝑇),𝐇𝑞2(Ω)) +‖∂tU‖𝐿𝑝((0,𝑇),𝐇𝑞0(Ω))
≤ 𝐶𝑒𝜆1𝑇(‖𝑈0‖
𝐵𝑞,𝑝2+2(1−
1
𝑝)(Ω)× 𝐵,𝑝2(1−
1 𝑝)(Ω)2
+ ‖𝐹‖𝐿
𝑝((0,𝑇),𝐻𝑞(Ω)2 × 𝐿𝑞(Ω)2).
Stokes equations
Let Ω be a uniformly 𝐶3 domain with boundary Γ in the N dimensional Euclidean space 𝑹𝑁 (𝑁 ≥ 2), and let 𝐧 be the unit outer normal to Γ. This section deals with the linear problem:
𝜕𝑡 𝐯 − Div(𝜇𝐃(𝐯) − 𝐩𝐈) = 𝐅, div 𝐯 = 𝐆 = 𝑑𝑖𝑣 𝐆 in Ω × (0, 𝑇), 𝜕𝑡𝜌 + 𝐴𝜎⋅ ∇Γ 𝜌 − 𝐯 ⋅ 𝐧 + Ƒ 𝐯 = 𝐷 on Γ × (0, 𝑇), (𝜇𝐃(𝐯) − 𝐩𝐈) – (𝐵 + 𝛿ΔΓ)𝜌)𝐈)𝐧 = 𝐇 on Γ × (0, 𝑇), (𝐯, 𝜌)|{𝑡=0} = (𝐯0, 𝜌0 ) on Ω × Γ. }
(6)
We assume that
𝑚0≤ 𝜇(𝑥), 𝛿(𝑥) ≤ 𝑚1, |∇(𝜇(𝑥), 𝛿(𝑥))| ≤ 𝑚1 for all 𝑥 ∈ Ω
‖Ƒ (𝐯)‖𝐻
𝑞(Ω)2 ≤ 𝑚1‖Ƒ (𝐯)‖𝐻
𝑞1(Ω) , ‖𝐵𝜌‖
𝑊𝑞1−
1 𝑞 (Γ)
≤ 𝑚1‖𝜌‖
𝑊𝑞2−
1 𝑞 (Γ)
for some positive constants 𝑚0 and 𝑚1. Moreover, we assume that 𝐴0 = 0 and that for any 𝜎 ∈ (0,1), 𝐴𝜎 satisfies the assumptions:
|𝐴𝜎(𝑥)| ≤ 𝑚2,
No. 3
|𝐴𝜎(𝑥) − 𝐴𝜎(𝑦)| ≤ 𝑚2|𝑥 − 𝑦|𝑎 for any 𝑥, 𝑦 ∈ Γ,
‖Aσ‖
𝑊𝑟2−
1 𝑟(Γ)
≤ 𝑚3 𝜎−𝑏
for some positive constants 𝑚2, 𝑚3, 𝑎 and 𝑏 that are independent of 𝜎 ∈ (0,1), where 𝑟 is an exponent for which 𝑁 < 𝑟 < ∞. Similar with the first part, to prove the maximal 𝐿𝑝− 𝐿𝑞 regularity, our approach is to construct 𝑅-bounded solution operator for the generalized resolvent problem and to apply the Weis
operator valued Fourier multiplier.
Since the pressure term 𝐩 has no time evolution, we eliminate 𝐩 and the divergence equation: div 𝐮 = 𝑔 = div 𝐠 following the idea due to Grubb and Solonnikov. For this purpose, we introduce the reduced Stokes equations. Given 𝐮 ∈ 𝐻𝑞2(Ω)𝑁 and ℎ ∈ 𝑊𝑞3−
1
𝑞(Γ), let 𝐾(𝐮, ℎ) be a unique solution of the weak Dirichlet problem:
(∇ 𝐾(𝐮, ℎ), ∇𝜑)Ω= (Div(𝜇𝐃(𝐮)) − ∇ div 𝐮, ∇𝜑)Ω for any 𝜑 ∈ 𝐻̂
𝑞′,0 1 (Ω) subject to
𝐾(𝐮, ℎ) = < 𝜇𝐃(𝐮)𝐧, 𝐧 > − (𝐵 + 𝛿ΔΓ)ℎ − div 𝐮 on Γ.
Then, we consider the reduced Stokes equations:
𝜆𝐮 − Div(𝜇𝐃(𝐮) − 𝐾(𝐮, ℎ)𝐈) = 𝐟 in Ω, 𝜆ℎ + 𝐴𝜎⋅ ∇Γ ℎ − 𝐮 ⋅ 𝐧 + Ƒ 𝐮 = 𝑑 on Γ, (𝜇𝐃(𝐮) − 𝐾(𝐮, ℎ)𝐈 − ((𝐵 + 𝛿ΔΓ)ℎ)𝐈)𝐧 = 𝐡 on Γ.
} (7) and show the equivalence between the resolvent problem associated with the problem (6) and the reduced Stokes equations (7).
The next step is to prove the 𝑅-bounded solution operators of the reduced Stokes equation in the whole space, half space and a bent half space. We employ similar argumentation concerning the procedure of the proof of the R-bounded solution operators to those in thermoelastic plate problem. Finally, the maximal 𝐿𝑝− 𝐿𝑞 regularity is obtained by using the R-bounded solution operator combined with Weis' operator value Fourier multiplier theorem.
The thesis is organized as follows. In Chapter 2, we prove the maximal 𝐿𝑝− 𝐿𝑞 regularity for linearized thermoelastic plate equation. In addition, we prove the 𝑅-bounded solution operators for the problem which plays important role in this chapter. Moreover, we discuss the local and global existence of some nonlinear thermoelastic plate equation as the reader can find the result in Chapter 3.
In Chapter 4, we study the maximal 𝐿𝑝− 𝐿𝑞 for the Stokes equations (5). Similar to previous chapter, to prove the maximal 𝐿𝑝− 𝐿𝑞 regularity, we use the R-bounded solution operators. The main result of this chapter is shown in Theorem 4.1.4 and Theorem 4.1.6. Moreover, we discuss a reduced Stokes operator to eliminate the pressure term 𝐩 since it has no time evolution.
In Appendix A, a unique existence theorem for the weak Dirichlet problem is proved in 𝑹𝑁 and 𝑹+𝑁. In Appendix B, the regularity theorem for the weak Dirichlet problem is proved. Notice that the uniqueness of strong solutions does not hold in general. In Appendix C, some Poincaré type inequality is proved. Finally, in Appendix D, several properties of uniform 𝐶3 domains are proved.
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
Suma Inna
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As of October, 2018
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