Free boundary problems for flows with phase transitions consistent with thermodynamics based on maximal regularity theorem

2016 Fiscal Year Final Research Report

• Project/Area Number: 24340025
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University (2016); Shizuoka University (2012-2015)
• Principal Investigator: Shimizu Senjo 京都大学, 人間・環境学研究科(研究院), 教授 (50273165)
• Co-Investigator(Kenkyū-buntansha): 田中 直樹 静岡大学, 理学部, 教授 (00207119) ; 菊地 光嗣 静岡大学, 工学部, 教授 (50195202) ; 小林 孝行 大阪大学, 基礎工学研究科, 教授 (50272133) ; 久保 隆徹 筑波大学, 数理物質科学研究科(系), 講師 (90424811)
• Co-Investigator(Renkei-kenkyūsha): OGAWA Takayoshi 東北大学, 大学院理学研究科, 教授 (20224107) ; KUMURA Hironori 静岡大学, 理学部, 准教授 (30283336)
• Project Period (FY): 2012-04-01 – 2017-03-31
• Keywords: 数学解析 / Navier-Stokes方程式 / 自由境界問題 / 最大正則性 / 相転移 / 適切性 / 安定性

Outline of Final Research Achievements

We study the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics in the case of constant but non-equal densities of the phases. We employ the direct mapping approach to transform the problem locally in time to a fixed domain. The proof of local well-posedness is based on maximal regularity of the underlying principal linearization and the contraction mapping principle. We extend our well-posedness result to general geometries, study the stability of the equilibria of the problem, and show that a solution which does not develop singularities exist globally, and if its limit set conatins a stable equilibrium it converge to this equilibrium as time goes to infinity.

Free Research Field

偏微分方程式論