Domain perturbation problem for the Stokes equations and its application

• Project/Area Number: 26800073
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Yokohama National University (2016-2017); Tamagawa University (2014-2015)
• Principal Investigator: Ushikoshi Erika 横浜国立大学, 大学院環境情報研究院, 講師 (20714041)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: ストークス方程式 / アダマール変分公式 / スリップ境界条件 / NP作用素 / 固有値 / Hadamard変分公式 / Stokes方程式 / Green関数

Outline of Final Research Achievements

In this research, our aim is to analyze the domain dependency of the Green function and the eigenvalues of the Stokes equations with the Dirichlet boundary condition.
As a result, we succeeded to establish the Hadamard variational formula for the Green function for the velocity and pressure under the general second order domain perturbations. Moreover, we composed the variational formula for the multiple eigenvalues, and we succeeded to apply that formula to clarify the topological type of the domain.

Research Products

• [Journal Article] Hadamard variational formula for eigenvalues of the Stokes equations and its application (2017)
  • Author(s): Shuichi Jimbo, Hideo Kozono, Yoshiaki Teramoto, Erika Ushikoshi
  • Journal Title: Mathematische Annalen Volume: 印刷中 Issue: 1-2 Pages: 877-884
  • DOI: 10.1007/s00208-016-1410-5
  • Peer Reviewed
• [Journal Article] Hadamard variational formula for the multiple eigenvalues of the Stokes operator with the Dirichlet boundary conditions (2015)
  • Author(s): S.Jimbo, E. Ushikoshi
  • Journal Title: Far East J. Math. Sci. Volume: 98
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] On the Hadamard's formula of the Stokes equations (2018)
  • Author(s): 牛越惠理佳
  • Organizer: 第15回日独流体数学国際研究集会
  • Int'l Joint Research / Invited
• [Presentation] Hadamard variational formula for the eigenvalue of the Stokes equations with the slip boundary condition (2018)
  • Author(s): 牛越惠理佳
  • Organizer: Hirosaki Workshop for Mathematical Fluid Mechanics 2018
  • Int'l Joint Research / Invited
• [Presentation] Hadamard variational formula for the eigenvalues of the Stokes equations and its application (2017)
  • Author(s): 牛越惠理佳
  • Organizer: Vorticity, Rotation and Symmetry (IV)-Complex Fluids and Issue of Regularity
  • Int'l Joint Research / Invited
• [Presentation] Hadamard variational formula for the eigenvalues of the Stokes equations and its application (2017)
  • Author(s): 牛越惠理佳
  • Organizer: The 5th Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDE's
  • Int'l Joint Research / Invited
• [Presentation] ストークス方程式の固有値に対するアダマール変分公式とその応用 (2017)
  • Author(s): 牛越惠理佳
  • Organizer: 第8回大阪駅前セミナー
  • Invited
• [Presentation] Hadamard variational formula for the eigenvalues of the Stokes equations and its applicaiton (2016)
  • Author(s): 牛越惠理佳
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学(茨城県・つくば市)
  • Year and Date: 2016-03-18
• [Presentation] ストークス方程式に対するアダマール変分公式とその応用 (2016)
  • Author(s): 牛越惠理佳
  • Organizer: 第8回福島応用数学研究集会
  • Place of Presentation: コラッセ福島(福島県・福島市)
  • Year and Date: 2016-03-11
  • Invited
• [Presentation] ストークス方程式の固有値に対するアダマール変分公式とその応用について (2016)
  • Author(s): 牛越惠理佳
  • Organizer: 応用解析研究会
  • Place of Presentation: 早稲田大学
  • Invited
• [Presentation] ストークス方程式に対するアダマール変分公式とその応用 (2016)
  • Author(s): 牛越惠理佳
  • Organizer: 第５回岐阜数理科学研究会
  • Place of Presentation: 飛騨高山まちの博物館
  • Invited
• [Presentation] ストークス方程式に対するアダマール変分公式とその応用について (2016)
  • Author(s): 牛越惠理佳
  • Organizer: 横浜国立大学における微分方程式セミナー
  • Place of Presentation: 横浜国立大学
  • Invited
• [Presentation] ストークス方程式の固有値に対するアダマール変分公式とその応用 (2016)
  • Author(s): 牛越惠理佳
  • Organizer: 第46回 金沢解析セミナー
  • Place of Presentation: 金沢大学
  • Invited
• [Presentation] ストークス方程式の固有値に対するアダマール変分公式とその応用 (2016)
  • Author(s): 牛越惠理佳
  • Organizer: 三大学偏微分方程式セミナー
  • Place of Presentation: 中央大学
  • Invited
• [Presentation] Hadamard variational formula for the eigenvalue of the Stokes equations and its application (2016)
  • Author(s): 牛越惠理佳
  • Organizer: Toya workshop
  • Place of Presentation: Crystal Hall, Hotel Toya Sun Palace
  • Int'l Joint Research / Invited
• [Presentation] Stokes方程式の固有値に対するアダマール変分公式とその応用 (2016)
  • Author(s): 牛越惠理佳
  • Organizer: Seminar on Nonlinear Analysis at O-okayama
  • Place of Presentation: 東京工業大学
  • Invited
• [Presentation] Hadamard variational formula for eigenvalues of the Stokes equations and its application (2015)
  • Author(s): 牛越惠理佳
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 東京大学(東京都・目黒区)
  • Year and Date: 2015-12-20
  • Invited
• [Presentation] ディリクレ境界条件を課したラプラス方程式のグリーン関数に対する高次アダマール変分公式 (2015)
  • Author(s): 牛越惠理佳
  • Organizer: 第20回さいたま数理解析セミナー
  • Place of Presentation: 埼玉大学サテライトキャンパス(埼玉県・さいたま市)
  • Year and Date: 2015-11-28
  • Invited
• [Presentation] The application of the Hadamard variational formula for the Stokes equations (2015)
  • Author(s): 牛越惠理佳
  • Organizer: International Conference, SPP 1506 Transport Processes at Fluidic Interfaces, IRTG 1529 Mathematical Fluid Dynamics, JSPS Program of the Japanese-German Graduate Externship
  • Place of Presentation: ダルムシュタット工科大学(ドイツ・ダルムシュタット)
  • Year and Date: 2015-10-08
  • Int'l Joint Research / Invited
• [Presentation] ディリクレ境界条件を課したラプラス方程式のグリーン関数に対する高次アダマール変分公式について (2015)
  • Author(s): 牛越惠理佳
  • Organizer: 日本数学会2015年秋季総合分科会
  • Place of Presentation: 京都産業大学(京都府・京都市)
  • Year and Date: 2015-09-14
• [Presentation] Hadamard variational formula for the multipleeigenvalue of the Stokes operator with the Dirichlet boundary conditions (2015)
  • Author(s): 牛越惠理佳
  • Organizer: 環瀬戸内ワークショップ
  • Place of Presentation: 愛媛大学(愛媛県)
  • Year and Date: 2015-02-20
  • Invited
• [Presentation] Hadamard variational formula for the eigenvalue of the Stokes equations with the Dirichlet boundary conditions (2014)
  • Author(s): 牛越惠理佳
  • Organizer: 偏微分方程式セミナー
  • Place of Presentation: 北海道大学(北海道)
  • Year and Date: 2014-11-10
  • Invited
• [Presentation] Hadamard variational formula for the Stokes equations (2014)
  • Author(s): 牛越惠理佳
  • Organizer: RIMS研究集会「偏微分方程式の解の形状と諸性質」
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2014-11-06
  • Invited
• [Presentation] Hadamard variational formula for the eigenvalue of the Stokes equations (2014)
  • Author(s): Erika Ushikoshi
  • Organizer: Autumn School and Workshop on Mathematical Fluid Dynamics
  • Place of Presentation: Bad Boll(ドイツ)
  • Year and Date: 2014-10-28
• [Presentation] Hadamard variational formula for the eigenvalue of the Stokes equations with the Dirichlet boundary conditions (2014)
  • Author(s): 牛越惠理佳
  • Organizer: 第１２８回神楽坂解析セミナー
  • Place of Presentation: 東京理科大学(東京都)
  • Year and Date: 2014-10-25
  • Invited
• [Presentation] ストークス作用素の固有値に対するアダマール変分公式について (2014)
  • Author(s): 牛越惠理佳
  • Organizer: 日本数学会2014年秋季総合分科会
  • Place of Presentation: 広島大学(広島県)
  • Year and Date: 2014-09-27
• [Presentation] Hadamard variational formula for the Green function of the Stokes equations (2014)
  • Author(s): 牛越惠理佳
  • Organizer: 第39回偏微分方程式論札幌シンポジウム
  • Place of Presentation: 北海道大学(北海道)
  • Year and Date: 2014-08-25