Soga Kohei

写真a

Affiliation

Faculty of Science and Technology, Department of Mathematics (Yagami)

Position

Associate Professor

 

Papers 【 Display / hide

  • More on Convergence of Chorin’s Projection Method for Incompressible Navier–Stokes Equations

    Maeda M., Soga K.

    Journal of Mathematical Fluid Mechanics (Journal of Mathematical Fluid Mechanics)  24 ( 2 )  2022.05

    ISSN  14226928

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    Kuroki–Soga (Numer. Math. 146:401–433, 2020) proved that Chorin’s fully discrete finite difference projection method, originally introduced by Chorin (Math. Comput. 23:341–353, 1969), is unconditionally solvable and convergent within an arbitrary fixed time interval to a Leray–Hopf weak solution of the incompressible Navier–Stokes equations on a bounded domain with an arbitrary external force. This paper is a continuation of Kuroki–Soga’s work to further exhibit mathematical aspects of the method. We show time-global solvability and convergence of the scheme; L2-error estimates for the scheme in the class of smooth exact solutions; application of the method to the problem with a time-periodic external force to investigate time-periodic (Leray–Hopf weak) solutions, long-time behaviors, error estimates, etc.

  • Weak KAM theory for action minimizing random walks

    Soga K.

    Calculus of Variations and Partial Differential Equations (Calculus of Variations and Partial Differential Equations)  60 ( 5 )  2021.10

    ISSN  09442669

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    We introduce a class of controlled random walks on a grid in Td and investigate global properties of action minimizing random walks for a certain action functional together with Hamilton–Jacobi equations on the grid. This yields an analogue of weak KAM theory, which recovers a part of original weak KAM theory through the hyperbolic scaling limit.

  • On convergence of Chorin’s projection method to a Leray–Hopf weak solution

    Kuroki H., Soga K.

    Numerische Mathematik (Numerische Mathematik)  146 ( 2 ) 401 - 433 2020.10

    ISSN  0029599X

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    The projection method to solve the incompressible Navier–Stokes equations was first studied by Chorin (Math Comput, 1969) in the framework of a finite difference method and Temam (Arch Ration Mech Anal, 1969) in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions assuming existence of a C5-solution, while Temam demonstrated an abstract argument to obtain a Leray–Hopf weak solution in problems on a bounded domain with the no-slip boundary condition. In the present paper, the authors extend Chorin’s result with full details to obtain convergent finite difference approximation of a Leray–Hopf weak solution to the incompressible Navier–Stokes equations on an arbitrary bounded Lipschitz domain of R3 with the no-slip boundary condition and an external force. We prove unconditional solvability of our implicit scheme and strong L2-convergence (up to a subsequence) under the scaling condition h3-α≤ τ (no upper bound is necessary), where h, τ are space, time discretization parameters, respectively, and α∈ (0 , 2] is any fixed constant. The results contain a compactness method based on a new interpolation inequality for step functions.

  • STOCHASTIC AND VARIATIONAL APPROACH TO FINITE DIFFERENCE APPROXIMATION OF HAMILTON-JACOBI EQUATIONS

    SOGA K.

    Mathematics of Computation (Mathematics of Computation)  89 ( 323 ) 1135 - 1159 2020.05

    ISSN  00255718

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    Previously, the author presented a stochastic and variational ap- proach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear Hamiltonians in the one-dimensional periodic set-ting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. 84 (2015), 629–651]. In the current paper, we extend these re-sults to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equa-tions, their spatial derivatives and the backward characteristic curves at the same time, within an arbitrary time interval. The proof is based on stochastic calculus of variations with random walks, a priori boundedness of minimizers of the variational problems that verifies a CFL type stability condition, and the law of large numbers for random walks under the hyperbolic scaling limit. Convergence of approximation and the rate of convergence are obtained in terms of probability theory. The idea is reminiscent of the stochastic and vari-ational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs 5 (1969) 515–530].

  • Weak KAM theory for discounted Hamilton–Jacobi equations and its application

    Mitake H., Soga K.

    Calculus of Variations and Partial Differential Equations (Calculus of Variations and Partial Differential Equations)  57 ( 3 )  2018.06

    ISSN  09442669

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    Weak KAM theory for discounted Hamilton–Jacobi equations and corresponding discounted Lagrangian/Hamiltonian dynamics is developed. Then it is applied to error estimates for viscosity solutions in the vanishing discount process. The main feature is to introduce and investigate the family of α-limit points of minimizing curves, with some details in terms of minimizing measures. In error estimates, the family of α-limit points is effectively exploited with properties of the corresponding dynamical systems.

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Papers, etc., Registered in KOARA 【 Display / hide

Presentations 【 Display / hide

  • A Finite Difference Method in Hamilton-Jacobi Equations and Weak KAM Theory

    Kohei Soga

    12th AIMS Conference in Taipei (Taiwan ) , 

    2018.07

    Oral presentation (invited, special)

  • On convergence of Chorin's projection method to a Leray-Hopf weak solution -Bounded Lipschitz domain case-

    Kohei Soga

    Conference on Mathematical Fluid Dynamics Bad Boll (Germany) , 

    2018.05

    Oral presentation (invited, special)

  • ハミルトン・ヤコビ方程式のディスカウント近似に対する選択問題:収束率

    SOGA KOHEI

    日本数学会2017年度年会 (首都大学東京) , 

    2017.03

    Oral presentation (general)

  • 弱KAM理論の応用1 ーHJ方程式の放物型近似・差分近似・discount近似と対応する力学系

    SOGA KOHEI

    RIMS研究集会: 力学系とその関連分野の連携探索 (京都大学) , 

    2016.06

    Oral presentation (invited, special)

  • 古典KAM理論・弱KAM理論入門

    SOGA KOHEI

    RIMS研究集会: 力学系とその関連分野の連携探索 (京都大学) , 

    2016.06

    Oral presentation (invited, special)

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Research Projects of Competitive Funds, etc. 【 Display / hide

  • 力学系・流体力学の応用解析的研究

    2018.04
    -
    2022.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Early-Career Scientists , Principal investigator

  • 応用解析としての非線形問題の研究

    2015.04
    -
    2019.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Young Scientists (B), Principal investigator

 

Courses Taught 【 Display / hide

  • INDEPENDENT STUDY ON FUNDAMENTAL SCIENCE AND TECHNOLOGY

    2022

  • GRADUATE RESEARCH ON FUNDAMENTAL SCIENCE AND TECHNOLOGY 2

    2022

  • GRADUATE RESEARCH ON FUNDAMENTAL SCIENCE AND TECHNOLOGY 1

    2022

  • BACHELOR'S THESIS

    2022

  • GRADUATE RESEARCH ON FUNDAMENTAL SCIENCE AND TECHNOLOGY 1

    2021

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Courses Previously Taught 【 Display / hide

  • 関数論第1同演習

    Keio University

    2014.04
    -
    2015.03

    Autumn Semester

  • 数学解析第2

    Keio University

    2014.04
    -
    2015.03

    Autumn Semester

  • 関数方程式第1同演習

    Keio University

    2014.04
    -
    2015.03

    Autumn Semester