KEIO RESEARCHERS INFORMATION SYSTEM

Career 【 Display / hide 】

Career 【 Display / hide 】

• 2008.04

  -

  2011.03

  日本学術振興会 特別研究員 DC1

• 2011.04

  -

  2013.08

  早稲田大学, 基幹理工学部 数学科, 助教（任期付）

• 2013.04

  -

  2013.08

  津田塾大学, 学芸学部 数学科, 非常勤講師

• 2013.09

  -

  2014.08

  École Normale Supérieure de Lyon, Unitė de Mathėmatiques Pures et Appliquėes, ポスドク研究員

• 2014.09

  -

  2019.03

  慶應義塾大学, 理工学部数理科学科, 専任講師（任期無）

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Academic Background 【 Display / hide 】

Academic Background 【 Display / hide 】

• 2002

  -

  2006

  Waseda University, 理工学部, 応用物理学科

  University, Graduated

• 2006

  -

  2008

  Waseda University, 大学院理工学研究科, 物理学及応用物理学専攻

  Graduate School, Completed, Master's course

• 2008

  -

  2011

  Waseda University, 大学院基幹理工学研究科, 数学応用数理専攻

  Graduate School, Completed, Doctoral course

Academic Degrees 【 Display / hide 】

Academic Degrees 【 Display / hide 】

• 博士（理学）, Waseda University, Coursework, 2011.03

  Theoretical and numerical analysis on the regular motions of Hamiltonian dynamics

Research Areas 【 Display / hide 】

Research Areas 【 Display / hide 】

• Natural Science / Mathematical analysis (偏微分方程式論　非線形解析学)

Research Keywords 【 Display / hide 】

Research Keywords 【 Display / hide 】

• 数理流体力学　Hamilton力学系　粘性解理論

Papers 【 Display / hide 】

Papers 【 Display / hide 】

• Mathematical analysis of the velocity extension level set method

  Dieter Bothe, Kohei Soga

  Journal of Differential Equations (Elsevier)  468 2026

  Research paper (scientific journal), Joint Work, Corresponding author, Accepted

  　View Summary

  The level set method relies on the representation of sharp interfaces as zero level sets of an appropriate class of level set functions. The advection of such a sharp interface within a flow field v is then governed by the linear transport equation in the simplest setting. A convenient choice for the level set function is, at least locally, the signed distance to the interface as its gradient has unit length. It thus gives the interface normal field from which further geometric quantities such as the curvature can be computed. While the signed distance of the interface hence is a geometrically convenient level set function, its time evolution is not governed by the linear transport equation. Several modifications of the linear level set equation have been proposed in order to compute the signed distance function or to stabilize the norm of the gradient of a level set function on the interface. The velocity extension level set method is a prominent approach used for efficient numerical approximation of the local signed distance function of the moving interface. We present a rigorous mathematical formulation of the velocity extension level set method and prove that it indeed provides the smooth local signed distance function of the interface. A key is to derive a first-order fully nonlinear PDE that is equivalent to the linear transport equation with extended velocity. The main challenge to be overcome is the fact that the velocity extension level set method on the one hand requires enough regularity near the interface to have basic geometric quantities well-defined, while on the other hand the problem should have a unique solution in the full spacial domain so that it is a wellposed problem for numerical methods. For this purpose, we combine techniques of viscosity solutions to Hamilton-Jacobi equations with the classical method of characteristics, and develop a novel local comparison principle to obtain partial regularity of viscosity solutions, confirming that the viscosity solution is locally smooth and consistent with the local signed distance function in a time-global tubular neighborhood of the interface.

  • Access to Document (DOI)

• Finite Difference Methods for Linear Transport Equations with Sobolev Velocity Fields

  Kohei Soga

  Journal of Mathematical Fluid Mechanics (Springer Nature)  27 ( 6 )  2024.11

  Research paper (scientific journal), Single Work, Lead author, Last author, Corresponding author, Accepted

  　View Summary

  DiPerna-Lions (Invent. Math., 1989) established the existence and uniqueness results for weak solutions to linear transport equations with Sobolev velocity fields.
  Motivated by fluid mechanics, this paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields.
  The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is L^p-strongly convergent in the class of DiPerna-Lions weak solutions.
  The second method is based on an implicit scheme with L^2-estimates, where the discrete Helmholtz-Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and L^2-strongly convergent in the class of DiPerna-Lions weak solutions.
  The key point for both of the methods is to obtain fine L^2-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna-Lions.
  Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method for sharp interfaces involving transport equations, where rigorous discrete approximation of level-sets and their geometric quantities is discussed.

  • Access to Document (DOI)

• Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations

  Kohei Soga

  Numerische Mathematik (Springer Nature)   2024.07

  Research paper (scientific journal), Single Work, Lead author, Last author, Corresponding author, Accepted

  　View Summary

  This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (i.e., non-constant density and viscosity) incompressible Navier-Stokes system on a bounded domain. The proposed method is classical in the sense that it consists of a version of the Lax-Friedrichs explicit scheme for the transport equation and a version of Ladyzhenskaya's implicit scheme for the Navier-Stokes equations. Under the condition that the initial density profile is strictly away from zero, the scheme is proven to be strongly convergent to a weak solution (up to a subsequence) within an arbitrary time interval, which can be seen as a proof of existence of a weak solution to the system. The results contain a new Aubin-Lions-Simon type compactness method with an interpolation inequality between strong norms of the velocity and a weak norm of the product of the density and velocity.

  • Access to Document (DOI)

• Existence of global weak solutions of inhomogeneous incompressible Navier–Stokes system with mass diffusion

  Kacedan E., Soga K.

  Zeitschrift fur Angewandte Mathematik und Physik (Springer Nature)  75 ( 2 )  2024.04

  ISSN  00442275

  　View Summary

  This paper proves existence of a global weak solution to the inhomogeneous (i.e., non-constant density) incompressible Navier–Stokes system with mass diffusion. The system is well-known as the Kazhikhov–Smagulov model. The major novelty of the paper is to deal with the Kazhikhov–Smagulov model possessing the non-constant viscosity without any simplification of higher order nonlinearity. Every global weak solution is shown to have a long time behavior that is consistent with mixing phenomena of miscible fluids. The results also contain a new compactness method of Aubin–Lions–Simon type.

  • Access to Document (DOI)

• Mathematical analysis of modified level-set equations

  Bothe D., Fricke M., Soga K.

  Mathematische Annalen (Springer Nature)   2024

  ISSN  00255831

  　View Summary

  The linear transport equation allows to advect level-set functions to represent moving sharp interfaces in multiphase flows as zero level-sets. A recent development in computational fluid dynamics is to modify the linear transport equation by introducing a nonlinear term to preserve certain geometrical features of the level-set function, where the zero level-set must stay invariant under the modification. The present work establishes mathematical justification for a specific class of modified level-set equations on a bounded domain, generated by a given smooth velocity field in the framework of the initial/boundary value problem of Hamilton–Jacobi equations. The first main result is the existence of smooth solutions defined in a time-global tubular neighborhood of the zero level-set, where an infinite iteration of the method of characteristics within a fixed small time interval is demonstrated; the smooth solution is shown to possess the desired geometrical feature. The second main result is the existence of time-global viscosity solutions defined in the whole domain, where standard Perron’s method and the comparison principle are exploited. In the first and second main results, the zero level-set is shown to be identical with the original one. The third main result is that the viscosity solution coincides with the local-in-space smooth solution in a time-global tubular neighborhood of the zero level-set, where a new aspect of localized doubling the number of variables is utilized.

  • Access to Document (DOI)

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

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

• Research on dynamical systems and fluid mechanics in terms of applied analysis

  Soga, Kohei

  科学研究費補助金研究成果報告書 2023

• Mathematical analysis of numerical methods in fluid dynamics

  Soga, Kohei

  福澤諭吉記念慶應義塾学事振興基金事業報告集 (福澤基金運営委員会)  2022

• Applied analysis for nonlinear problems

  Soga, Kōhei

  科学研究費補助金研究成果報告書 2018

Presentations 【 Display / hide 】

Presentations 【 Display / hide 】

• Mathematical analysis of modified level-set equations

  Kohei Soga

  [International presentation]  94th GAMM @ Magdeburg (Magdeburg, Germany) , 

  2024.03

  , 

  Oral presentation (invited, special)

• Tonelli’s calculus of variations and related topics

  Kohei Soga

  [International presentation]  Kolloquium@Technische Universität Darmstadt, 

  2022.10

  , 

  Oral presentation (invited, special)

• Weak KAM theory for discretized Hamilton-Jacobi equations

  Kohei Soga

  [International presentation]  New trends in Hamilton-Jacobi: PDE, Control, Dynamical Systems and Geometry (Shanghai) , 

  2019.07

  , 

  Oral presentation (invited, special)

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

  Kohei Soga

  [International presentation]  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

  [International presentation]  Conference on Mathematical Fluid Dynamics Bad Boll (Germany) , 

  2018.05

  , 

  Oral presentation (invited, special)

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

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

• 力学系のレア・イベント解析と数論、幾何学、統計物理学への展開

  2026.04

  -

  2031.03

  文部科学省・日本学術振興会, 科学研究費助成事業, 基盤研究(A) , Coinvestigator(s)

• 流体力学における数値解法の数学解析と解析力学における古典KAM理論の数学解析

  2022.04

  -

  2027.03

  MEXT,JSPS, Grant-in-Aid for Scientific Research, 基盤研究(C), Principal investigator

• 流体力学における数値解法の数学解析

  2022.04

  -

  2023.03

  慶應義塾大学, 福澤基金 （福澤諭吉記念慶應義塾学事振興基金）, Principal investigator

• 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 】

Courses Taught 【 Display / hide 】

• MATHEMATICAL SCIENCES PRACTICAL RESEARCH ACTIVITY E

  2026

• BACHELOR'S THESIS

  2026

• GRADUATE RESEARCH ON MATHEMATICAL AND PHYSICAL SCIENCES 1

  2026

• GRADUATE RESEARCH ON FUNDAMENTAL SCIENCE AND TECHNOLOGY 1

  2026

• INDEPENDENT STUDY ON FUNDAMENTAL SCIENCE AND TECHNOLOGY

  2026

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Courses Previously Taught 【 Display / hide 】

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

Memberships in Academic Societies 【 Display / hide 】

Memberships in Academic Societies 【 Display / hide 】

• 日本数学会, 

  2007

  -

  Present

Committee Experiences 【 Display / hide 】

Committee Experiences 【 Display / hide 】

• 2025.07

  -

  Present

  Editor, Nonlinear Analysis: Real World Applications (Elsevier)

• 2026.06

  -

  Present

  Associate Editor, Partial Differential Equations in Applied Mathematics (Elsevier)

• 2020.04

  -

  2022.03

  Editor, Tokyo Journal of Mathematics