KEIO RESEARCHERS INFORMATION SYSTEM

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

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  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.

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• 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

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  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.

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• 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

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  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.

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• 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

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  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.

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• Mathematical analysis of modified level-set equations

  Bothe D., Fricke M., Soga K.

  Mathematische Annalen (Springer Nature)   2024

  ISSN  00255831

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  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.

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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 】

• 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)

• 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

  [Domestic presentation]  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

  [Domestic presentation]  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 】

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

• 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

• 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 】

• MATHEMATICS 3B

  2026

• ESSENTIALS OF MATHEMATICAL SCIENCES B

  2026

• MATHEMATICAL SCIENCES PRACTICAL RESEARCH ACTIVITY B

  2026

• MATHEMATICAL SCIENCES PRACTICAL RESEARCH ACTIVITY D

  2026

• MATHEMATICAL SCIENCES PRACTICAL RESEARCH ACTIVITY E

  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

Committee Experiences 【 Display / hide 】

Committee Experiences 【 Display / hide 】

• 2025.07

  -

  Present

  Editor, Nonlinear Analysis: Real World Applications