論文 - 曽我　幸平

Finite Difference Methods for Linear Transport Equations with Sobolev Velocity Fields

Kohei Soga

Journal of Mathematical Fluid Mechanics （Springer Nature）  27 （ 6 ）  2024年11月

研究論文（学術雑誌）, 単著, 筆頭著者, 最終著者, 責任著者, 査読有り

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

Kohei Soga

Numerische Mathematik （Springer Nature）  2024年07月

研究論文（学術雑誌）, 単著, 筆頭著者, 最終著者, 責任著者, 査読有り

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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年04月

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年

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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A Remark on Tonelli’s Calculus of Variations

Soga K.

Russian Journal of Nonlinear Dynamics （Institute of Computer Science Izhevsk）  19 （ 2 ） 239 - 248 2023年

ISSN  26585324

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This paper provides a quite simple method of Tonelli’s calculus of variations with positive definite and superlinear Lagrangians. The result complements the classical literature of calculus of variations before Tonelli’s modern approach. Inspired by Euler’s spirit, the proposed method employs finite-dimensional approximation of the exact action functional, whose minimizer is easily found as a solution of Euler’s discretization of the exact Euler – Lagrange equation. The Euler – Cauchy polygonal line generated by the approximate minimizer converges to an exact smooth minimizing curve. This framework yields an elementary proof of the existence and regularity of minimizers within the family of smooth curves and hence, with a minor additional step, within the family of Lipschitz curves, without using modern functional analysis on absolutely continuous curves and lower semicontinuity of action functionals.

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Hamilton-Jacobi方程式に対する有限差分法の数学解析―確率論と変分法を用いた枠組―

曽我幸平

学会誌「応用数理」 （日本応用数理学会）  32 （ 3 ） 127 - 138 2022年09月

研究論文（学術雑誌）, 単著, 筆頭著者, 査読有り

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

Maeda M., Soga K.

Journal of Mathematical Fluid Mechanics （Springer Nature）  24 （ 2 ）  2022年05月

ISSN  14226928

　概要を見る

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.

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Weak KAM theory for action minimizing random walks

Soga K.

Calculus of Variations and Partial Differential Equations （Springer Nature）  60 （ 5 ）  2021年10月

筆頭著者, 最終著者, 責任著者,  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.

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

Kuroki H., Soga K.

Numerische Mathematik （Springer Nature）  146 （ 2 ） 401 - 433 2020年10月

ISSN  0029599X

　概要を見る

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.

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

SOGA K.

Mathematics of Computation （American Mathematical Society）  89 （ 323 ） 1135 - 1159 2020年05月

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].

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

Mitake H., Soga K.

Calculus of Variations and Partial Differential Equations （Springer Nature）  57 （ 3 ）  2018年06月

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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Selection problems of Z^2-periodic entropy solutions and viscosity solutions

Kohei Soga

Calculus of Variation and Partial Differential Equations （Springer Nature）  56 （ 4 ）  2017年

研究論文（学術雑誌）, 単著, 査読有り

More on stochastic and variational approach to the Lax-Friedrichs scheme

SOGA KOHEI

Mathematics of Computation （American Mathematical Society）  2016年02月

研究論文（学術雑誌）, 単著, 筆頭著者, 最終著者, 責任著者, 査読有り

On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

Takayuki Kubo, Yoshihiro Shibata, Kohei Soga

Discrete and Continuous Dynamical Systems. Series A （AIMS）  36 （ 7 ） 3741 - 3774 2016年

研究論文（学術雑誌）, 査読有り

Thermodynamically consistent modeling for dissolution/growth of bubbles in an incompressible solvent

Dieter Bothe, Kohei Soga

Adv. Math. Fluid Mech. （Birkhauser/Springer）     111 - 134 2016年

研究論文（国際会議プロシーディングス）, 共著, 査読有り

Stochastic and variational approach to the Lax-Friedrichs scheme

曽我 幸平

Mathematics of Computation （American Mathematical Society）  84   629 - 651 2015年03月

研究論文（学術雑誌）, 単著, 査読有り

On the ℛ-boundedness for the two phase problem: compressible-incompressible model problem

Takayuki Kubo, Yoshihiro Shibata, Kohei Soga

Boundary Value Problems （Springer Open）  2014:141 2014年09月

研究論文（学術雑誌）, 共著, 査読有り

Space–time continuous limit of random walks with hyperbolic scaling

曽我 幸平

Nonlinear Analysis: Theory, Methods & Applications （Elsevier）  102   264 - 271 2014年06月

研究論文（学術雑誌）, 単著, 査読有り

Difference approximation to Aubry-Mather sets of the forced Burgers equation

Takaaki Nishida, Kohei Soga

Nonlinearity （IOP）  25   2401 - 2422 2012年

研究論文（学術雑誌）, 共著, 査読有り

A point-wise criterion for quasi-periodic motions in the KAM theory

Kohei Soga

Nonlinear Analysis （Elsevier）  73   3151 - 3161 2010年

研究論文（学術雑誌）, 査読有り