arXiv subjects

Kohei Soga

Publications and source records attributed to Kohei Soga.

16 recordsLinked to original sources

arXiv2026

Co-moving volumes and Reynolds transport theorem in DiPerna-Lions theory

Co-moving volumes and Reynolds transport theorem along a fluid flow are fundamental tools to derive balance laws in fluid mechanics, where the classical theory on flow maps of ODEs associated to smooth vector fields plays a central role. Related to weak solutions of Navier-Stokes equations in Sobolev classes, DiPerna-Lions (Invent. Math. 1989) generalized the classical notion of ODEs and flow maps in the case of vector fields belonging to Sobolev classes. DiPerna-Lions theory also clarifies evolution of measure of the inverse image of each Borel measurable set under generalized flow maps in terms of the divergence of vector fields. On the other hand, the image of each measurable set under generalized flow maps, which corresponds to co-moving volumes in the classical theory, is not necessarily measurable. Hence, formulation of Reynolds transport theorem would not make sense. In this paper, we show that the image of each Borel measurable set trimmed with a suitable null set is measurable possessing measure consistent with the classical case without trimming. Then, defining co-moving volumes with such trimming, we prove Reynolds transport theorem for generalized flow maps. We also formulate Reynolds transport theorem in terms of the inverse image.

math.AP↗

arXiv2024

Mathematical analysis of the velocity extension level set method

A passively advected sharp interface can be represented as the zero level set of a level set function $f$. The linear transport equation $\partial_tf+v\cdot \nabla f =0$ is the simplest governing equation for such a level set function. While the signed distance of the interface is a geometrically convenient function, e.g., the norm of the gradient is everywhere one, its time evolution is not governed by the linear transport equation. In computational fluid dynamics, several modifications of the simplest case 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 method is a prominent method used for efficient numerical approximation of the local signed distance function of the interface. Our current paper presents a rigorous mathematical formulation of the velocity extension method and proves that the method provides the local signed distance function of the moving interface. A key is to derive a first-order fully nonlinear PDE that is equivalent to the linear transport equation with extended velocity. Then, wellposedness of the PDE is established in the class of $C^2$-smooth solutions, global in time and local in space, with the local signed distance property, as well as in the class of $C^0$-viscosity solutions, global in time and space. Furthermore, partial regularity of the viscosity solution is proven, thus confirming that, if initial data is smooth near the initial interface, the viscosity solution is smooth in a time-global tubular neighborhood of the interface, coinciding with the local-in-space $C^2$-smooth solution.

math.AP↗

arXiv2023

Mathematical analysis of modified level-set equations

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.

math.AP↗

arXiv2023

Existence of global weak solutions of inhomogeneous incompressible Navier-Stokes equations with mass diffusion

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. Any 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.

math.AP↗

arXiv2023

A remark on Tonelli's calculus of variations

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.

math.CA↗

arXiv2023

A finite difference method for inhomogeneous incompressible Navier-Stokes equations

This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (non-constant density and viscosity) incompressible Navier-Stokes system on a bounded domain. The proposed method consists of a version of 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 $0$, 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.

math.NA↗

arXiv2022

Finite difference methods for linear transport equations

DiPerna-Lions (Invent. Math. 1989) established the existence and uniqueness results for linear transport equations with Sobolev velocity fields. 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. 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. The key point for both of our 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 involving transport equations, where rigorous discrete approximation of geometric quantities of level sets is discussed.

math.NA↗

arXiv2020

Weak KAM theory for action minimizing random walks

We introduce a class of controlled random walks on a grid in $\mathbb{T}^d$ 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.

math.DS↗

arXiv2020

More on convergence of Chorin's projection method for incompressible Navier-Stokes equations

Kuroki and Soga [Numer. Math. 2020] proved that a version of Chorin's fully discrete projection method, originally introduced by A. J. Chorin [Math. Comp. 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. We show time-global solvability and convergence of our scheme; $L^2$-error estimates for the scheme in the class of smooth exact solutions; application of the scheme to the problem with a time-periodic external force to investigate time-periodic (Leray-Hopf weak) solutions, long-time behaviors, error estimates, etc.

math.AP↗

arXiv2019

On convergence of Chorin's projection method to a Leray-Hopf weak solution

The projection method to solve the incompressible Navier-Stokes equations was first studied by Chorin [Math. Comp., 1969] in the framework of a finite difference method and Temam [Arch. Rational Mech. and Anal., 1969] in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with the periodic boundary condition assuming existence of a $C^5$-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 $\mathbb{R}^3$ with the no-slip boundary condition and an external force. We prove unconditional solvability of our implicit scheme and strong $L^2$-convergence (up to subsequence) under the scaling condition $ h^{3-α}\leτ$ (no upper bound is necessary), where $h,τ$ are space, time discretization parameters, respectively, and $α\in(0,2]$ is any fixed constant. The results contain a compactness method based on a new interpolation inequality for step functions.

math.AP↗

arXiv2018

Stochastic and variational approach to finite difference approximation of Hamilton-Jacobi equations

The author presented a stochastic and variational approach 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 setting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. (2015)]. In the current paper, we extend these results to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equations, 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; 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 variational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs (1969)].

math.NA↗

arXiv2016

Weak KAM theory for discount Hamilton-Jacobi equations and its application

Weak KAM theory for discount Hamilton-Jacobi equations and corresponding discount 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.

math.AP↗

arXiv2015

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

Z^2-periodic entropy solutions of hyperbolic scalar conservation laws and Z^2-periodic viscosity solutions of Hamilton-Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton-Jacobi equations. Ugo Bessi ('03) investigated the convergence of approximate Z^2-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural Z^2-periodic solution with the aid of Aubry-Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which exhibits difference in characteristics between the two approximation techniques.

math.AP↗

arXiv2014

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

We derive mathematical models of the elementary process of dissolution/growth of bubbles in a liquid under pressure control. The modeling starts with a fully compressible version, both for the liquid and the gas phase so that the entropy principle can be easily evaluated. This yields a full PDE system for a compressible two-phase fluid with mass transfer of the gaseous species. Then the passage to an incompressible solvent in the liquid phase is discussed, where a carefully chosen equation of state for the liquid mixture pressure allows for a limit in which the solvent density is constant. We finally provide a simplification of the PDE system in case of a dilute solution.

physics.flu-dyn↗

arXiv2012

More on Stochastic and Variational Approach to the Lax-Friedrichs Scheme

A stochastic and variational aspect of the Lax-Friedrichs scheme was applied to hyperbolic scalar conservation laws by Soga [arXiv: 1205.2167v1]. The results for the Lax-Friedrichs scheme are extended here to show its time-global stability, the large-time behavior, and error estimates. The proofs essentially rely on the calculus of variations in the Lax-Friedrichs scheme and on the theory of viscosity solutions of Hamilton-Jacobi equations corresponding to the hyperbolic scalar conservation laws. Also provided are basic facts that are useful in the numerical analysis and simulation of the weak Kolmogorov-Arnold-Moser (KAM) theory. As one application, a finite difference approximation to KAM tori is rigorously treated.

math.NA↗

arXiv2012

Stochastic and Variational Approach to the Lax-Friedrichs Scheme

We present a stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws. This is a finite difference version of Fleming's results ('69) that the vanishing viscosity method is characterized by stochastic processes and calculus of variations. We convert the difference equation into that of the Hamilton-Jacobi type and introduce corresponding calculus of variations with random walks. The stability of the scheme is obtained through the calculus of variations. The convergence of approximation is derived from the law of large numbers in hyperbolic scaling limit of random walks. The main advantages due to our approach are the following: Our framework is basically pointwise convergence, not $L^1$ as usual, which yields uniform convergence except "small" neighborhoods of shocks; The convergence proof is verified for arbitrarily large time interval, which is hard to obtain in the case of flux functions of general types depending on both space and time; The approximation of characteristics curves is available as well as that of PDE-solutions, which is particularly important for applications of the Lax-Friedrichs scheme to the weak KAM theory.

math.NA↗