論文

This study explains a stochastic and variational approach to the finite difference method applied to first order Hamilton-Jacobi equations generated by Hamiltonians of the Tonelli class. In Section 1, the connection between Hamilton-Jacobi equations and Hamiltonian dynamics is stated, which implies the necessity to introduce weak solutions. In Section 2, the representation formula of a local classical solution obtained via the method of characteristics is extended to a globally defined function employing Tonelli’s calculus of variations. This function is called a value function in optimal control theory, and it is a weak solution of the Hamilton-Jacobi equation in the sense of viscosity solutions. In Section 3, reasoning similar to the previous section is demonstrated for the discretized Hamilton-Jacobi equations with the most elementary finite difference method, where a stochastic feature appears due to numerical viscosity. The convergence of approximation is proven in terms of stochastic and variational techniques, yielding a viscosity solution, its derivative and characteristic curves all at once. This approach is substantially applied to weak Kolmogorov-Arnold-Moser (KAM) theory.

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-bounded solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.

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.

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal - regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).

We consider space-time inhomogeneous one-dimensional random walks which move by ±∆x in each time interval ∆t with arbitrary transition probabilities depending on position and time. Unlike Donsker's theorem, we study the continuous limit of the random walks as ∆x,∆t →0 under hyperbolic scaling λ_1 ≥∆t/∆x≥λ_0 > 0 with fixed numbersλ_1 and λ_0. Our aim is to present explicit formulas and estimates of probabilistic quantities which characterize asymptotics of the random walks as ∆x,∆t → 0. This provides elementary proofs of several limit theorems on the random walks. In particular, if transition probabilities satisfy a Lipschitz condition, the random walks converge to solutions of ODEs. This is the law of large numbers. The results here will be foundations of a stochastic and variational approach to finite difference approximation of nonlinear PDEs of hyperbolic types.

Jauslin–Kreiss–Moser and W E made clear the connection between the Aubry–Mather theory and the inviscid forced Burgers equation with a Z^2-periodic forcing term and established the smooth approximation of Z^2-periodic entropy solutions of the PDE. This paper presents results of a difference approximation to the Aubry–Mather sets. We prove the convergence of the Lax–Friedrichs scheme for the -periodic entropy solutions. This result leads to difference approximations of the corresponding effective Hamiltonian and Z^2-periodic viscosity solutions of the Hamilton–Jacobi equation. We numerically construct the Aubry–Mather sets through the approximate entropy solutions, based on the dynamical properties of the Aubry–Mather sets.

In this paper 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 a.e. pointwise convergence with characterization of “a.e.”, which yields uniform convergence except “small” neighborhoods of shocks; The stability and convergence proofs are verified for arbitrarily large time interval, which are hard to obtain in the case of flux functions of general types depending on both space and time; the approximation of characteristic 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.

A stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws and Hamilton-Jacobi equations generated by space-time dependent flux functions of the Tonelli type was clarified by Soga (2015). The results for the Lax-Friedrichs scheme are extended here to show its time-global stability, the large-time behavior, and error estimates. Also provided is a weak KAM-like theorem for discrete equations that is useful in the numerical analysis and simulation of the weak KAM theory. As one application, a finite difference approximation to effective Hamiltonians and KAM tori is rigorously treated. 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.

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. Bessi (Commun Math Phys 235:495–511, 2003) 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 may depend on the approximation parameter.

We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rüssmann’s nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.

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.

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.

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.

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.

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].

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.

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.

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.

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.

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.