Hamilton-Jacobi Type Equation Research Articles

Solution of a Parabolic Hamilton–Jacobi Type Equation Determined by a Simple Boundary Singularity

Solution of a Parabolic Hamilton–Jacobi Type Equation Determined by a Simple Boundary Singularity

Group analysis of a Hamilton–Jacobi type equation

In this paper, we apply Lie group theory to a Hamilton-Jacobi type equation, which is a nonlinear partial differential equation of the first order. We then employ the local inverse theorem to build the Lie invariance condition for this equation. The determining equations are then obtained by using this condition. We split and solve these obtained equations to obtain the symmetries of the equation under study. We obtain the largest solvable Lie algebra. We also obtain the symmetry algebra and make a group classification of this equation. Further, some exact solutions are deduced and represented graphically.

A Method to Solve Hamilton–Jacobi Type Equation on Unstructured Meshes

A new method is developed to approximate a first-order Hamilton–Jacobi equation. The constant motion of an interface in the normal direction is of interest. The interface is captured with the help of a “Level-Set” function approximated through a finite-volume Godunov-type scheme. Contrarily to most computational approaches that consider smooth Level-Set functions, the present one considers sharp “Level-Set”, the numerical diffusion being controlled with the help of the Overbee limiter (Chiapolino et al. in J Comput Phys 340:389–417, 2017). The method requires gradient computation that is addressed through the least squares approximation. Multidimensional results on fixed unstructured meshes are provided and checked against analytical solutions. Geometrical properties such as interfacial area and volume computation are addressed as well. Results show excellent agreement with the exact solutions.

A Level Set Method for the Dirichlet k-Partition Problem

We propose a simple level set method for the Dirichlet k-partition problem which aims to partition an open domain into K different subdomains as to minimize the sum of the smallest eigenvalue of the Dirichlet Laplace operator in each subdomain. We first formulate the problem as a nested minimization problem of a functional of the level set function and the eigenfunction defined in each subdomain. As an approximation, we propose to simply replace the eigenfunction by the level set function so that the nested minimization can then be converted to a single minimization problem. We apply the standard gradient descent method so that the problem leads to a Hamilton–Jacobi type equation. Various numerical examples will be given to demonstrate the effectiveness of our proposed method.

A multiple level-set approach for modelling containerless freezing process

We propose a multiple level-set model to represent the physics governing the three-phase solidification problem. The model couples thermal characteristics of the freezing front with the dynamics of droplet interface. To deal with liquid, solid, and gas phases, two distinct level-sets are used. The liquid-gas interface which is represented by a level-set moves with an external velocity field that is obtained from the Navier-Stokes equations. The solid-liquid interface, on the other hand, evolves according to the freezing rate of the liquid. The solid-liquid level-set is comprised of an active and passive part. The active segment of the level-set evolves based on temperature gradients and the latent heat of fusion. The passive segment, however, is merely utilized to impose an angle at the tri-junction point. We propose a Hamilton-Jacobi type equation to impose constant or variable angles at the tri-junction point. In order to consider the effect of volume expansion we modify the continuity, and the energy equation. Importantly, in the case of expansion during solidification we can capture the pointy shape on top of the freezing droplet. For validation we compare numerical results with the analytical Stefan problem with and without the density expansion. In addition, we use experimental results of water droplet freezing, available in the literature, to examine the accuracy of the freezing rate, and the droplet morphological.

The localized radial basis functions for parameterized level set based structural optimization

The current work is mainly about the implementation of the local radial basis functions (LRBFs) within the level set framework for structural optimization of two-dimensional linear elastic problems. The implicit representation of the structural geometry is accomplished through the LRBFs based level set function (LSF), and the geometry modification during the optimization process is carried out through an update of the LSF using a system of coupled ordinary differential equations (ODEs) instead of the Hamilton-Jacobi (HJ) type equation. This new implementation of LRBFs and level set method (LSM) allows essential topological changes automatically, i.e hole nucleation, hole merging with each other, and with the boundary to obtain the optimal structures. The coefficient matrix of the LRBFs is less sensitive to the values of shape parameter as well as the proposed method is mesh independent as explored during the numerical experiments performed. This new implementation is applied to distinct benchmark problems for minimum compliance with single and multiple load cases. The numerical experiments performed reveal significant insightful of the use of radial basis functions (RBFs) within the LSM framework, which may be considered as a step forward for further exploration of their intrinsic capabilities and improved performance for the solution of structural optimization problems.

Weakly nonlinear surface waves in magnetohydrodynamics. I

This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the system of ideal incompressible magnetohydrodynamics. Current vortex sheets are piecewise smooth solutions that satisfy suitable jump conditions on the (free) discontinuity surface. In this article, we complete an earlier work by Alì and Hunter (Quart. Appl. Math. 61(3) (2003) 451–474) and construct approximate solutions at any arbitrarily large order of accuracy to the three-dimensional free boundary problem when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to ‘surface waves’ on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime which we consider here, their leading amplitude is governed by a nonlocal Hamilton–Jacobi type equation known as the ‘HIZ equation’ (standing for Hamilton–Il’insky–Zabolotskaya (J. Acoust. Soc. Amer. 97(2) (1995) 891–897)) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. We exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on mere algebra and arguments of combinatorial analysis, namely a Leibniz type formula which we have not been able to find in the literature. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the current vortex sheet equations. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves (C. R. Math. Acad. Sci. Paris 349(23–24) (2011) 1239–1244) does not arise in the same way for the current vortex sheet problem.

On the Coincidence of the Minimax Solution and the Value Function in a Time-Optimal Game with a Lifeline

We consider time-optimal differential games with a lifeline. In such games, as usual, there is a terminal set to which the first player tries to guide the system as fast as possible, and there is also a set, called a lifeline, such that the second player wins when the system attains this set. The payoff is the result of applying Kruzhkov’s variable change to the time when the system reaches the terminal set. We also consider Hamilton-Jacobi equations corresponding to such games. The existence of a minimax solution of a boundary value problem for a Hamilton-Jacobi type equation is proved. For this, we introduce certain strong assumptions on the dynamics of the game near the boundary of the game domain. More exactly, the first and second players can direct the motion of the system to the terminal set and the lifeline, respectively, if the system is near the corresponding set. Under these assumptions, the value function is continuous in the game domain. The coincidence of the value function and the minimax solution of the boundary value problem is proved under the same assumptions.

L2-contraction of large planar shock waves for multi-dimensional scalar viscous conservation laws

We consider a L2-contraction (a L2-type stability) of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift by using the relative entropy methods. Quite different from the previous results, we find a new way to determine the shift function, which depends both on the time and space variables and solves a viscous Hamilton-Jacobi type equation with source terms. Moreover, we do not impose any conditions on the anti-derivative variables of the perturbation around the shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in L2-norm, then the L2-contraction holds true for the viscous shock wave up to a suitable shift function. Note that BV-norm or the L∞-norm of the initial perturbation and the shock wave strength can be arbitrarily large. Furthermore, as the time t tends to infinity, the L2-contraction holds true up to a (spatially homogeneous) time-dependent shift function. In particular, if we choose some special initial perturbations, then L2-convergence of the solutions towards the associated shock profile can be proved up to a time-dependent shift.

On Basic Equation of Differential Games for Neutral-Type Systems

For a conflict-controlled dynamical system described by neutral-type functional differential equations in Hale’s form, a differential game is considered in the classes of control strategies with a guide for a minimax-maximin of the quality index, which evaluates the system’s motion history implemented by the terminal time moment. The differential game is associated with the Cauchy problem for a functional Hamilton-Jacobi type equation in coinvariant derivatives. It has been proven that the game value functional coincides with the minimax solution of this problem. A method of constructing the optimal strategies of players is given. The approximation by ordinary Hamilton-Jacobi equations in partial derivatives is proposed for this functional Hamilton-Jacobi equation in coinvariant derivatives.

On Hamilton-Jacobi-Bellman-Isaacs Equation for Time-Delay Systems

The paper deals with a two-person zero-sum differential game for a dynamical system which motion is described by a delay differential equation under an initial condition defined by a piecewise continuous function. For the value functional of this game, we derive the Hamilton-Jacobi type equation with coinvariant derivatives. It is proved that, if the solution of this equation satisfies certain smoothness conditions, then it coincides with the value functional. On the other hand, it is proved that, at the points of coinvariant differentiability, the value functional satisfies the derived Hamilton-Jacobi equation. Therefore, this equation can be called the Hamilton-Jacobi-Bellman-Isaacs equation for time-delay systems.

Construction of Singular Sets in a Velocity Control Problem with Nonconvex Target

Abstract The problem of occurrence of singular sets of solution of the velocity control problem for one class of dynamic problems on a plane with a nonconvex target set is studied. The theoretical apparatus is developed for determining pseudo-vertices of a target set in case when its boundary is a curve with a minimum order of smoothness. Finding pseudo-vertices is a necessary element of the procedure for constructing branches of the singular set of the optimal result function. Necessary conditions for the existence of pseudo-vertices, expressed in terms of one-sided partial limits of differential relations dependent on properties of local diffeomorphisms that determine these singular points, are obtained. Examples of constructing a nonsmooth solution to the velocity control problem are given. The developed procedures for determining the disturbance of smoothness of the solution to the dynamic control problem are also applicable when constructing generalized solutions of Hamilton-Jacobi type equations, as well as when forming a generalized eikonal in geometrical optics.

Experiments on Frequency Dependence of the Deflection of Light in Yang-Mills Gravity

In Yang-Mills gravity based on flat space-time, the eikonal equation for a light ray is derived from the modified Maxwell’s wave equations in the geometric-optics limit. One obtains a Hamilton-Jacobi type equation, GLµv∂µΨ∂vΨ = 0 with an effective Riemannian metric tensor GLµv. According to Yang-Mills gravity, light rays (and macroscopic objects) move as if they were in an effective curved space-time with a metric tensor. The deflection angle of a light ray by the sun is about 1.53″ for experiments with optical frequencies ≈ 1014Hz. It is roughly 12% smaller than the usual value 1.75″. However, the experimental data in the past 100 years for the deflection of light by the sun in optical frequencies have uncertainties of (10-20)% due to large systematic errors. If one does not take the geometric-optics limit, one has the equation, GLµv[∂µΨ∂vΨcosΨ+ (∂µ∂vΨ)sinΨ] = 0, which suggests that the deflection angle could be frequency-dependent, according to Yang-Mills gravity. Nowadays, one has very accurate data in the radio frequencies ≈ 109Hz with uncertainties less than 0.1%. Thus, one can test this suggestion by using frequencies ≈ 1012 Hz, which could have a small uncertainty 0.1% due to the absence of systematic errors in the very long baseline interferometry.

On Hamilton-Jacobi equations for neutral-type differential games

We consider a two-person zero-sum differential game in which a motion of the dynamical system is described by neutral-type functional-differential equations in Hale’s form and the quality index estimates a motion history realized up to the terminal instant of time and includes integral estimations of control realizations of the players. The formalization of the game in the class of pure positional strategies is given, the corresponding notions of the value functional and optimal control strategies of the players are defined. For the value functional, we derive a Hamilton-Jacobi type equation with coinvariant derivatives. It is proved that, if a solution of this equation satisfies certain smoothness conditions, then it coincides with the value functional. On the other hand, it is proved that, at the points of coinvariant differentiability, the value functional satisfies the derived Hamilton-Jacobi equation. Therefore, this equation can be called the Hamilton-Jacobi-Bellman-Isaacs equation for neutral-type systems.

Guest Editorial: Shape Analysis Beyond the Eikonal Equation

“Shape Analysis” is an interesting research area which contains many thought-provoking problems and mathematical challenges, especially at the representation level. Shape of an object is considered as the enclosing surface (the boundary) or the enclosed interior. One of the major challenges is to define representations of “shapes” that can be manipulated by image processing methods. This is an important need in practice because more often than not shapes need to be extracted or recovered from raw images; moreover, the process of recovering shapes from raw images highly benefits from prior knowledge of candidate shapes. One of the key means of shape representation is the Euclidean Distance Transform (EDT), which is governed by the Eikonal Equation. The values of the EDT at each interior point give the distance to the nearest boundary point. This distance notion is used in almost any problem that requires integrating shape and image information; e.g., a rich variety of shape guided image-processing or shape extraction tasks. Especially in the 1990s, the idea of embedding shapes as zeros of the signed EDT is highly popularized by the level-set framework and its fast implementation, leading to a booming research area in implicit shape representations. However, as challenging problems are developed, formulations that move beyond the classical Eikonal Equation emerged. This special issue consists of six papers each exploring an interesting direction moving beyond the Eikonal formulation in the field of Shape Analysis. The first two papers, “Analysis and Approximation of some Shape–from–Shading Models for non-Lambertian Surfaces” and “A Unified Variational Volume Registration Method based on Automatically Learned Brain Structures”, deal with two key applications that jointly involve shapes and images. In the first paper, Silvia Tozza and Maurizio Falcone present a unified mathematical formulation of some non-Lambertian models, which lead to more complex stationary nonlinear partial differential equations of Hamilton-Jacobi type which can be regarded as a generalization of the classical Eikonal Equation corresponding to the Lambertian case. In the second paper, Carl Lederman et al. present an active volume technique employing level sets of the distance function. Unlike the common practice, the level sets in their formulation are only utilized on their respective regions and not on the whole domain, making the computations feasible as the number of regions increases. The next four papers propose alternative distance notions or computations. They are briefly discussed as follows. In “Affine Invariant Distance Using Multiscale Analysis”, Luis Alvarez et al. present a distance transform for planar shapes that is invariant under any planar projection of the

Euclidean signature semi-classical methods for bosonic field theories: interacting scalar fields

Elegant 'microlocal' methods have long since been extensively developed for the analysis of conventional Schroedinger eigenvalue problems. For technical reasons though these methods have not heretofore been applicable to quantum field theories. In this article however we initiate a 'Euclidean signature semi-classical' program to extend the scope of these analytical techniques to encompass the study of self-interacting scalar fields in 1 + 1, 2 + 1 and 3 + 1 dimensions. The basic microlocal approach entails, first of all, the solution of a single, nonlinear equation of Hamilton-Jacobi type followed by the integration (for both ground and excited states) of a sequence of linear 'transport' equations along the 'flow' generated by the 'fundamental solution' to the aforementioned Hamilton-Jacobi equation. Using a combination of the direct method of the calculus of variations, elliptic regularity theory and the Banach space version of the implicit function theorem we establish, in a suitable function space setting, the existence, uniqueness and global regularity of this needed 'fundamental solution' to the relevant, Euclidean signature Hamilton-Jacobi equation for the systems under study. Our methods are applicable to (massive) scalar fields with polynomial self-interactions of renormalizable type. They can, as we shall show elsewhere, also be applied to Yang-Mills fields in 2 + 1 and 3 + 1 dimensions.

Exact Space-Time Gauge Symmetry of Gravity, Its Couplings and Approximate Internal Symmetries in a Total-Unified Model

The gravitational field is the manifestation of space-time translational (T_4) gauge symmetry, which enables the gravitational interaction to be unified with the strong and the electroweak interactions. Such a total-unified model is based on a generalized Yang-Mills framework in flat space-time. Following the idea of Glashow-Salam-Ward-Weinberg, we gauge the groups T_4 × (SU_3)_(color) × SU_2 × U_1 × U_(1b) on equal-footing, so that we have the total-unified gauge covariant derivative δ_μ = ∂μ - igϕ _μ^νpv + ig_sG _μ^a (λ^a/2) + ifW_μ^kt^k + if'U_μt_o + ig_bB_μ. The generators of the external T_4 group have the representation p_μ = i∂_μ, which differs from the other generators of all the other internal groups, which have constant matrix representations. Consequently, the total-unified model leads to the following new results: (a) All internal (SU_3)_(color), SU_2, U_1, and baryonic U_(1b) gauge symmetries have extremely small violations due to the gravitational interaction. (b) The T_4 gauge symmetry remains exact and dictates the universal coupling of gravitons. (c) Such a gravitational violation of internal gauge symmetries leads to modified eikonal and Hamilton-Jacobi type equations, which are obtained in the geometric-optics limit and involve effective Riemann metric tensors. (d) The rules for Feynman diagrams involving new couplings of the photon-graviton, gluon-graviton, and quark-graviton are obtained.

On the blow-up of the solution of an equation related to the Hamilton-Jacobi equation

A new model three-dimensional third-order equation of Hamilton-Jacobi type is derived. For this equation, the initial boundary-value problem in a bounded domain with smooth boundary is studied and local solvability in the strong generalized sense is proved; in addition, sufficient conditions for the blow-up in finite time and sufficient conditions for global (in time) solvability are obtained.

Periodic homogenization of G-equations and viscosity effects

G-equations are well-known front propagation models in combustion and are Hamilton–Jacobi type equations with convex but non-coercive Hamiltonians. Viscous G-equations arise from numerical discretization or modeling dissipative mechanisms. Although viscosity helps to overcome non-coercivity, we prove homogenization of an inviscid G-equation based on approximate correctors and attainability of controlled flow trajectories. We verify the attainability for two-dimensional mean zero incompressible flows, and demonstrate asymptotically and numerically that viscosity reduces the homogenized Hamiltonian in cellular flows. In the case of one-dimensional compressible flows, we found an explicit formula of homogenized Hamiltonians, as well as necessary and sufficient conditions for wave trapping (effective Hamiltonian vanishes identically). Viscosity restores coercivity and wave propagation.

Minimax and viscosity solutions in optimization problems for hereditary systems

For a dynamical system with discrete and distributed time delays, a control problem under disturbance or counteraction is considered. The problem is formalized in the context of the game-theoretical approach in the class of control strategies with memory. The problem is associated with a functional Hamilton-Jacobi type equation with coinvariant derivatives. The minimax and viscosity approaches to a generalized solution to this equation are discussed. It is shown that, under the same condition at the right endpoint, the minimax and viscosity solutions coincide, thereby uniquely defining the functional of optimal guaranteed result in the control problem.