scholarly journals Global propagation of singularities for discounted Hamilton-Jacobi equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Cui Chen ◽  
Jiahui Hong ◽  
Kai Zhao
Keyword(s):  
Viscosity Solution ◽  
Jacobi Equation ◽  
Technical Difficulty ◽  
Singular Locus ◽  
Locally Lipschitz ◽  
Propagation Of Singularities ◽  
Aubry Set ◽  
Fixed Constant ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

<p style='text-indent:20px;'>The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE333"> \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with fixed constant <inline-formula><tex-math id="M1">\begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}</tex-math></inline-formula>. We reduce the problem for equation <inline-formula><tex-math id="M2">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of <inline-formula><tex-math id="M3">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> propagate along locally Lipschitz singular characteristics <inline-formula><tex-math id="M4">\begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> can extend to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>. Essentially, we use <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-compactness of the Euclidean space which is different from the original construction in [<xref ref-type="bibr" rid="b4">4</xref>]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of <inline-formula><tex-math id="M8">\begin{document}$ u $\end{document}</tex-math></inline-formula> and the complement of Aubry set using the basic idea from [<xref ref-type="bibr" rid="b9">9</xref>].</p>

scholarly journals Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Qinbo Chen
Keyword(s):  
Contact Problem ◽  
Viscosity Solution ◽  
Unknown Function ◽  
Jacobi Equation ◽  
Specific Solution ◽  
Convergence Of Solutions ◽  
Limit Solution ◽  
Peierls Barrier ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

Abstract Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Let H ⁢ ( x , p , u ) {H(x,p,u)} be a continuous Hamiltonian which is strictly increasing in u, and is convex and coercive in p. For each parameter λ > 0 {\lambda>0} , we denote by u λ {u^{\lambda}} the unique viscosity solution of the Hamilton–Jacobi equation H ⁢ ( x , D ⁢ u ⁢ ( x ) , λ ⁢ u ⁢ ( x ) ) = c . H\big{(}x,Du(x),\lambda u(x)\big{)}=c. Under quite general assumptions, we prove that u λ {u^{\lambda}} converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equation H ⁢ ( x , D ⁢ u ⁢ ( x ) , 0 ) = c {H(x,Du(x),0)=c} . We also characterize the limit solution in terms of Peierls barrier and Mather measures.

scholarly journals Particle dynamics inside shocks in Hamilton–Jacobi equations

2010 ◽  
Vol 368 (1916) ◽  
pp. 1579-1593 ◽  
Author(s):  
Konstantin Khanin ◽  
Andrei Sobolevski
Keyword(s):  
Velocity Field ◽  
Jacobi Equation ◽  
Particle Dynamics ◽  
Velocity Fields ◽  
Vanishing Viscosity ◽  
Effective Velocity ◽  
Lagrangian Action ◽  
The Moment ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

The characteristic curves of a Hamilton–Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth ‘viscosity’ solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the ‘dissipative anomaly’ in the limit of vanishing viscosity.

scholarly journals Averaging of the Hamilton-Jacobi equation in infinite dimensions and an application

2000 ◽  
Vol 41 (3) ◽  
pp. 372-385
Author(s):  
Shihong Wang ◽  
Zuoyi Zhou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Viscosity Solution ◽  
Jacobi Equation ◽  
Value Function ◽  
Limit Problem ◽  
Infinite Dimensions ◽  
The Value Function ◽  
Hamilton Jacobi Equation

AbstractWe study the averaging of the Hamilton-Jacobi equation with fast variables in the viscosity solution sense in infinite dimensions. We prove that the viscosity solution of the original equation converges to the viscosity solution of the averaged equation and apply this result to the limit problem of the value function for an optimal control problem with fast variables.

The Hamilton-Jacobi Equations for a Relativistic Charged Particle

1963 ◽  
Vol 6 (3) ◽  
pp. 341-350 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Classical Particle ◽  
Quantum Mechanical ◽  
Order Partial Differential Equation ◽  
First Order ◽  
Relativistic Charged Particle ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

scholarly journals Instability of the Dirichlet problem for Hamilton-Jacobi equation

1997 ◽  
Vol 55 (2) ◽  
pp. 311-319
Author(s):  
Kewei Zhang
Keyword(s):  
Dirichlet Problem ◽  
Jacobi Equation ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation ◽  
General Conditions

We show the instability of solutions of the Dirichlet problem for Hamilton-Jacobi equations under quite general conditions.

scholarly journals THE MATHER MEASURE AND A LARGE DEVIATION PRINCIPLE FOR THE ENTROPY PENALIZED METHOD

2011 ◽  
Vol 13 (02) ◽  
pp. 235-268 ◽  
Author(s):  
D. A. GOMES ◽  
A. O. LOPES ◽  
J. MOHR
Keyword(s):  
Jacobi Equation ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Deviation Function ◽  
Deviation Principle ◽  
Probability Densities ◽  
Aubry Set ◽  
Dynamical Properties ◽  
Hamilton Jacobi Equation

We present the rate function and a large deviation principle for the entropy penalized Mather problem when the Lagrangian is generic (it is known that in this case the Mather measure μ is unique and the support of μ is the Aubry set). We assume the Lagrangian L(x, v), with x in the torus 𝕋N and v∈ℝN, satisfies certain natural hypotheses, such as superlinearity and convexity in v, as well as some technical estimates. Consider, for each value of ϵ and h, the entropy penalized Mather problem [Formula: see text] where the entropy S is given by [Formula: see text], and the minimization is performed over the space of probability densities μ(x, v) on 𝕋N×ℝN that satisfy the discrete holonomy constraint ∫𝕋N×ℝN φ(x + hv) - φ(x) dμ = 0. It is known [17] that there exists a unique minimizing measure μϵ, h which converges to a Mather measure μ, as ϵ, h→0. In the case in which the Mather measure μ is unique we prove a Large Deviation Principle for the limit lim ϵ, h→0ϵ ln μϵ, h(A), where A ⊂ 𝕋N×ℝN. In particular, we prove that the deviation function I can be written as [Formula: see text], where ϕ0 is the unique viscosity solution of the Hamilton – Jacobi equation, [Formula: see text]. We also prove a large deviation principle for the limit ϵ→ 0 with fixed h. Finally, in the last section, we study some dynamical properties of the discrete time Aubry–Mather problem, and present a proof of the existence of a separating subaction.

A METRIC APPROACH TO PLASTICITY VIA HAMILTON–JACOBI EQUATION

2010 ◽  
Vol 20 (09) ◽  
pp. 1617-1647
Author(s):  
FERDINANDO AURICCHIO ◽  
ELENA BONETTI ◽  
ANTONIO MARIGONDA
Keyword(s):  
Jacobi Equation ◽  
Constitutive Relations ◽  
Yield Function ◽  
Generalized Gradients ◽  
Jacobi Equations ◽  
Thermodynamical Consistency ◽  
Hamilton Jacobi Equation ◽  
New Framework

Thermodynamical consistency of plasticity models is usually written in terms of the so-called "maximum dissipation principle". In this paper, we discuss constitutive relations for dissipative materials written through suitable generalized gradients of a (possibly non-convex) metric. This new framework allows us to generalize the classical results providing an interpretation of the yield function in terms of Hamilton–Jacobi equations theory.

scholarly journals Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Albert Fathi
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Jacobi Equation ◽  
Compact Manifold ◽  
Complete Riemannian Manifold ◽  
Time Dependent ◽  
Jacobi Equations ◽  
Uniformly Continuous ◽  
Hamilton Jacobi Equation

AbstractIf $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

scholarly journals Local singular characteristics on $$\mathbb {R}^2$$

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng
Keyword(s):  
Jacobi Equation ◽  
Differential Inclusions ◽  
Space Dimension ◽  
Uniqueness Result ◽  
Singular Set ◽  
Uniqueness Criterion ◽  
The Moment ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation ◽  
Singular Characteristics

AbstractThe singular set of a viscosity solution to a Hamilton–Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on $$\mathbb {R}^2$$ R 2 , two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861–885, 2016].

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