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 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].

scholarly journals The spacetime models with dust matter that admit separation of variables in Hamilton–Jacobi equations of a test particle

2016 ◽  
Vol 31 (06) ◽  
pp. 1650027 ◽  
Author(s):  
Konstantin Osetrin ◽  
Altair Filippov ◽  
Evgeny Osetrin
Keyword(s):  
Velocity Field ◽  
Energy Density ◽  
Test Particle ◽  
Jacobi Equation ◽  
Functional Form ◽  
Separation Of Variables ◽  
Complete Separation ◽  
Coordinate Systems ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

The characteristics of dust matter in spacetime models, admitting the existence of privilege coordinate systems are given, where the single-particle Hamilton–Jacobi equation can be integrated by the method of complete separation of variables. The resulting functional form of the 4-velocity field and energy density of matter for all types of spaces under consideration is presented.

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 Brownian fluctuations of flame fronts with small random advection

2020 ◽  
Vol 30 (07) ◽  
pp. 1375-1406
Author(s):  
Christopher Henderson ◽  
Panagiotis E. Souganidis
Keyword(s):  
Velocity Field ◽  
White Noise ◽  
Turbulent Combustion ◽  
Jacobi Equation ◽  
Hamilton Jacobi Equation

We study the effect of small random advection in two models in turbulent combustion. Assuming that the velocity field decorrelates sufficiently fast, we (i) identify the order of the fluctuations of the front with respect to the size of the advection; and (ii) characterize them by the solution of a Hamilton–Jacobi equation forced by white noise. In the simplest case, the result yields, for both models, a front with Brownian fluctuations of the same scale as the size of the advection. That the fluctuations are the same for both models is somewhat surprising, in view of known differences between the two models.

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 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.

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