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.

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 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.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

Sign in / Sign up

Export Citation Format

Share Document