Constructing Continuous Solutions

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Vector Field ◽  
Small Parameter ◽  
Error Term ◽  
Main Lemma ◽  
Smooth Vector ◽  
Continuous Solutions ◽  
High Term ◽  
Smooth Vector Field ◽  
Divergence Equation ◽  
Energy Increment

This chapter demonstrates how the preceding construction, combined with a few estimates from Part V, can be used to prove the Main Lemma for continuous solutions. The first step is to mollify the velocity, followed by mollification of the stress. The lifespan is then chosen, preferring a small parameter to ensure that the first term in the parametrix for the High–High term is controlled. The chapter proceeds by discussing the bounds for the new stress and solving the divergence equation, along with the bounds for the corrections and finally, control of the energy increment. The equation for the energy increment includes a smooth vector field and involves bounding the error term.

The Divergence Equation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Vector Fields ◽  
Classical Mechanics ◽  
Main Lemma ◽  
Constant Vector ◽  
Smooth Vector ◽  
Continuous Solutions ◽  
All Solutions ◽  
Divergence Equation ◽  
Conservation Of Momentum ◽  
Special Solutions

This chapter introduces the divergence equation. A key ingredient in the proof of the Main Lemma for continuous solutions is to find special solutions to this divergence equation, which includes a smooth function and a smooth vector field on ³, plus an unknown, symmetric (2, 0) tensor. The chapter presents a proposition that takes into account a condition relating to the conservation of momentum as well as a condition that reflects Newton's law, which states that every action must have an equal and opposite reaction. This axiom, in turn, implies the conservation of momentum in classical mechanics. In view of Noether's theorem, the constant vector fields which act as Galilean symmetries of the Euler equation are responsible for the conservation of momentum. The chapter shows proof that all solutions to the Euler-Reynolds equations conserve momentum.

scholarly journals Tails of exit times from unstable equilibria on the line

2020 ◽  
Vol 57 (2) ◽  
pp. 477-496
Author(s):  
Yuri Bakhtin ◽  
Zsolt Pajor-Gyulai
Keyword(s):  
Vector Field ◽  
Malliavin Calculus ◽  
Elementary Proof ◽  
Exit Time ◽  
Exit Times ◽  
Smooth Vector ◽  
Small Noise ◽  
One Dimensional ◽  
Smooth Vector Field ◽  
Exit Distribution

AbstractFor a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits. We also discuss our program on rare transitions in noisy heteroclinic networks.

scholarly journals On Singular "Semifocus" Type Point Bifurcations of Piecewise Smooth Dynamical System

2018 ◽  
pp. 57-70
Author(s):  
V. Sh. Roitenberg
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Piecewise Smooth ◽  
Dimensional Manifold ◽  
Smooth Vector ◽  
Weak Focus ◽  
Smooth Vector Field ◽  
Small Parameters ◽  
Line Of Discontinuity ◽  
Piecewise Smooth Vector Field

For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

scholarly journals Geodesic Vector Fields on a Riemannian Manifold

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 137 ◽  
Author(s):  
Sharief Deshmukh ◽  
Patrik Peska ◽  
Nasser Bin Turki
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Vector Fields ◽  
Euclidean Spaces ◽  
Smooth Vector ◽  
Smooth Vector Field ◽  
Geodesic Vector ◽  
Integral Curves

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.

scholarly journals Birth of limit cycles from a 3D triangular center of a piecewise smooth vector field

2017 ◽  
Vol 82 (3) ◽  
pp. 561-578
Author(s):  
Tiago Carvalho ◽  
Rodrigo D. Euzébio ◽  
Marco Antonto Teixeira ◽  
Durval José Tonon
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Piecewise Smooth ◽  
Smooth Vector ◽  
Smooth Vector Field ◽  
Piecewise Smooth Vector Field

scholarly journals Tri-partitions and Bases of an Ordered Complex

2020 ◽  
Vol 64 (3) ◽  
pp. 759-775
Author(s):  
Herbert Edelsbrunner ◽  
Katharina Ölsböck
Keyword(s):  
Vector Field ◽  
Planar Graph ◽  
Betti Number ◽  
Hodge Decomposition ◽  
Reduction Algorithm ◽  
Smooth Vector ◽  
Polyhedral Complex ◽  
Canonical Bases ◽  
Smooth Vector Field ◽  
Connected Planar Graph

Abstract Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.

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