LINEAR TRANSPORT OF PARTICLES ON NETWORKS

1996 ◽  
Vol 06 (02) ◽  
pp. 279-294 ◽  
Author(s):  
LUIGI BARLETTI
Keyword(s):  
Boundary Conditions ◽  
Initial Value Problem ◽  
Connected Graph ◽  
Well Posedness ◽  
Linear Boundary ◽  
Initial Value ◽  
One Dimensional ◽  
Linear Transport ◽  
Streaming Transport

We introduce a model of transport of particles in a network, which is represented by a connected graph with m vertices and n edges. Each edge represents a one-dimensional conductor of particles, whose behavior is described by means of a linear Boltzmann-like equation. In graph vertices, a system of linear boundary conditions is given which takes into account the exchanges of particles between the edges. The well-posedness of the initial value problem is studied into an abstract L1-like setting and the structure of the solution is given for simplest case of pure streaming transport.

scholarly journals AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340015 ◽  
Author(s):  
DAVID HILDITCH
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Initial Value ◽  
Initial Boundary Value

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

scholarly journals ENERGY CRITICAL NLS IN TWO SPACE DIMENSIONS

2009 ◽  
Vol 06 (03) ◽  
pp. 549-575 ◽  
Author(s):  
J. COLLIANDER ◽  
S. IBRAHIM ◽  
M. MAJDOUB ◽  
N. MASMOUDI
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Energy Space ◽  
Well Posedness ◽  
Initial Value ◽  
Exponential Nonlinearity ◽  
Supercritical Case ◽  
Two Space Dimensions

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with exponential nonlinearity [Formula: see text] We identify subcritical, critical, and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.

scholarly journals Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

Universe ◽  
2019 ◽  
Vol 5 (12) ◽  
pp. 224 ◽  
Author(s):  
Stefano Vignolo
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Junction Conditions ◽  
Field Equations ◽  
Initial Value ◽  
The Cauchy Problem ◽  
General Conditions

We discuss the Cauchy problem and the junction conditions within the framework of f ( R ) -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable f ( R ) -models with torsion.

The non-autonomous wave equation with general Wentzell boundary conditions

2005 ◽  
Vol 135 (2) ◽  
pp. 317-329 ◽  
Author(s):  
Angelo Favini ◽  
Ciprian G. Gal ◽  
Gisèle Ruiz Goldstein ◽  
Jerome A. Goldstein ◽  
Silvia Romanelli
Keyword(s):  
Hilbert Space ◽  
Wave Equation ◽  
Boundary Conditions ◽  
Abstract Cauchy Problem ◽  
Regularity Conditions ◽  
Well Posedness ◽  
One Dimensional ◽  
Wentzell Boundary Conditions ◽  
Image Position ◽  
Dimensional Wave

We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation utt = A(t)u with general Wentzell boundary conditions Here A(t)u := (a(x, t)ux)x, a(x, t) ≥ ε > 0 in [0, 1] × [0, + ∞) and βj(t) > 0, γj(t) ≥ 0, (γ0(t), γ1(t)) ≠ (0,0). Under suitable regularity conditions on a, βj, γj we prove the well-posedness in a suitable (energy) Hilbert space

Sign in / Sign up

Export Citation Format

Share Document