Continuous Dependence on the Initial-Time Geometry in Generalized Heat Conduction

1997 ◽  
Vol 07 (01) ◽  
pp. 125-138 ◽  
Author(s):  
L. E. Payne ◽  
J. C. Song
Keyword(s):  
Heat Conduction ◽  
Initial Data ◽  
Continuous Dependence ◽  
Conduction System ◽  
Initial Time ◽  
Generalized Heat Conduction

In this paper we investigate continuous dependence on the initial-time geometry for solutions of a generalized heat conduction system. Assuming the initial data to be measured on a surface t = εF(x), for |F| < 1, and assigned at t = 0, we examine the effects of this error in the initial-time geometry on the solution both forward and backward in time.

scholarly journals Dependence on the Initial Data for the Continuous Thermostatted Framework

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 602
Author(s):  
Bruno Carbonaro ◽  
Marco Menale
Keyword(s):  
Complex System ◽  
Initial Data ◽  
External Force ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Complete Information ◽  
Complete Picture ◽  
Order Moment ◽  
Lack Of Information ◽  
Activity Variable

The paper deals with the problem of continuous dependence on initial data of solutions to the equation describing the evolution of a complex system in the presence of an external force acting on the system and of a thermostat, simply identified with the condition that the second order moment of the activity variable (see Section 1) is a constant. We are able to prove that these solutions are stable with respect to the initial conditions in the Hadamard’s sense. In this connection, two remarks spontaneously arise and must be carefully considered: first, one could complain the lack of information about the “distance” between solutions at any time t ∈ [ 0 , + ∞ ) ; next, one cannot expect any more complete information without taking into account the possible distribution of the transition probabiliy densities and the interaction rates (see Section 1 again). This work must be viewed as a first step of a research which will require many more steps to give a sufficiently complete picture of the relations between solutions (see Section 5).

scholarly journals Macroscopic limit of the Becker–Döring equation via gradient flows

2019 ◽  
Vol 25 ◽  
pp. 22 ◽  
Author(s):  
André Schlichting
Keyword(s):  
Initial Data ◽  
Time Scale ◽  
Continuous Dependence ◽  
Monomer Concentration ◽  
Small Cluster ◽  
Gradient Structure ◽  
Gradient Flows ◽  
Cluster Distribution ◽  
Wagner Equation

This work considers gradient structures for the Becker–Döring equation and its macroscopic limits. The result of Niethammer [J. Nonlinear Sci. 13 (2003) 115–122] is extended to prove the convergence not only for solutions of the Becker–Döring equation towards the Lifshitz–Slyozov–Wagner equation of coarsening, but also the convergence of the associated gradient structures. We establish the gradient structure of the nonlocal coarsening equation rigorously and show continuous dependence on the initial data within this framework. Further, on the considered time scale the small cluster distribution of the Becker–Döring equation follows a quasistationary distribution dictated by the monomer concentration.

Boundary flow for the minimal surfaces in Rn with Plateau boundary condition

2005 ◽  
Vol 135 (3) ◽  
pp. 537-562
Author(s):  
Kung-Ching Chang ◽  
Jia-Quan Liu
Keyword(s):  
Boundary Condition ◽  
Initial Data ◽  
Global Existence ◽  
Minimal Surfaces ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Boundary Flow ◽  
Global Existence And Uniqueness

We introduce the notion of the boundary flow for minimal surfaces in Rn with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence on the initial data.

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