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Open AccessArticle

Dependence on the Initial Data for the Continuous Thermostatted Framework

by Bruno Carbonaro 1 and Marco Menale 1,2,*
1
Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy
2
Laboratoire Quartz EA 7393, École Supérieure d’Ingénieurs en Génie Électrique, Productique et Management Industriel, 95092 Cergy Pontoise Cedex, France
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(7), 602; https://doi.org/10.3390/math7070602
Received: 6 June 2019 / Revised: 2 July 2019 / Accepted: 4 July 2019 / Published: 6 July 2019
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). View Full-Text
Keywords: kinetic theory; integro-differential equations; complex systems; stability; evolution equations kinetic theory; integro-differential equations; complex systems; stability; evolution equations
MDPI and ACS Style

Carbonaro, B.; Menale, M. Dependence on the Initial Data for the Continuous Thermostatted Framework. Mathematics 2019, 7, 602.

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