Dependence on the Initial Data for the Continuous Thermostatted Framework

Bruno Carbonaro and Marco Menale
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Bruno Carbonaro: Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy
Marco Menale: Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy

Mathematics, 2019, vol. 7, issue 7, 1-11

Abstract: 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).

Keywords: kinetic theory; integro-differential equations; complex systems; stability; evolution equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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