An asymptotic convergence result for a system of P.D.E.s with hysteresis
by
Michela Eleuteri
WIAS - Weierstrass Institute for Applied Analysis and Stochastics
Coauthors: Pavel Krejci (WIAS - Weierstrass Institute for Applied Analysis and Stochastics)

The aim of this talk is to present an asymptotic convergence result for the following system of partial differential equations

ì ï í ï î ∂ ∂t (a u + b w)- \triangle u = f w = F æ è u -g  ∂w ∂t ö ø        in W×(0, T),

(1)

where W is an open bounded set of R^N, N ≥ 1, F is a continuous rate independent invertible hysteresis operator, f is a given function, g, a and b are given positive constants.

This system can be obtained by coupling in a suitable way the Maxwell equations, the Ohm law and a constitutive relation between the magnetic field and the magnetic induction, provided we neglect the displacement current. The meaning of the parameter g is to take into account in the constitutive relation also a rate dependent component of the memory. The introduction of the parameter g regularizes the resulting P.D.E. system.

Our aim is to justify this regularization by proving that in the case , the solutions of converge as g→ 0 to the (unique) strong solution of the system

ì ï í ï î ∂ ∂t (a u + b w) -\triangle u = f w = F(u).

(2)

Therefore a rate independent problem can be derived as a limit of rate dependent problems.

Bibliography

[1] M. ELEUTERI, P. KREJCí: "An asymptotic convergence result for a system of partial differential equations with hysteresis", Communications on Pure and Applied Analysis, to appear.

Date received: June 14, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caum-97.