Uniqueness and continuous dependence of solutions to the incompressible micropolar ¯ows forward and backward in time S. Chirit ß a * Faculty of Mathematics, University of Ias ßi, 6600-Ias ßi, Romania Received 6 November 2000; accepted 12 January 2001 Abstract This paper studies the continuous dependence of the solutions for the boundary-initial and boundary- ®nal value problems associated with the incompressible micropolar ¯ows. For the incompressible micro- polar ¯ows forward in time, the continuous dependence of solutions with respect to the changes in the body force and body couple and in the initial data is established by means of a method based on a Gronwall-type inequality, while an adapted version of the logarithmic convexity method is used to study the continuous dependence of solutions for the incompressible micropolar ¯ows backward in time. As a direct conse- quence, some uniqueness results are obtained. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Micropolar ¯ows; Forward and backward in time problems; Continuous dependence 1. Introduction The theory of micropolar ¯uids is due to Eringen [1±5] whose approach allows for the presence of microstructure or particles in the ¯uid by additionally accounting for the motion of microel- ements or of particles. Such theories have been applied for modelling rheologically complex liquids such as blood and dilute polymeric suspensions. The mathematical model of micropolar ¯uids introduces a new kinematic variable called mi- crorotation describing the individual rotation of particles within the continuum, independent of the velocity ®eld. Micropolar ¯uids have been studied intensively in the literature. Various reviews of the subject are given by Ariman et al. [6,7], Eringen and Kafadar [8] and Brulin [9]. International Journal of Engineering Science 39 2001) 1787±1802 www.elsevier.com/locate/ijengsci * Tel.: +40-32-213-041; fax: +40-32-201-160. E-mail address: schirita@uaic.ro S. Chirit ß a). 0020-7225/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0020-722501)00029-5