Bounds for some non-standard problems in porous flow and viscous Green–Naghdi fluids BY R. QUINTANILLA 1 AND B. STRAUGHAN 2 1 Departamento Matematica Aplicada 2, E.T.S. d’Enginyers Industrials de Terrassa, Universidad Politecnica de Catalunya, Colo ´n 11, Terrassa, 08222 Barcelona, Spain (ramon.quintanilla@upc.edu) 2 Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK (brian.straughan@durham.ac.uk) A. E. Green, F. R. S. and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time tZ0 and at a later time tZT. Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time. In addition, we derive similar energy bounds for a solution to the Brinkman–Forchheimer equations of viscous flow in porous media. Keywords: Green–Naghdi equations; dipolar fluids; Camassa–Holm equations; Brinkman–Forchheimer equations; non-standard problem 1. Introduction Green & Naghdi (1991, 1992, 1993, 1995a–d, 1996) developed an analysis in a rational way to produce fully consistent theories of thermoelasticity which incorporates thermal pulse transmission in a very logical manner, and nonlinear fluid behaviour. We believe that only extensive mathematical and physical analyses of the developments of Green & Naghdi will reveal the usefulness of their theories and it is to this goal that the present paper is addressed. In the papers cited above, Green & Naghdi developed a theory for describing the behaviour of a continuous body which relies on an entropy balance law rather than an entropy inequality. Their thermodynamics introduces a quantity T which is the ‘empirical’ temperature and a term z Z ð t t 0 T ðx; tÞdt C z 0 ; ð1:1Þ Proc. R. Soc. A (2005) 461, 3159–3168 doi:10.1098/rspa.2005.1508 Published online 23 August 2005 Received 14 January 2005 Accepted 26 April 2005 3159 q 2005 The Royal Society Downloaded from https://royalsocietypublishing.org/ on 29 January 2023