Constructing a relativistic heat flow by transport time steps

An alternative construction to Andreu et al. (2005) [12] is given for L-w(1)([0, T], BV(Omega)) solutions to the relativistic heat equation (1) (see Brenier (2003) [14], Mihalas and Mihalas (1984) [37], Rosenau (1992) [40], Chertock et al. (2003) [20], Caselles (2007) [19]) under the assumption of initial data bounded from below and from above. For that purpose, we introduce a time discretized scheme in the style of Jordan et al. (1998) [30], Otto (1996) [38] involving an optimal transportation problem with a discontinuous hemispherical cost function. The limiting process is based on a monotonicity argument and on a bound of the Fisher information by an entropy balance characteristic of the minimization problem. (C) 2009 Elsevier Masson SAS. All rights reserved.