Objective The main goal of the present research proposal is to build up a general mathematical theory describing the motion of a compressible, viscous, and heat conductive fluid. Our approach is based on the concept of generalized (weak) solutions satisfying the basic physical principles of balance of mass, momentum, and energy. The energy balance is expressed in terms of a variant of entropy inequality supplemented with an integral identity for the total energy balance.We propose to identify a class of suitable weak solutions, where admissibility is based on a direct application of the principle of maximal entropy production compatible with Second law of thermodynamics. Stability of the solution family will be investigated by the method of relative entropies constructed on the basis of certain thermodynamics potentials as ballistic free energy.The new solution framework will be applied to multiscale problems, where several characteristic scales become small or extremely large. We focus on mutual interaction of scales during this process and identify the asymptotic behavior of the quantities that are filtered out in the singular limits. We also propose to study the influence of the geometry of the underlying physical space that may change in the course of the limit process. In particular, problems arising in homogenization and optimal shape design in combination with various singular limits are taken into account.The abstract approximate scheme used in the existence theory will be adapted in order to develop adequate numerical methods. We study stability and convergence of these methods using the tools developed in the abstract part, in particular, the relative entropies. Fields of science natural sciencesphysical sciencesthermodynamicsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicsapplied mathematicsnumerical analysissocial scienceslaw Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Topic(s) ERC-AG-PE1 - ERC Advanced Grant - Mathematical foundations Call for proposal ERC-2012-ADG_20120216 See other projects for this call Funding Scheme ERC-AG - ERC Advanced Grant Host institution MATEMATICKY USTAV AV CR V.V.I. EU contribution € 726 320,00 Address ZITNA 609/25 115 67 Praha Czechia See on map Region Česko Praha Hlavní město Praha Activity type Other Principal investigator Eduard Feireisl (Prof.) Administrative Contact Jiří Rákosník (Dr.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data Beneficiaries (1) Sort alphabetically Sort by EU Contribution Expand all Collapse all MATEMATICKY USTAV AV CR V.V.I. Czechia EU contribution € 726 320,00 Address ZITNA 609/25 115 67 Praha See on map Region Česko Praha Hlavní město Praha Activity type Other Principal investigator Eduard Feireisl (Prof.) Administrative Contact Jiří Rákosník (Dr.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data
