In this thesis we propose a mathematical framework modeling non-isothermal fluids.The framework is based on a coupling between non-equilibrium... Show moreIn this thesis we propose a mathematical framework modeling non-isothermal fluids.The framework is based on a coupling between non-equilibrium thermodynamics and an energetic variational approach for the mechanical parts of the system. From this general model we derive and analyze three separate systems.The first application is the Brinkman-Fourier model. This is related to the ideal gas system, where the pressure and internal energy depend linearly on the product of density and temperature. This is a subsystem to the general Navier-Stokes-Fourier system. We prove the existence of local-in-time weak solutions via compensated compactness arguments.The next model we study is a non-isothermal diffusion system involving chemical reactions. For a system close to chemical equilibrium we show the well-posedness of classical solution using a fixed-point argument involving theory of homogeneous Besov spaces.The third application of the general theory is for another general diffusion system with a Cahn-Hilliard energy. In this framework, we study in detail how the temperature can affect the system on different scales, leading to different models. For the analysis, we focus on one case and show the well-posedness of classical solutions. The proof relies on methods from the theory of Besov spaces and paradifferential calculus. Show less
