Evolutionary inequalities as generalized solution concepts

Many phenomena in science and technology are modeled by nonlinear partial differential equations. Since smooth solutions often do not exist or are not physically meaningful—particularly in the presence of shocks or turbulence—it is necessary to consider generalized solution concepts. In this thesis, several generalized formulations based on inequality principles are investigated. For thermodynamically consistent phase-field models, entropic solutions are introduced, relying on a global energy inequality and a local entropy inequality. This approach offers notable advantages over formulations based solely on a local energy balance, like an existing general weak-strong uniqueness principle. For anisotropic fluid and geophysical models, dissipative solutions are studied, which are defined through a relative energy inequality. Finally, a novel and unifying concept of energy-variational solutions is proposed in a general framework and applied to hyperbolic conservation laws, liquid crystal systems, and various viscoelastic fluid models. In selected cases, this new solution concept is compared with more established approaches.