Mathematical Fluid Dynamics – Hyperbolic Balance Laws across the Scales

Heinrich Freistühler is a professor at the University of Konstanz. His interests focus on hyperbolic conservation laws, shock waves, and dynamical phase boundaries. He has also organized various activities in research networks.Christian Rohde is a professor at the University of Stuttgart. His work concerns the analysis and numerics of hyperbolic balance laws, multiphase and porous media flow. He coordinates the current DFG Priority Programme Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness.Gerald Warnecke is a professor at the Otto von Guericke University Magdeburg. His research covers conservation laws and related applications, including numerical methods. He has coordinated the earlier DFG Priority Programme Analysis and Numerics of Conservation Laws.Gianluca Crippa is a professor at the University of Basel and has worked on partial differential equations, geometric measure theory, transport and flows with irregular velocity fields, fluid dynamics, mixing, turbulence, singular limits, and (local and nonlocal) conservation laws. Regarding mixing as a PDE theory of its own, Crippa has co-authored a book on corresponding results on significant low-regularity cases.Marcelo Disconzi is an associate professor at the Vanderbilt University and is interested in partial differential equations, mathematical fluid dynamics, mathematical general relativity, geometric analysis, and mathematical physics, which involves considering relativistic fluids in three spatial dimensions, with vorticity, without symmetry assumptions, and possibly allowing for the presence of free boundaries. He has co-authored a novel formulation of the relativistic dynamics of viscous, heat-conducting fluids.Theodore Drivas is an associate professor at the Stony Brook University and has worked on geometrical hydrodynamics, incompressible fluid dynamics and turbulence, compressible fluid dynamics, enhanced diffusion and convection, non-smooth dynamical systems, plasma physics, stochastic partial differential equations and general relativity. Furthermore, his exact mathematical theorems on turbulence complement the existing abundance of non-rigorous physical theories.Tai-Ping Liu is an emeritus professor at Stanford University and is a distinguished visiting professor at Academia Sinica. He has made numerous fundamental contributions to the theory of nonlinear hyperbolic first-order systems and shock waves, establishing, among more, a deterministic version of the Glimm scheme and an array of theorems on the asymptotic behaviour of conservative and relaxation systems. His analysis, wih Yu, of nonlinear shock and boundary layers for the Boltzmann equation, establishes a new, rigorous understanding of central phenomena in kinetic theory.Simon Markfelder is a junior professor of partial differential equations of mathematical physics at the University of Konstanz and works on the non-uniqueness in mathematical fluid dynamics and its relation to turbulence. He has constructed multiple solutions via the technique of convex integration and investigated/proposed admissibility criteria for weak and measure-valued generalized solutions.Hana Mizerová is an assistant professor at the Comenius University in Bratislava and a PostDoc at the Czech Academy of Sciences and she works on analytical and numerical methods for the evolutionary PDE systems of ideal, viscous, heat-conductive, or viscoelastic fluids and magnetohydrodynamics. She has also co-authored a new concept of dissipative measure-valued solutions to the Euler equations.Shih-Hsien Yu is a distinguished research fellow at Academia Sinica and has worked on compressible Navier-Stokes, existence, propagation, and stability of shock waves, H-Theorem and energy methods for the Boltzmann equation, and the Broadwell model. His micro-macro decomposition, obtained with Liu, has been instrumental for their refined understanding of the dissipation of momentum and energy from that of the microscopic part.

• Chapter 1. A general framework for convex integration and its application to the compressible Euler equations.- Chapter 2. Convergence of entropy-stable finite volume approximations of compressible Euler equations.- Chapter 3. Mathematical theorems on turbulence.- Chapter 4. Introduction to the theory of mixing for incompressible flows.- Chapter 5. Micro-micro decomposition: Positivity of Boltzmann shock profile and Green’s function.- Chapter 6. Coupling of boundary layer and fluid waves for the Boltzmann equation.- Chapter 7. Recent progress in relativistic viscous fluid dynamics.