Quasilineare symmetrisch hyperbolisch-hyperbolische Systeme zweiter Ordnung, mit Anwendungen in der relativistischen Fluiddynamik

• Antragsteller: Professor Dr. Heinrich Freistühler
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2019 bis 2023
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 423781085

• Erstellungsjahr: 2023

Zusammenfassung der Projektergebnisse

The project has studied a class of systems of partial differential equations which are characterized by the interplay between two hyperbolic operators, one of first, the other of second order. While each of the two operators generates a unitary group, it is assumed that their action on the other’s orbits is dissipative. It is the work done within this project that has identified this class of dissipative symmetric hyperbolic-hyperbolic systems as a consistent object of mathematical study: a finitespeed of propagation counterpart of the classical hyperbolic-parabolic setting. A main result of the project is a characterization of dissipativity in terms of algebraic conditions related to Fourier- Laplace modes in particular at small and large wave numbers. These conditions are finer than the Kawashima-Shizuta condition known from the hyperbolic-parabolic case, but also simple and appealing, and they entail the same asymptotic decay rates in Sobolev spaces. While that characterization of systems takes place at the linear constant-coefficients level, the theory developed in the context of the project studies nonlinear systems of this kind. Two main results of the project are proofs for the global existence of solutions corresponding to data that are small perturbations of a homogeneous reference state and for their time-asymptotic decay to that state. These results have been obtained in two steps, first for systems in which the second order operator is symmetric hyperbolic in the sense of Hughes-Kato-Marsden, i. e., simply said, its spatial part is elliptic, then for a broader class of systems, whose identification, while consistent with classical ideas of Leray, has again taken place within this project. As had earlier first been found by the P.I. and Temple, dissipative symmetric hyperbolic-hyperbolic systems can naturally be used in relativistic continuum dynamics, and this project pays particular attention to dissipative relativistic compressible fluid dynamics, which has classical and very new applications both in astrophysics (gaseous stars) and in laboratory situations (quark-gluon plasmas). The catchword is relativistic Navier-Stokes, and the results of the project contribute to an ongoing debate on how to properly formulate dissipative relativistic fluid dynamics. The original proposal by Freistühler and Templehas recently been somewhat refined as a similar description given by Bemfica, Disconzi, Noronha and Kovtun (‘BDNK’) – the project’s results on global existence and asymptotic stability apply to the BDNK description, which is not of Hughes-Kato-Marsden type. A particularly pointed expression of hyperbolic nonlinearity is the formation and propagation of shock waves. The project has taken a first step into a systematic investigation of shock waves in the setting of dissipative hyperbolic-hyperbolic systems. The main result here is a proof for the spectral stability of shock waves in the case of small amplitude. This result via the Evans function is obtained through applications of geometric singular perturbation theory to the occurring dynamics on a Grassmann manifold, and applies to a prototypical class of systems in which the second order operator is the d’Alembertian, i.e., of Hughes-Kato-Marsden type. Another result obtained here is that regarding shock waves the situation is different for other systems: The project has shown that shock waves may have oscillatory or even no smooth profiles in BDNK models – this prompts the question of whether shock waves can be stably represented in those models.

Projektbezogene Publikationen (Auswahl)

• Asymptotic stability in a second-order symmetric hyperbolic system modeling the relativistic dynamics of viscous heat-conductive fluids with diffusion. Journal of Differential Equations, 268(2), 825-851.
  Sroczinski, Matthias
  
• A class of uniformly dissipative symmetric hyperbolic-hyperbolic systems. Journal of Differential Equations, 288, 40-61.
  Freistühler, Heinrich & Sroczinski, Matthias
  
• Decay and subluminality of modes of all wave numbers in the relativistic dynamics of viscous and heat conductive fluids. Journal of Mathematical Physics, 62(5).
  Freistühler, Heinrich; Reintjes, Moritz & Temple, Blake
  
• Uniform dissipativity for mixed-order hyperbolic systems, with an application to relativistic fluid dynamics. Journal of Differential Equations, 325, 70-81.
  Freistühler, Heinrich; Reintjes, Moritz & Sroczinski, Matthias
  
• A generically singular type of saddle–node bifurcation that occurs for relativistic shock waves. Physica D: Nonlinear Phenomena, 453, 133829.
  Pellhammer, Valentin
  
• Global existence and decay of small solutions for quasi-linear second-order uniformly dissipative hyperbolic-hyperbolic systems. Journal of Differential Equations, 383, 130-162.
  Sroczinski, Matthias
  
• Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws. Archive for Rational Mechanics and Analysis, 248(6).
  Bärlin, Johannes