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Heinrich Freistuhler

Publications and source records attributed to Heinrich Freistuhler.

16 recordsLinked to original sources
arXiv2025

Consistency of some well-posed five-field theories of dissipative relativistic fluid dynamics

Within the FTBDNK family of formulations of relativistic Navier-Stokes (H. Freistühler and B. Temple, Proc. R. Soc. A 470, 20140055 (2014), Proc. R. Soc. A 473 (2017), 20160729; F. S. Bemfica, M. Disconzi, and J. Noronha, Phys. Rev. D 98, 104064 (2018), Phys. Rev. D 100, 104020 (2019); P. Kovtun, J. High Energy Phys. 2019, 034 (2019)), this paper collects some consistency properties for certain causal hyperbolic five-field theories obtained from the Landau-Lifshitz formulation via Eulerian gradient shifts, a family, EGS(L), of models that slightly generalize a class identified in H. Freistühler, J. Math.\ Phys. 61, 033101 (2020). With $ε$ the magnitude of the dissipation coefficients that quantify viscosity and heat conduction, the paper shows that any element of EGS(L) is $O(ε^2)$ equivalent to the Landau-Lifshitz formulation, has an $O(ε^3)$ excess entropy production, represents heterogeneous local thermodynamic equilibria cleanly, and admits regular heteroclinic profiles for all shock waves of sufficiently small amplitude.

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arXiv2025

Nonlinear Stability of First-Order Relativistic Viscous Hydrodynamics

This paper shows nonlinear stability of homogeneous states in second-order hyperbolic systems of partial differential equations that model the dynamics of dissipative relativistic fluids, by checking a dissipativity criterion formulated earlier by the authors and invoking a recent general result by the second author on long-time existence and time-asymptotic stability of small-data solutions to nonlinear hyperbolic systems. Version 3 differs from version 2 by a trivial correction (minus signs in front of six coefficients).

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arXiv2024

Linear Degeneracy in a Class of Nonlinear Second-Order Hyperbolic Systems

For a class of nonlinear hyperbolic systems of second order the paper shows that all Lax modes associated with their first-order formulations are linearly degenerate. This property holds for recently considered models of dissipative relativistic fluid dynamics, supporting the possibility that solutions to these models generally avoid singularity formation.

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arXiv2023

On Phase Boundaries in Relativistic Korteweg Fluids

This is the first of several planned papers that study the existence and local-in-time persistence of phase fronts in the Lorentz invariant Euler equations for gases of van der Waals type, aiming at transferring earlier results of Slemrod and of Benzoni-Gavage and collaborators to the context of the theory of relativity. While the later papers will examine more general admissibility criteria, this one uses and extends the author's Lorentz invariant formulation of Korteweg's theory of capillarity, establishing a family of phase fronts that have a regular heteroclinic profile with respect to the associated Euler-Korteweg equations.

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arXiv2023

On Shock Profiles in Four-Field Formulations of Dissipative Relativistic Fluid Dynamics

This paper shows that in second-order hyperbolic systems of partial differential equations proposed in authors' earlier paper (J. Math. Phys. 59 (2018)) for modelling the relativistic dynamics of barotropic fluids in the presence of viscosity and heat conduction, shock waves of arbitrary strength have smooth, monotone dissipation profiles. The results and arguments extend classical considerations of Weyl (Comm. Pure Appl. Math. 2 (1949)) and Gilbarg (Amer. J. Math. 73 (1951)) to the relativistic setting.

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arXiv2021

Remarks on Ruggeri's First Model of Dissipative Fluid Dynamics as a Symmetric Hyperbolic System

This note establishes properties of a model of dissipative fluid dynamics as a symmetric hyperbolic system by which Ruggeri once triggered the development of Rational Extended Thermodynamics. These properties, i.e., (1) convergence of solutions to those of the Navier-Stokes-Fourier equations and (2) nonlinear asymptotic stability of homogeneous reference solutions, also hold in the analogous situation for barotropic fluids.

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arXiv2021

A Galilei Invariant Version of Yong's Model

In a paper in Arch. Rational Mech. 214 (2014), Wen-An Yong has described the dynamics of a compressible fluid with Maxwell delayed viscosity as a symmetric hyperbolic system of balance laws, and shown that the solutions of this system tend to solutions of the Navier-Stokes equations. The purpose of the present note is to present a Galilei invariant version of Yong's model and communicate observations on its shock waves.

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arXiv2021

Non-Existence and Existence of Shock Profiles in the Bemfica-Disconzi-Noronha Model

This note studies a four-field hyperbolic PDE model that was recently introduced by Bemfica, Disconzi, and Noronha for the pure radiation fluid with viscosity, and asks whether shock waves admit continuous profiles in this description. The model containing two free parameters mu, nu and being causal whenever one chooses (mu,nu) from a certain range C subset R2, this paper shows that for any choice of (mu,nu) in the interior of C, there is a dichotomy in so far as (i) shocks of sufficiently small amplitude admit profiles and (ii) certain other, thus necessarily non-small, shocks do not. This finding does not preclude the possibility that if one chooses (mu,nu) from a specific part S of the boundary of C, the dichotomy disappears and all shocks have profiles; the parameter set S corresponds to the "sharply causal" case, in which one of the characteristic speeds of the dissipation operator is the speed of light.

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arXiv2017

Godunov Variables in Relativistic Fluid Dynamics

This note presents Godunov variables and 4-potentials for the relativistic Euler equations of barotropic fluids. The associated additional conservation/ production law has different interpretations for different fluids. In particular it refers to ENTROPY in the case of thermobarotropic fluids, and to MATTER in the case of isentropic fluids. The paper also presents an explicit formula for the generating function of the Euler equations in the case of ideal gases. It pursues ideas on symmetric hyperbolicity going back to Godunov (cf. also Lax and Friedrichs as well as Boillat) that were elaborated as Ruggeri and Strumia's theory of convex covariant density systems.

math-ph↗
arXiv2015

Emergence of Unstable Modes for Shock Waves in Ideal MHD

This note studies classical magnetohydrodynamic shock waves in an inviscid fluidic plasma that is assumed to be a perfect conductor of heat as well as of electricity. For this mathematically prototypical material, it identifies a critical manifold in parameter space, across which slow classical MHD shock waves undergo emergence of a complex conjugate pair of unstable transverse modes. In the reflectionally symmetric case of parallel shocks, this emergence happens at the spectral value 0, and the critical manifold possesses a simple explicit algebraic representation. Results of refined numerical treatment show that for only almost parallel shocks the unstable mode pair emerges from a pair of non-zero imaginary spectral values.

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arXiv2013

Reductions of the Navier-Stokes-Allen-Cahn and the Navier-Stokes-Cahn-Hilliard equations

This paper studies two well-known models for two-phase fluid flow at constant temperature, the isothermal Navier-Stokes-Allen-Cahn and the isothermal Navier-Stokes-Cahn-Hilliard equations, both of which consist of equations for the (total) fluid density rho, the (mass-averaged)velocity u and the concentration (of one of the phases,) c. Assuming in either case that both phases are incompressible with different densities, each of the models is shown to reduce to a system of evolution equations in rho and u alone. In the case of the Navier-Stokes-Allen-Cahn model, this reduced system is the classical Navier-Stokes-Korteweg model. The reduced system resulting from the Navier-Stokes-Cahn-Hilliard equations is a novel `integro'-differential system in which a non-local operator acts on the divergence of the velocity.

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arXiv2013

Diffuse planar phase boundaries in a two-phase fluid with one very dense phase

This note studies Navier-Stokes-Allen-Cahn models for compressible fluids that are mixtures of two incompressible phases whose density ratio eps=rho_1/rho_2 is very small. Under a natural assumption on the mixing energy, it shows the existence of diffuse planar phase boundaries for all 0=eps<eps_0. For eps=0, one recovers the Navier-Stokes-Korteweg model and its well-known diffuse phase boundaries.

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arXiv2013

Diffuse planar phase boundaries in a two-phase fluid with one incompressible phase

This note studies a family of Navier-Stokes-Allen-Cahn systems parameterized by temperature. Derived from an internal energy that corresponds to one incompressible and one compressible phase, this family is considered as a simple model for water. Decreasing temperature across a critical value, a transition takes places from a situation without towards one with planar diffuse phase boundaries.

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arXiv2011

The Lopatinski determinant of small shocks may vanish

The Kreiss-Majda Lopatinski determinant encodes a uniform stability property of shock wave solutions to hyperbolic systems of conservation laws in several space variables. This note deals with the Lopatinski determinant for shock waves of sufficiently small amplitude. The determinant is known to be non-zero for so-called extreme shock waves, i. e., shock waves which are asscoiated with either the slowest or the fastest mode the system displays for a given direction of propagation, if the mode is Metivier convex. The result of the note is that for arbitrarily small non-extreme shock waves associated with a Metivier convex mode, the Lopatsinki determinant may vanish.

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