arXiv subjects

Heinrich Freistühler

Publications and source records attributed to Heinrich Freistühler.

7 recordsLinked to original sources

arXiv2026

Dynamical stability of planar phase boundaries for hyperelastic materials of Hadamard type

The dynamical stability of laminates or planar phase boundaries for hyperelastic materials of Hadamard type in two space dimensions is studied. For that purpose, the stability function, known as the Lopatinskii determinant, is computed for states of deformation at both sides of the planar interface that account for the generalized Legendre-Hadamard conditions derived by Grabovsky and Truskinovsky (J. Elast. 123 (2016), 225--243). The sufficient conditions for the dynamical stability of such configurations are described in terms of the physical parameters of the model, such as the shear modulus, and computed under kinetic conditions across the interface of both Maxwell (conservation of energy) or Abeyaratne and Knowles (dissipation of energy) types.

math.AP↗

arXiv2020

Decay and Subluminality of Modes of all Wave Numbers in the Relativistic Dynamics of Viscous and Heat Conductive Fluids

To further confirm the causality and stability of a second-order hyperbolic system of partial differential equations that models the relativistic dynamics of barotropic fluids with viscosity and heat conduction (H. Freistühler and B. Temple, J. Math. Phys. 59 (2018)), this paper studies the Fourier-Laplace modes of this system and shows that all such modes, relative to arbitrary Lorentz frames, (a) decay with increasing time and (b) travel at subluminal speeds. Stability is also shown for the related model of non-barotropic fluids (H. Freistühler and B. Temple. Proc. R. Soc. A 470 (2014) and Proc. R. Soc. A 473 (2017)). Even though these properties had been known for a while in the sense of numerical evidence, the fully analytical proofs for the subluminality of modes of arbitrary wave numbers in arbitrary frames given here appear to be the first regarding any five-field formulation of dissipative relativistic fluid dynamics.

physics.flu-dyn↗

arXiv2018

Nonlinear waves and polarization in diffusive directed particle flow

We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave solutions for weak diffusion by singular perturbation techniques and provide evidence for their existence also for stronger diffusion. We discuss different types of wave fronts and their composition to more complex patterns and illustrate their emergence from generic initial data by simulations. We also investigate the dependence of the wave velocities on the model parameters.

math.AP↗

arXiv2017

The Blesgen and Lowengrub-Truskinovsky Descriptions of Two-Phase Compressible Fluid Flow: Interstitial Working and a Reduction to Korteweg Theory

Formulated by Blesgen and Lowengrub and Truskinovsky, the `Navier-Stokes-Allen-Cahn' (NSAC) equations and the `Navier-Stokes-Cahn-Hilliard' (NSCH) equations describe diphasic fluid flow, combining the conservation laws for mass, momentum, and energy with a balance law for the phases, which governs the concentration of one, and thus the other, phase as an order parameter. By contrast, the `Navier-Stokes-Korteweg' (NSK) theory for the dynamics of a one-phase capillary fluid, given by Dunn and Serrin following Korteweg and Slemrod, uses only the three said conservation laws and has the mass density itself as its order parameter. In a previous paper (ARMA, 2017), the authors considered the derivation of the NSAC, the NSCH, and the NSK equations, and showed that in the case that the two phase are incompressible with different specific volumes, both NSAC and NSCH reduce to versions of NSK. They realized only after that article was published that their assumptions tacitly corresponded to neglecting the so-called microforces. While the argumentation in that previous paper remains mathematically consistent, the microforces are a physical reality, and the purpose of the present note is to show that the reduction of NSAC and NSCH to NSK still holds when one does take them into account.

math.AP↗

arXiv2013

An analytical proof for the stability of Heimburg-Jackson pulses

This paper studies analytically the stability of solitary waves in a generalized Boussinesq equation with quadratic-cubic nonlinearity. For general values of two parameters a and b determining the system, unstable waves may occur. If however, as in a situation for which this Boussinesq equation was recently proposed as a model for pulse propagation in nerves, (a,b) belongs to a certain natural regime, then all possible waves are stable.

math.AP↗

arXiv2013

Symmetrizations of RMHD equations and stability of relativistic current-vortex sheets

We consider the equations of relativistic magnetohydrodynamics (RMHD) in the case of special relativity. For the fluid rest frame a nonconservative reformulation of the RMHD equations gives a symmetric system for the vector of primitive (physical) variables. By applying the Lorentz transformation to this system we find a concrete form of symmetric matrices in the LAB-frame. The resulting symmetric system in terms of primitive variables is important for the study of various initial boundary value problems for the RMHD equations. We also find a so-called secondary symmetrization whose direct consequence is the extension of the sufficient stability condition obtained earlier for non-relativistic planar current-vortex sheets to the relativistic case. As in non-relativistic settings, this implies the local-in-time existence of corresponding smooth nonplanar current-vortex sheets.

math.AP↗

arXiv2012

Convex entropy, Hopf bifurcation, and viscous and inviscid shock stability

We consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (i) can inviscid instability occur? (ii) can there occur viscous instability not detected by inviscid theory? (iii) can there occur the ---necessarily viscous--- effect of Hopf bifurcation, or "galloping instability"? and, perhaps most important from a practical point of view, (iv) as shock amplitude is increased from the (stable) weak-amplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (i) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (ii) and (iii) do occur for an artifically constructed system with convex viscosity-compatible entropy. We do not know of an example for which (iv) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state.

math.AP↗