Nonexistence and existence of shock profiles in the Bemfica-Disconzi-Noronha model
Heinrich Freistühler
Abstract
This paper 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 $\ensuremath{\mu}$, $\ensuremath{\nu}$ and being causal whenever one chooses $(\ensuremath{\mu},\ensuremath{\nu})$ from a certain range $\mathcal{C}\ensuremath{\subset}{\mathbb{R}}^{2}$, this paper shows that for any choice of $(\ensuremath{\mu},\ensuremath{\nu})$ in the interior of $\mathcal{C}$, there is a dichotomy in so far as (i) shocks of sufficiently small amplitude admit profiles and (ii) certain other, thus necessarily nonsmall, shocks do not. This finding does not preclude the possibility that if one chooses $(\ensuremath{\mu},\ensuremath{\nu})$ from a specific part $\mathcal{S}$ of the boundary of $\mathcal{C}$, the dichotomy disappears and all shocks have profiles; the parameter set $\mathcal{S}$ corresponds to the ``sharply causal'' case, in which one of the characteristic speeds of the dissipation operator is the speed of light.