Hyperbolic Dissipation and Lorentz Invariant Zero Viscosity Limits for Compressible Relativistic Fluids

Description

Fluid Dynamics is rich in applications, ranging from cosmological models of the universe to simulations of blood flow in Medicine. Shock waves—discontinuous wave fronts that form generically by compression—are central to Fluid Dynamics and pose many mathematical challenges. The vanishing viscosity method has played a fundamental role in the theory of classical shock waves, both by providing a mollified limit that identifies the correct physical shock waves, and as a guiding principle in the design of numerical difference schemes for simulating physical shock waves. However, for relativistic fluid flow, the underlying dissipation mechanism (based on the Laplace operator) violates the speed of light bound—the fundamental principle of Relativity. By this, Relativistic Fluid Dynamics is lacking a simple dissipation mechanism for the vanishing viscosity method to study relativistic shock waves as smooth viscous approximations (say, in numerical schemes), consistent with the laws of Relativity.To overcome this lack in Relativistic Fluid Dynamics, the Principal Investigator (PI) recently introduced a simple dissipation mechanism for relativistic compressible fluid flow, the HD Euler equations, (HD stands for hyperbolic dissipation), based on the (hyperbolic) wave operator. The HD Euler equations are Lorentz invariant and thus obey the principles of Relativity, including the speed of light bound. The PI here proposes to investigate whether the HD Euler equations are on par with the classical (Laplacian based) dissipation mechanism of the vanishing viscosity method for the study of shock waves, (featuring global well-posedness, non-linear stability and physical shock waves as zero viscosity limits), but consistent with the laws of Relativity.In preliminary work, the PI established basic consistency of the HD Euler equations as a dissipative model for the vanishing viscosity method, by proving decay of Fourier Laplace modes (dissipation) and existence of 1-D shock profiles selecting the physical shock waves. This analysis revealed a significant simplification over that of classical dissipation mechanisms, due to the refined geometric structure of Relativity and the speed of light bound. This raises the question, as a potential follow-up project, whether the HD Euler equations are more efficient than classical dissipation mechanisms. In particular, could zero viscosity limits of the HD Euler equations hold the potential to generate multi-dimensional shock waves, and could these limits be restrictive enough to rule out spurious (unphysical) weak solutions of the compressible Euler equations? This could be transformative for Relativistic Fluid Dynamics and its numerous applications, ranging from Astrophysics to the problem of plasma confinement in nuclear fusion reactors.