Gradient Catastrophe for Solutions to the Hyperbolic Navier-Stokes Equations

TL;DR

This paper investigates the formation of singularities in solutions to the one-dimensional hyperbolic Navier-Stokes equations, demonstrating gradient blow-up due to hyperbolization mechanisms.

Contribution

It establishes the hyperbolic nature of these equations and proves the existence of gradient blow-up through analysis of nonlinear eigenvalues.

Findings

Proved the hyperbolic character of the equations.

Established the presence of two genuinely nonlinear eigenvalues.

Demonstrated gradient blow-up of solutions.

Abstract

This paper studies local existence and the singularity formation of the solutions of the one-dimensional hyperbolic Navier-Stokes equations, in particular proving the gradient blow-up of the derivatives of the solutions. The underlying model introduces a relaxation mechanism that leads to hyperbolization, achieved both through a nonlinear Cattaneo law for heat conduction and through Maxwell-type constitutive relations for the stress tensor. Our main approach is to prove that the hyperbolic Navier-Stokes equations are indeed hyperbolic, and to prove that they possess two genuinely nonlinear eigenvalues, thereby establishing the blow-up of the gradient of the solution. In addition, we provide a derivation of the equation of state for the hyperbolic Navier-Stokes equations in the appendix.