Lifespan of smooth solutions to the two dimensional repulsive Hookean viscoelastic equations with zero shear viscosity

In this paper, we investigate the initial value problem for the two-dimensional repulsive Hookean viscoelastic equations with zero shear viscosity. We apply the Hodge decomposition to decouple the system into compressible and incompressible parts. For the compressible parts, we derive the $ L^2 $ estimates via the standard energy method and obtain $ L^2 $ and $ L^\infty $ decay estimates using the Green's function techniques. However, for the incompressible parts, the linearized system for the equations is hyperbolic. Since the quadratic nonlinearity satisfies the null condition, we establish the $ L^2 $ estimates by employing the ghost weight energy method. Subsequently, based on the local energy decay theorem, we derive the weight energy estimates for the incompressible components. Finally, we determine the lifespan of smooth solutions via a bootstrap argument.