Well‐Posedness and Vanishing Viscosity Limit for Viscoelastic Fluids in a 3D Periodic Slab

ABSTRACT

This paper investigates the well‐posedness and vanishing viscosity limit for three‐dimensional incompressible viscoelastic fluids in a 3D periodic slab. The fluid velocity is subject to Navier‐slip boundary conditions, while the deformation tensor satisfies a specific boundary condition. We first establish the existence of global weak solutions. By deriving uniform estimates in Sobolev spaces that are independent of the viscosity coefficients, we analyze the convergence of the viscous solutions to the inviscid ones as the viscosities tend to zero. Furthermore, for sufficiently small initial data, we prove the global existence and uniqueness of classical solutions for both the viscoelastic Navier–Stokes equations and the elastodynamics system. A key result for the inviscid system is the exponential decay of its ‐norm. As an application of this decay property, we establish time‐independent convergence rates for the vanishing viscosity limit.