Global vanishing viscosity limit for the two dimensional incompressible viscoelasticity in Lagrangian coordinates

This paper is concerned with the global in time vanishing viscosity limit for the two dimensional incompressible viscoelasticity. We show that for sufficiently small initial displacements in certain weighted Sobolev spaces, the Cauchy problem of the systems for the two dimensional incompressible viscoelasticity admits unique global classical solutions. Moreover, the estimate is uniform in the viscosity parameter, which would imply the existence of global solutions to the two dimensional incompressible elastodynamics. The result has been obtained by Lin et al. (Commun Pure Appl Math 72(10):2063–2120, 2019) under Eulerian coordinates. In this paper, we try to improve the understanding of the problem in Lagrangian coordinates by using a different method.