GLOBAL WELL-POSEDNESS FOR VISCOELASTIC FLUID SYSTEM IN BOUNDED DOMAINS

In this paper, the initial boundary value problem of the incompressible viscoelatic fluid system is considered. In the contrast to the previous work [Y. Chen and P. Zhang, Comm. Partial Differential Equations, 31 (2006), pp. 1793-1810], [F. H. Lin and P. Zhang, Comm. Pure Appl. Math., 61 (2008), pp. 539-558], [F. H. Lin, C. Liu, and P. Zhang, Comm. Pure Appl. Math., 18 (2005), pp. 1437-1471], and [Z. Lei, C. Liu, and Y. Zhou, Arch. Ration. Mech. Anal., 188 (2008), pp. 371-398], our main ingredient to solve the problem is to reformulate the original system along the particle trajectory. It is shown that in the so-called Lagrangian coordinate, the system is decoupled, and the well-posedness problem is reduced to the solvability of standard damped-wave equations with a no-slip boundary condition. Finally, we establish the global existence and exponential decay estimate for the equivalent system with the small and smooth initial data.