Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows

We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space Rᵈ ($d=2,3$).By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of global smooth solutions near equilibrium. Then under additional assumptions that the initial data belong to L¹ and their Fourier modes do not degenerate at low frequencies, we obtain the optimal L² decay rates for the global smooth solutions and their spatial derivatives. At last, we establish the weak-strong uniqueness property in the class of finite energy weak solutions for the incompressible viscoelastic system.