Global existence and optimal decay estimates of the compressible viscoelastic flows in Lᵖ critical spaces

In this paper, we are concerned with the compressible viscoelastic flows in whole space Rⁿ with n≥2. We aim at extending the global existence in energy spaces (see [18] by Hu & Wang and [30] by Qian & Zhang) such that it holds in more general Lᵖ critical spaces, which allows to the case of large highly oscillating initial velocity. Precisely, We define 'two effective velocities' which are used to eliminate the coupling between the density, velocity and deformation tensor. Consequently, the global existence in the Lᵖ critical framework is constructed by elementary energy approaches. In addition, the optimal time-decay estimates of strong solutions are firstly shown in the Lᵖ framework, which improve recent decay efforts for compressible viscoelastic flows.