Local Existence and Uniqueness of Solutions for Non Stationary Compressible Fluid

Abstract

This work is devoted to the study of a compressible viscoelastic fluids satisfying the Oldroyd-B model in a regular bounded domain. We prove the local existence of solutions and uniqueness of flows by a classical fixed point argument.

Strong solutions of 3D compressible Oldroyd‐B fluids

“…Following the ideas of Cho, Choe, and Kim [21,22] in the study of Navier-Stokes equations, we establish the existence and uniqueness of local strong solutions to (7). The initial data 0 and 0 here are of slightly lower regularities compared with that in [1] and [18]. Moreover, we give a blow-up criterion for the strong solution obtained earlier.…”

“…They converted the original system into a dimensionless one and improved the results in to the case of with the aid of some boundary estimates of div u . We would like to mention that both in and (see also ), the initial density ρ 0 is assumed to be a small perturbation of a positive constant and thus bounded away from zero. Finally, it is worth noting that, for the Oldroyd‐type model based on the deformation tensor, Hu and Wang recently obtained a blow‐up criterion for the compressible viscoelastic fluids in terms of the temporal integral of the velocity gradient and then, Du, Liu, and Zhang proved can be replaced by where F is the deformation tensor.…”