| Abstract: |
| In this talk, we mainly focus on the multi-dimensional viscoelastic flows of Oldroyd-B type. Considering a system of equations related to the compressible viscoelastic fluids of Oldroyd-B type with the general pressure law, $P^\prime(\bar{\rho})+\alpha>0$, with $\alpha >0$ being the elasticity coefficient of the fluid, we prove the global existence and uniqueness of the strong solution in the critical Besov spaces when the initial data $u_0$ and the low frequency part of $\rho_0$, $\tau_0$ are small enough compared to the viscosity coefficients. The proof we display here does not need any compatible conditions. In addition, we also obtain the optimal decay rates of the solution in the Besov spaces. At Last, considering the multi-dimensional compressible Oldroyd-B model, which is derived by Barrett, Lu, and Suli (Comm. Math. Sci. 2017) through the micro-macro analysis of the compressible Navier-Stokes-Fokker-Planck system in the case of Hookean bead-spring chains. We would provide a unified method to study the system with the background polymer number density $\eta_\infty\geq0$, including the vanishing case and the nonvanishing case, and establish the global-in-time existence of the strong solution for the associated Cauchy problem when the initial data are small in the critical Besov spaces. |
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