Globally analytical solutions of the compressible Oldroyd-B model without retardation

Abstract

In this paper, we prove the global existence of analytical solutions to the compressible Oldroyd-B model without retardation near a non-vacuum equilibrium in ${\mathbb R}^n$ $(n=2,3)$. Zero retardation results in zero dissipation in the velocity equation, which is the main difficulty that prevents us to obtain the long time well-posedness of solutions. Through dedicated analysis, we find that the linearized equations of this model have damping effects, which ensures the global-in-time existence of small data solutions. However, the nonlinear quadratic terms have one more order derivative than the linear part and no good structure is discovered to overcome this derivative loss problem. So we can only build the result in the analytical energy space rather than Sobolev space with finite order derivatives.