Existence and Stability of Global Solutions to a regularized Oldroyd-B Model in its Vorticity Formulation

Abstract

We present a new regularized Oldroyd-B model in three dimensions which satisfies an energy estimate analogous to that of the standard model, and maintains the positive semi-definiteness of the conformation tensor. This results in the unique existence and stability of global solutions in a periodic domain. To be precise, given an initial velocity $u_0$ and initial conformation tensor $σ_0$, both with components in $H^2$, we obtain a velocity $u$ and conformation tensor $σ$ both with components in $C([0, T]; H^2)$ for all $T>0$. Assuming better regularity for the initial data allows us to obtain better regularity for the solutions. We treat both the diffusive and non-diffusive cases of the model. Notably, the regularization in the equation for the conformation tensor in our new model has been applied only to the velocity, rather than to the conformation tensor, unlike other available regularization techniques \cite{barrett;suli2009}. This is desired since the stress, and thus the conformation tensor, is typically less regular than the velocity for the creeping flow of non-Newtonian fluids. In \cite{constantinnote} the existence and regularity of solutions to the non-regularized two dimensional diffusive Oldroyd-B model was established. However, the proof cannot be generalized to three dimensions nor to the non-diffusive case. The proposed regularization overcomes these obstacles. Moreover, we show that the solutions in the diffusive case are stable in the $H^2$ norm. In the non-diffusive case, we are able to establish that the solutions are stable in the $L^2$ norm. Furthermore, we show that as the diffusivity parameter goes to zero, our solutions converge in the $L^2$ norm to the non-diffusive solution.