Existence and stability of global solutions to a regularized OldroydB model in its vorticity formulation
Abstract
We present a new regularized OldroydB model in three dimensions which satisfies an energy estimate analogous to that of the standard model, and maintains the positive semidefiniteness 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 nondiffusive 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 [5]. This is desired since the stress, and thus the conformation tensor, is typically less regular than the velocity for the creeping flow of nonNewtonian fluids. In [9] the existence and regularity of solutions to the nonregularized two dimensional diffusive OldroydB model was established. However, the proof cannot be generalized to three dimensions nor to the nondiffusive 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 nondiffusive 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 nondiffusive solution.
 Publication:

Journal of Differential Equations
 Pub Date:
 August 2022
 DOI:
 10.1016/j.jde.2022.04.027
 arXiv:
 arXiv:2111.12030
 Bibcode:
 2022JDE...327..259J
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics