Global solutions to the Oldroyd-B model with a class of large initial data

TL;DR

This paper proves the existence and uniqueness of global solutions for the incompressible Oldroyd-B model with certain large initial vertical velocities, under smallness conditions in specific Besov spaces.

Contribution

It establishes global well-posedness for the Oldroyd-B model with large initial vertical velocity, extending previous results to larger initial data in critical Besov spaces.

Findings

Unique global solution exists under smallness conditions in Besov spaces.

The result applies even with large initial vertical velocity.

Provides conditions for global well-posedness of the Oldroyd-B model.

Abstract

Consider a global wellposed problem for the incompressible Oldroyd-B model. It is shown that this set of equations admits a unique global solution provided the initial horizontal velocity $u^h_0$, the product $\om u^d_0$ of the coupling parameter $\om$ and initial the vertical velocity $u^d_0$, and initial symmetric tensor of constrains $\tau_0$ are sufficient small in the scaling invariant Besov space $\dot{B}^{\fr{d}{2}-1}_{2,1}\times\dot{B}^{\fr{d}{2}}_{2,1}, d\ge2$. In particular, the result implies the global well-posedness of Oldroyd-B model with large initial vertical velocity $u_0^d$.