On the global well-posedness of interface dynamics for gravity Stokes flow

Abstract

In this paper, we establish the global-in-time well-posedness for an arbitrary $C^{1,\gamma }$, $0<\gamma <1$, initial internal periodic wave for the free boundary gravity Stokes system in two dimensions. This classical well-posedness result is complemented by a weak solvability result in the case of $C^{\gamma }$ or Lipschitz interfaces. In particular, we show new cancellations that prevent the so-called two-dimensional Stokes paradox, despite the polynomial growth of the Stokeslet in this horizontally periodic setting. The bounds obtained in this work are exponential in time, which are in strong agreement with the growth of the solutions obtained in [22]. Additionally, these new cancellations are used to establish global-in-time well-posedness for the Stokes-transport system with initial densities in $L^p$ for $2<p<\infty$. Furthermore, we also propose and analyze several one-dimensional models that capture different aspects of the full internal wave problem for the gravity Stokes system, showing that all of these models exhibit finite-time singularities. This fact evidences the fine structure of the nonlinearity in the full system, which allows the free boundary problem to be globally well-posed, while simplified versions blow-up in finite time.