A Regularized Interface Method for Fluid-Poroelastic Structure Interaction Problems with Nonlinear Geometric Coupling

TL;DR

This paper develops a new regularized interface method to prove the existence of weak solutions for complex fluid-poroelastic structure interaction problems with nonlinear geometric coupling and low-regularity interfaces.

Contribution

It introduces a regularization technique and a modified weak formulation to establish existence results for nonlinear moving boundary problems with irregular interfaces.

Findings

Proved existence of weak solutions for the regularized problem.

Established uniform a priori bounds and compactness for moving domains.

Demonstrated the effectiveness of regularization in handling low-regularity interfaces.

Abstract

We introduce a new regularized interface method for proving existence of weak solutions to nonlinear moving boundary problems with low-regularity interfaces. We study a fluid-poroelastic structure interaction (FPSI) problem coupling the Navier-Stokes equations for an incompressible viscous fluid with the Biot system for a bulk poroelastic medium. The two phases occupy domains of the same spatial dimension, separated by a moving interface defined by the trace of the poroelastic displacement, which exhibits low regularity and strong geometric nonlinearities. Despite its importance in applications, no existence theory has been available for this nonlinear moving-domain setting, primarily because the lack of interface regularity precludes even the formulation of a weak solution framework. To address this gap, we (1) introduce a regularization of the Biot displacement via spatial convolution at scale $\delta > 0$, which defines regularized moving domains and interface, and (2) modify the weak formulation in a way that preserves energy consistency with the original problem. For each fixed $\delta > 0$, we prove existence of a weak solution to the resulting regularized interface problem. The proof strategy involves inserting a thin plate of thickness $h > 0$ at the interface, applying a time-discretization via a Lie operator splitting scheme, establishing uniform a priori bounds, and employing Aubin-Lions compactness on moving domains. The analysis is particularly involved, partly because the thin plate allows displacements in all spatial directions. Passing to the limit $h \to 0$ with uniform-in-h estimates and compactness arguments yields a regularized interface weak solution. The regularization introduced in this manuscript is essential to maintain uniform geometric control of the moving interface and to accommodate vector-valued structural displacements.