Weighted H2 regularity for a Stokes--Lam\'{e} fluid-structure interaction problem with interface--boundary intersections
Abstract
We establish weighted spatial regularity for unsteady Stokes flow coupled to inertial linear elasticity on fixed two-dimensional polygons. The fluid--solid interface meets the exterior boundary at its endpoints, where corner singularities may preclude unweighted H2 regularity. Under suitable data and compatibility conditions, we prove existence and uniqueness of a weak solution whose fluid velocity and solid displacement remain bounded in weighted H2 spaces on every finite time interval. The pressure is bounded in a weighted H1 space, and the solution satisfies corresponding temporal energy estimates. The domains may be nonconvex, the exterior solid boundary is traction free, and nonhomogeneous fluid boundary velocities are allowed. Their second time derivative is measured in the dual of a weighted trace space. The admissible weights satisfy the spectral conditions of the separate mixed Stokes and Lam\'e problems and agree at the interface endpoints. The proof uses an interface-compatible lifting, weighted trace estimates, and transposition to control the differentiated boundary data. An equation for the time-integrated fluid velocity then yields weighted regularity without assuming additional interface regularity. The estimates provide a basis for numerical analysis on meshes refined towards the corners.
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BibTeX
@misc{dong2026weighted,
title = {{Weighted $\mathbf H^2$ regularity for a Stokes--Lam\'e fluid-structure interaction problem with interface--boundary intersections}},
author = {Zhaonan Dong and Tiantian Huang and Buyang Li},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://inria.hal.science/hal-05507626v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weighted H^2 regularity for a Stokes--Lam\'{e} fluid-structure interaction problem with interface--boundary intersections
AU - Zhaonan Dong
AU - Tiantian Huang
AU - Buyang Li
PY - 2026
DA - 2026/10/31
PB - HAL
UR - https://inria.hal.science/hal-05507626v2
AB - We establish weighted spatial regularity for unsteady Stokes flow coupled to inertial linear elasticity on fixed two-dimensional polygons. The fluid--solid interface meets the exterior boundary at its endpoints, where corner singularities may preclude unweighted H^2 regularity. Under suitable data and compatibility conditions, we prove existence and uniqueness of a weak solution whose fluid velocity and solid displacement remain bounded in weighted H^2 spaces on every finite time interval. The pressure is bounded in a weighted H^1 space, and the solution satisfies corresponding temporal energy estimates. The domains may be nonconvex, the exterior solid boundary is traction free, and nonhomogeneous fluid boundary velocities are allowed. Their second time derivative is measured in the dual of a weighted trace space. The admissible weights satisfy the spectral conditions of the separate mixed Stokes and Lam\'e problems and agree at the interface endpoints. The proof uses an interface-compatible lifting, weighted trace estimates, and transposition to control the differentiated boundary data. An equation for the time-integrated fluid velocity then yields weighted regularity without assuming additional interface regularity. The estimates provide a basis for numerical analysis on meshes refined towards the corners.
ER -