Global strong solutions without contact for a viscous fluid interacting with a purely elastic beam

Abstract

We consider a two-dimensional incompressible viscous fluid confined in a periodic channel between a fixed flat wall and a purely elastic periodic beam. For compatible initial fluid velocity in $H^1$, beam height in $H^3$, and beam velocity in $H^1$, with strictly positive initial height, we establish global existence and uniqueness of strong solutions. In particular, there is no contact of the beam and bottom, which is built on a uniform distance bound derived by the mechanical energy and fluid dissipation through a height-dependent degenerate elliptic regularization of the classical distance test. Combining tangential differentiation, a fractional estimate for the height, and a negative-order tangential estimate for the accelerations, we establish a global regularity estimate. Moreover, quadratic time-weighted estimates that quantify the higher regularity at positive times are justified. As a consequence, the work gives an affirmative answer to the problem proposed by Grandmont and Hillairet (Arch. Ration. Mech. Anal. 220(3): 1283--1333, 2016) of whether fluid dissipation alone is sufficient to derive the distance estimate.