On the interaction between a rigid-body and a viscous-fluid: existence of a weak solution and a suitable Théorème de Structure
Paolo Maremonti, Filippo Palma
TL;DR
This work extends the classical Leray structure theory to the fluid–rigid body interaction problem by working in a body-attached frame, where the term $\omega\times x\cdot \nabla u$ poses novel analytical challenges. The authors construct a weak solution via mollified convective terms and an invading-domain Galerkin framework, obtaining a Hopf-type energy inequality and uniform estimates that enable a weak limit to solve the original IBVP for $t>0$. A key outcome is a Leray-type partial regularity result: after a finite time $\theta$, the solution becomes regular in the usual Navier–Stokes sense, while the finite-time interval $[0,\theta)$ may remain nonregular and is analyzed using vorticity estimates and extension arguments. The results rely on detailed a priori bounds, vorticity propagation outside moving balls, and a carefully constructed extension scheme that guarantees convergence to a weak solution satisfying the energy inequality and a Large-Time regularity regime. Overall, the paper provides a rigorous framework for existence and partial regularity in fluid–structure interaction models, highlighting how mollification and energy methods can yield Leray-type structure results in exterior-domain FSI problems with rigid bodies.
Abstract
In this paper, we prove the existence and a partial regularity of a weak solution to the system governing the interaction between a rigid body and a viscous incompressible Newtonian fluid. The evolution of the system body-fluid is studied in a frame attached to the body. The choice of this special frame becomes critical from an analytical point of view due to the presence of the term $ω\times x\cdot\nabla u$ in the balance of momentum equation for the fluid. As a consequence, we are forced to look for a technique that is different from the ones usually employed both for the existence and for the partial regularity of a weak solution to the Navier-Stokes problem. Hence, we prove the existence of a weak solution in an original way and give a new proof of the celebrated Théorème de Structure due to Leray. However, the regularity obtained for our weak solution is only for large times, hence our result is weaker compared to the one obtained by Leray.