Well-posedness of the motion of a rigid body immersed in a compressible inviscid fluid

We consider a rigid body freely moving in a compressible inviscid fluid within a bounded domain $Ω\subset\mathbb{R}^3$. The fluid is thereby governed by the non necessarily isentropic compressible Euler equations, while the rigid body obeys the conservation of linear and angular momentum. This forms a coupled system comprising an ODE and the initial boundary value problem (IBVP) of a hyperbolic system with characteristic boundary in a moving domain, where the fluid velocity matches the solid velocity along the normal direction of the solid boundary. We establish the existence of a unique local classical solution to this coupled system. To construct the solution, we first perform a change of variables to reformulate the problem in a fixed spatial domain, and then analyze an approximate system with a non-characteristic boundary. For this nonlinear approximate system, we use the better regularity for the trace of the pressure on the boundary to contruct a solution by a fixed-point argument in which the fluid motion and the solid motion are updated in successive steps. We are then able to derive estimates independent of the regularization parameter and to pass to the limit by a strong compactness arguments.

💡 Research Summary

The paper addresses the initial‑boundary value problem describing the interaction between a compressible inviscid fluid and a rigid body fully immersed in it, both confined in a bounded three‑dimensional domain Ω. The fluid dynamics are governed by the non‑isentropic compressible Euler equations written in terms of pressure p, velocity u, and entropy s, while the rigid body obeys Newton’s laws for linear and angular momentum. The coupling occurs through the kinematic condition that the normal component of the fluid velocity equals that of the solid on the solid’s surface, together with an impermeability condition on the outer boundary ∂Ω.

A major difficulty stems from the fact that the fluid domain F(t)=Ω∖S(t) moves with the solid, and the boundary conditions are characteristic for the hyperbolic Euler system. Classical well‑posedness results for hyperbolic IBVPs rely on non‑characteristic boundaries, so a direct treatment is impossible. The authors overcome this by first constructing an artificial velocity field that interpolates smoothly between the solid velocity on ∂S(t) and zero on ∂Ω. The flow map generated by this field provides a time‑dependent diffeomorphism Φ(t,·) which transports the moving fluid region onto a fixed reference domain F(0). Under this change of variables the equations retain a quasilinear hyperbolic structure but now live on a stationary spatial domain.

Even after fixing the domain, the boundary remains characteristic. To regain control of the pressure trace, the authors introduce a non‑characteristic regularization: a small parameter ε>0 is added to the fluid equations as a transport term that makes the normal component of the characteristic field non‑degenerate. The resulting ε‑regularized system admits the standard energy estimates for first‑order hyperbolic systems with non‑characteristic boundaries.

The core of the analysis is a high‑order a‑priori estimate. Using the conserved total energy, the authors first bound tangential and time derivatives of (p,u,s) in conormal Sobolev spaces X^m. To recover normal derivatives, they exploit the transport equations satisfied by entropy and vorticity, together with the structure of the Euler equations, converting time and tangential derivatives into normal ones for the acoustic variables near the boundary. This yields uniform bounds (independent of ε) for all derivatives up to order m≥3, as well as C^m bounds for the solid translational velocity l(t) and angular velocity ω(t), which are obtained from the pressure trace via the solid momentum equations.

With these uniform estimates, a fixed‑point scheme is set up. For a given solid trajectory, the regularized fluid problem is solved; the resulting pressure trace is then used to update the solid ODEs. The mapping from the guessed trajectory to the updated one is shown to be a contraction on a sufficiently small time interval, providing existence and uniqueness of a solution to the ε‑regularized coupled system.

Finally, letting ε→0, the authors employ strong compactness arguments (Aubin–Lions lemma, Rellich–Kondrachov theorem) to extract a convergent subsequence. Because the estimates are ε‑independent, the limit satisfies the original characteristic system, thereby establishing the existence of a unique local classical solution to the original fluid–solid interaction problem.

The main theorem states that for initial data (p₀,u₀,s₀,l₀,ω₀) in H^m(F(0))³×ℝ³×ℝ³, satisfying compatibility conditions up to order m−1 and lying inside a physically admissible state set U (where density and sound speed are positive), there exists a time T>0 such that the transformed unknowns belong to X^m(