My research field is computational fluid-structure interaction. Mathematically it belongs to free boundary problem. I know there are some closed-form solutions for the elasticity equation and the Navier-Stokes equation [1]. But I don't know one closed-form solution for fluid-structure interaction or more generally free-boundary problem. So I want to ask my friends: is there any closed-form solution for a free boundary problem? (and it is better if the solution is related to fluid-structure interaction.)
Bibliography:
[1]. Masatsuka, Katate. I do Like CFD, vol. 1. Vol. 1. Lulu. com, 2013. I do like CFD
From my understanding, finding closed form solutions for PDEs arising from fluid mechanics are extremely impossible (let alone for free boundary problems), even the global existence and uniqueness of (weak) solutions for compressible/incompressible Euler's equations or N-S equation are still not well understood in general among the fluid community.