On weak solutions to the problem of a rigid body with a cavity filled with a compressible fluid, and their asymptotic behavior

Abstract

We prove the existence of a weak solution to the equations describing the inertial motions of a coupled system constituted by a rigid body containing a viscous compressible fluid. We then provide a weak-strong uniqueness result that allows us to completely characterize, under certain physical assumptions, the asymptotic behavior in time of the weak solution corresponding to smooth data of restricted "size," and show that it tends to a uniquely determined steady-state. 2"small" in size. As explained in the introductory section in [10] the analogous study in the larger class of weak solution corresponding to data possessing just finite energy appears, currently, out of reach. The main reason is ascribed to the fact that a necessarily detailed study of the Ω-limit requires that weak solutions become eventually smooth enough, a property that, while true in the incompressible case [5], need not be generally valid if the fluid is compressible [16].This drawback notwithstanding, one may wonder if, at least, a weak solution with smooth and sufficiently "small" initial data could satisfy the above property. The latter is certainly true, if one can show a weak-strong uniqueness result, which would, in particular, ensure that the weak solution is strong for all positive times.Objective of this article is to prove that this is, in fact, the case. More precisely, we show that, for generic initial data with finite energy, there corresponds at least one weak solution (see Theorem 3.2) suitably defined (see Definition 3). Furthermore, we show that if there is a strong solution (also suitably defined) possessing the same data, then necessarily it should coincide with the weak one (see Theorem 5.2). In view of this result, we may then conclude that, for smooth and "small" initial data, the weak solution must coincide with that constructed and used in [10], and this will ensure that, as established in [10], under the assumption of suitable "mass distribution" and "small" Mach numbers, the ultimate state reached by the weak solution must be a uniform rotation around a central axis of inertia.The plan of the paper is the following. In Section 2 we give a formulation of the problem and recall the relevant equations. Section 3 is dedicated to the definition of weak solutions and the proof of their existence (Theorem 3.2). In achieving the latter, we appropriately adjust to the case at hand an approximation procedure developed in [23,6]. Successively, in Section 4, we state results of local and global existence and uniqueness of strong solutions proved in [10]; see Theorems 4.1 and 4.2. The following Section 5 is devoted to the proof that the weak and the strong solution, dealt with before, corresponding to the same data must, in fact, coincide; see Theorem 5.2. In the proof of this result, we adapt the arguments of [7,18,2]; see also [18]. Once the weak-strong coincidence has been established, we dedicate the last two sections to recall and formulate the results proved in [10] concerning the asymptotic behavior of a strong solution for...

Measure-valued solutions and weak-strong uniqueness for the incompressible inviscid fluid-rigid body interaction

“…In a case of rigid body with a cavity filled by incompressible fluid the weak-strong uniqueness was shown in [11]. For an analogous result for a cavity filled by compressible fluid see [14]. We also refer to [9], [26] for problems on moving domains.…”

Moving boundary problems

“…Usually, to handle this difficulty, the problem is mapped onto a fixed domain using a mapping that depends on the regularity of solutions, so strong solutions are easier to deal with. In 2015, however, uniqueness of weak solution for a fluid-rigid body system in the two-dimensional case was obtained by Glass and Sueur in [65] for the no-slip case, and by Bravin in 2019 for the slip case [17], while uniqueness results of weak-strong type were recently published in [27,40,54]. In [40] the authors studied a rigid body with its cavity filled with fluid, while in [27] a rather high regularity for strong solutions was required for the uniqueness result to hold (the time derivative and second spatial derivatives of the fluid velocity were required to be in L 2 ).…”

“…In [40] the authors studied a rigid body with its cavity filled with fluid, while in [27] a rather high regularity for strong solutions was required for the uniqueness result to hold (the time derivative and second spatial derivatives of the fluid velocity were required to be in L 2 ). In [54] Galdi, Mácha, and Nečasová studied a rigid body with its cavity filled with a compressible fluid and showed a weak-strong uniqueness property using a relative entropy inequality. The most resent result by Muha, Nečasová, and Radošević [108] generalizes these results, since they prove, for both the slip and no-slip cases, a generalization of the well-known weak-strong uniqueness result for the Navier-Stokes equations to the fluid-rigid body system.…”

On the Motion of a Pendulum with a Cavity Filled with a Compressible Fluid

“…At this point, it must be emphasized that in all the papers indicated above, the fluid is supposed to be viscous and incompressible. Thus, more recently, in [13,14] we began to investigate the case where the fluid is still viscous but compressible. This study has a two-fold motivation.…”