A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity

Mazzone, Giusy; Prüß, Jan; Simonett, Gieri

doi:10.1007/s00021-019-0449-y

Cited by 17 publications

(29 citation statements)

References 23 publications

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“…One readily verifies that K has finite rank, and then it defines a compact linear operator on L q,σ (Ω). In [18], it has been also proved that (I + K) is invertible on L q,σ (Ω).…”

Section: Local Well-posedness and Critical Spacesmentioning

confidence: 96%

“…The mathematical model features a combination of conservative and dissipative properties as can be observed by the conservation of angular momentum (1.3) and the energy inequality (4.5). These are distinctive characteristics for this type of fluid-solid interactions (see also [5,17,8,18]). We prove the existence of weak solutions á la Leray-Hopf to (1.1) corresponding to initial data with arbitrary (finite) kinetic energy.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

“…In contrast, there is a large literature dealing with the motion of fluid-filled rigid bodies with no-slip boundary conditions, it spans from the early work by Stokes [38], Zhukovskii [42], Hough [11], Poincaré [26], and Sobolev [35] to more recent contributions mostly concerned with stability problems ( [32,33,21,4,15,13,12,34,16,9,5,17]). A comprehensive study of the motion of fluid-filled rigid bodies has recently been given in [8] (in an L 2 framework), and in [18] (in a more general L q framework). It is shown that equilibria correspond to permanent rotations (rotations with constant angular velocity around the central axes of inertia) of S with the fluid at a relative rest with respect to the solid.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

“…Equation (1.1) 4 represents a partial-slip boundary condition. Dividing both sides of (1.1) 4 by ζ and taking the limit as ζ → ∞ one formally obtains the case of no-slip boundary condition v = 0 on Γ considered in [18]. Throughout this paper, we use the notation (·|·) for the Euclidean inner product in R 3 and | · | for the associated norm.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

See 3 more Smart Citations

On the Motion of a Fluid-Filled Rigid Body with Navier Boundary Conditions

Mazzone¹,

Prüß²,

Simonett³

2019

SIAM J. Math. Anal.

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We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution à la Leray-Hopf to the equations of motion converges to an equilibrium at an exponential rate in the Lq-topology for every fluid-solid configuration.A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology.

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“…One readily verifies that K has finite rank, and then it defines a compact linear operator on L q,σ (Ω). In [18], it has been also proved that (I + K) is invertible on L q,σ (Ω).…”

Section: Local Well-posedness and Critical Spacesmentioning

confidence: 96%

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

See 2 more Smart Citations

On the Motion of a Fluid-Filled Rigid Body with Navier Boundary Conditions

Mazzone¹,

Prüß²,

Simonett³

2019

SIAM J. Math. Anal.

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“…However, it is only over the past few years, that a rigorous mathematical analysis has been initiated, with the objective of investigating a fundamental property of such coupled systems, namely, the characterization of their "ultimate dynamics" [24,11,5,12,15,9,14,21]. In fact, as shown by both experiment and qualitative analysis [27,3,4], the viscous liquid acts as a damper on the rigid body to the point, in some cases, of even bringing it to rest (see [13,15] for a rigorous mathematical explanation).…”

Section: Introductionmentioning

confidence: 99%

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

Galdi

Mácha

Nečasová

2020

International Journal of Non-Linear Mechanics

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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...

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$$H^\infty $$-Calculus for the Surface Stokes Operator and Applications

Simonett

Wilke

2022

J. Math. Fluid Mech.

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We consider a smooth, compact and embedded hypersurface $$\Sigma $$ Σ without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded $$H^\infty $$ H ∞ -calculus with angle smaller than $$\pi /2$$ π / 2 . As an application, we consider critical spaces for the Navier–Stokes equations on the surface $$\Sigma $$ Σ . In case $$\Sigma $$ Σ is two-dimensional, we show that any solution with a divergence-free initial value in $$L_2(\Sigma , \textsf{T}\Sigma )$$ L 2 ( Σ , T Σ ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.

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A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity

Cited by 17 publications

References 23 publications

On the Motion of a Fluid-Filled Rigid Body with Navier Boundary Conditions

On the Motion of a Fluid-Filled Rigid Body with Navier Boundary Conditions

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

$$H^\infty $$-Calculus for the Surface Stokes Operator and Applications

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