On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy

Bravin, Marco

doi:10.5802/crmath.36

Cited by 6 publications

(6 citation statements)

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“…Notice that a velocity field that has non-zero circulation around the body behaves as 1/|x| when |x| goes to +∞, in particular it is not L 2 so it does not enter in the theory of Hopf-Leray weak solutions, see also the comments in section 2 of [28]. Let recall that an existence result in this direction is available in [3] where the author considers initial data for the velocity field of the type Let now move to the proof of Theorem 3. The remaining part of the paper is divided in two main sections.…”

Section: Theorem 3 Let (U ε ε ω ε ) Be a Sequence Of Hopf-leray Sol...mentioning

confidence: 99%

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On the Velocity of a Small Rigid Body in a Viscous Incompressible Fluid in Dimension Two and Three

Bravin

Nečasová

2023

J Dyn Diff Equat

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In this paper we study the evolution of a small rigid body in a viscous incompressible fluid, in particular we show that a small particle is not accelerated by the fluid in the limit when its size converges to zero under a lower bound on its mass. This result is based on a new a priori estimate on the velocities of the centers of mass of rigid bodies that holds in the case when their masses are also allowed to decrease to zero.

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Section: Theorem 3 Let (U ε ε ω ε ) Be a Sequence Of Hopf-leray Sol...mentioning

confidence: 99%

“…The existence of weak solutions for the system (1)-( 2)-( 3) is now classical and can be found for example in [15]- [35]- [32]- [9]. Theorem 1 For initial data S in , ρ in , u in , in and ω in satisfying the hypothesis (4) and such that ũin ∈ L 2 (R d ), there exist a Hopf-Leray weak solution (u, , ω) of the system (1)-( 2)- (3).…”

Section: Introductionmentioning

confidence: 98%

On the Velocity of a Small Rigid Body in a Viscous Incompressible Fluid in Dimension Two and Three

Bravin

Nečasová

2023

J Dyn Diff Equat

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“…1 In the case of the fluid -rigid body system theory is much less developed. 2D case is studied in [2,3,19] where existence and uniqueness of global weak solution is proved provided that rigid body does not touch the boundary. Moreover, they show that these solutions are strong away from t = 0.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Moreover, global-in-time existence (or existence up to the time of contact) of Leray -Hopf type weak solution were studied in [6,9,14,21,13]. We also mention the existence results in the case of slip boundary conditions [17,5,2,3,34]. Uniqueness of weak solutions is still an open problem, but results of weak-strong uniqueness type were proved in both slip and no-slip case ( [4,20]) which state that the weak solution satisfying additional condition on the fluid velocity is unique in the larger class of weak solutions.…”

Section: Literature Reviewmentioning

confidence: 99%

On the regularity of weak solutions to the fluid-rigid body interaction problem

Muha¹,

Nečasová²,

Ana³

2022

Preprint

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We study a 3D fluid-rigid body interaction problem. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations describing conservation of linear and angular momentum. Our aim is to prove that any weak solution satisfying certain regularity conditions is smooth. This is a generalization of the classical result for the 3D incompressible Navier-Stokes equations, which says that a weak solution that additionally satisfy Prodi -Serrin L r − L s condition is smooth. We show that in the case of fluid -rigid body the Prodi -Serrin conditions imply W 2,p and W 1,p regularity for the fluid velocity and fluid pressure, respectively. Moreover, we show that solutions are C ∞ if additionally we assume that the rigid body acceleration is bounded almost anywhere in time variable. * The research of B.M. leading to these results has been supported by Croatian Science Foundation under the project IP-2018-01-3706† The research of Š.N. leading to these results has received funding from the Czech Sciences Foundation (GA ČR), 22-01591S. Moreover, Š. N. has been supported by Praemium Academiae of Š. Nečasová. CAS is supported by RVO:67985840.‡ The research of A.R. leading to these results has been supported by Croatian Science Foundation under the project IP-2019-04-1140. Moreover, The research of A.R. leading to these results has received funding from the Czech Sciences Foundation (GA ČR) 22-01591S, and by Praemium Academiae of Š. Nečasová.

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“…But, as we shall see below (see Theorem 20), the reasoning extends to L 2 initial data. An alternative approach relying on Leraytype arguments is also provided in [1] in case of Navier-type slip boundary conditions on the fluid/solid interface. Since our results rely strongly on decay estimates of the semi-group A we shall stick to this mild-solution approach herein.…”

Section: Introductionmentioning

confidence: 99%

Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data

Ferriere¹,

Hillairet²

2023

Comptes Rendus. Mathématique

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In this paper, we analyse the long-time behavior of solutions to a coupled system describing the motion of a rigid disk in a 2D viscous incompressible fluid. Following previous approaches in [4,15,17] we look at the problem in the system of coordinates associated with the center of mass of the disk. Doing so, we introduce a further nonlinearity to the classical Navier Stokes equations. In comparison with the classical nonlinearities, this new term lacks time and space integrability, thus complicating strongly the analysis of the long-time behavior of solutions.We provide herein two refined tools: a refined analysis of the Gagliardo-Nirenberg inequalities and a thorough description of fractional powers of the so-called fluid-structure operator [2]. On the basis of these two tools we extend decay estimates obtained in [4] to arbitrary initial data and show local stability of the Lamb-Oseen vortex in the spirit of [7,8].

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On the 2D “viscous incompressible fluid + rigid body” system with Navier conditions and unbounded energy

Abstract: On the 2D "viscous incompressible fluid + rigid body" system with Navier conditions and unbounded energy Sur le mouvement d'un corps rigide dans un écoulement bidimensionel d'un fluide visqueux incompressible avec conditions au bord de Navier et énergie infinie

Cited by 6 publications

References 15 publications

On the Velocity of a Small Rigid Body in a Viscous Incompressible Fluid in Dimension Two and Three

On the Velocity of a Small Rigid Body in a Viscous Incompressible Fluid in Dimension Two and Three

On the regularity of weak solutions to the fluid-rigid body interaction problem

Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data

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