Strong Solutions to the Navier-Stokes Equations Around a Rotating Obstacle

Galdi, Giovanni P.; Silvestre, Ana L.

doi:10.1007/s00205-004-0348-z

Cited by 63 publications

(52 citation statements)

References 9 publications

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“…As for the nonstationary problem, the existence of global weak solutions is proved by Borchers [2], and the unique existence of time-local regular solutions is shown by Hishida [20] and Geissert, Heck, and Hieber [18], while the global strong solutions for small data are obtained by Galdi and Silvestre [15]. The spectrum of the linear operator arising in this problem is studied by Farwig and Neustupa [10]; see also the linear analysis by Hishida [21].…”

Section: Theorem 13 There Exists a Positive Constant δ Such That If mentioning

confidence: 99%

“…On th other hand, the asymptotic profile of these steady flows at spatial infinity is studied by Farwig and Hishida [8,9] and Farwig, Galdi, and Kyed [6]. The stability of these steady solutions is proved in [15] and Hishida and Shibata [23] in the L 2 and L q functional framework, respectively. All results mentioned above are in the three-dimensional case, and there are still few results for the flow around a rotating obstacle in the two-dimensional case.…”

Section: Theorem 13 There Exists a Positive Constant δ Such That If mentioning

confidence: 99%

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On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

Maekawa

2017

Arch Rational Mech Anal

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We study the stability of some exact stationary solutions to the two-dimensional Navier-Stokes equations in an exterior domain to the unit disk. These stationary solutions are known as a simple model of the flow around a rotating obstacle, while their stability has been open due to the difficulty arising from their spatial decay in a scale-critical order. In this paper we affirmatively settle this problem for small solutions. That is, we will show that if these exact solutions are small enough then they are asymptotically stable with respect to small L 2 perturbations.Keywords Navier-Stokes equations · two-dimensional exterior flows · scale-criticality · stability of stationary flows · flow around a rotating obstacle

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Section: Theorem 13 There Exists a Positive Constant δ Such That If mentioning

confidence: 99%

Section: Theorem 13 There Exists a Positive Constant δ Such That If mentioning

confidence: 99%

On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

Maekawa

2017

Arch Rational Mech Anal

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“…On the existence, uniqueness and regularity of solutions of (1), there are many papers, such as [1][2][3][4][5][6][7][8] and references therein. However, few results about the numerical analysis are developed.…”

Section: Introductionmentioning

confidence: 99%

The boundary integral method for the steady rotating Navier–Stokes equations in exterior domain (I): the existence of solution

2009

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper, we apply the boundary integral method to the steady rotating Navier-Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and a infinite domain, we obtain a coupled problem by the steady rotating Navier-Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence of solution in a convex set. Mathematics Subject Classification (2000). 76M15, 76U05.

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“…The classic paper of Weinberger [9] concerning the steady fall of a body in a Navier-Stokes liquid starts with the following definition: "We say a body undergoes a steady falling motion in an infinite viscous fluid if the motion of the fluid as seen by an observer attached to the body is independent of time. "One of the interesting possible cases is a body that is falling steadily, and is rotating around an axis that is parallel to the direction in which the body is falling.A first proof of the existence of such solutions for this case has been given only recently in the three papers by Galdi and Silvestre [4,3,2]. Their method for solving the problem is to consider the equations, as proposed by Weinberger, in a frame attached to the body, where the flow is stationary.…”

mentioning

confidence: 99%

“…A first proof of the existence of such solutions for this case has been given only recently in the three papers by Galdi and Silvestre [4,3,2]. Their method for solving the problem is to consider the equations, as proposed by Weinberger, in a frame attached to the body, where the flow is stationary.…”

mentioning

confidence: 99%

Time Periodic Solutions of the Navier–Stokes Equations with Nonzero Constant Boundary Conditions at Infinity

Baalen

Wittwer

2011

SIAM J. Math. Anal.

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Abstract. We construct solutions for the Navier-Stokes equations in three dimensions with a time periodic force which is of compact support in a frame that moves at constant speed. These solutions are related to solutions of the problem of a body which moves within an incompressible fluid at constant speed and rotates around an axis which is aligned with the motion. In contrast to other authors who analyze stationary solutions in a frame of reference attached to the body, the analysis for the present problem is done in a frame which is moving at constant speed but is not rotating. This avoids the unpleasant unbounded linear terms which are present in a description with respect to a rotating frame. 1. Introduction. The classic paper of Weinberger [9] concerning the steady fall of a body in a Navier-Stokes liquid starts with the following definition: "We say a body undergoes a steady falling motion in an infinite viscous fluid if the motion of the fluid as seen by an observer attached to the body is independent of time."One of the interesting possible cases is a body that is falling steadily, and is rotating around an axis that is parallel to the direction in which the body is falling.A first proof of the existence of such solutions for this case has been given only recently in the three papers by Galdi and Silvestre [4,3,2]. Their method for solving the problem is to consider the equations, as proposed by Weinberger, in a frame attached to the body, where the flow is stationary. In this frame the Navier-Stokes equations have an additional linear term with unbounded coefficients, which is due to the transformation into the rotating frame. This complicates the problem considerably when compared to the situation without rotation.It is important to note that even without the rotation the problem is difficult because of the slow decay of the vorticity in the downstream region. This leads to a very strong asymmetry in the behavior at infinity, and the main difficulty is to encode this behavior when choosing function spaces.Once the existence of a solution is established one is interested in giving detailed information concerning its behavior at infinity. As in related problems, this behavior is expected to be independent of the details of the body. It turns out to be possible to use this fact in order to simplify the analysis of the asymptotic behavior, by considering first the problem in the whole space, and to mimic the body by a smooth force of compact support (see the end of this section for details). The case with a body, i.e., the case of an exterior domain, can then be treated in a second step, once the behavior at infinity is understood. For a related problem in two dimensions, this strategy has

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Strong Solutions to the Navier-Stokes Equations Around a Rotating Obstacle

Cited by 63 publications

References 9 publications

On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

The boundary integral method for the steady rotating Navier–Stokes equations in exterior domain (I): the existence of solution

Time Periodic Solutions of the Navier–Stokes Equations with Nonzero Constant Boundary Conditions at Infinity

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