L p -L q Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle

Hishida, Toshiaki; Shibata, Yoshihiro

doi:10.1007/s00205-008-0130-8

Cited by 83 publications

(80 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result is actually used in the study of the stability of the stationary solution, see Hishida and Shibata [29,Section 9]. q; y ðDÞ, we see from (2.7) that kv n wk 3=2; y þ kv n wk q; y a kvk 3; y ðkwk 3; y þ kwk q Ã ; y Þ ð6:9Þ This combined with (6.13) implies the estimate of ðu; pÞ ¼ ðv þ b; pÞ in (6.7), which completes the proof.…”

Section: Navier-stokes Problemmentioning

confidence: 98%

“…When o is small enough and f ¼ div F satisfies the decay estimates jxj 2 jF ðxÞj þ jxj 3 j f ðxÞj þ jxj 4 jdiv f ðxÞj a c 0 for some small c 0 > 0, Galdi [19] derived the pointwise estimates jxj juðxÞj þ jxj 2 ðj'uðxÞj þ j pðxÞjÞ þ jxj 3 j'pðxÞj a C ð1:4Þ of a unique solution. These decay properties are important in the study of stability ( [3], [20], [29]), but, at first glance, rather surprising in view of (1.1). In [21] Galdi and Silvestre have recently extended a part of [19]; that is, they have shown jxj juðxÞj a C for a small force f ¼ div F A L 2 with jxj 2 jF ðxÞj a c 0 when the translation of the obstacle is also taken into account.…”

Section: Introductionmentioning

confidence: 99%

“…We finally mention that the stability of the stationary flow obtained in Theorem 1.1 is studied by [29] within the same framework of Lorentz spaces as adopted in the present paper. This paper is organized as follows.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Stationary Navier-Stokes Flow Around a Rotating Obstacle

Farwig

Hishida

2007

Self Cite

View full text Add to dashboard Cite

Abstract. Consider a viscous incompressible fluid filling the whole 3-dimensional space exterior to a rotating body with constant angular velocity o. By using a coordinate system attached to the body, the problem is reduced to an equivalent one in a fixed exterior domain. The reduced equation involves the crucial drift operator ðo5xÞ Á ', which is not subordinate to the usual Stokes operator. This paper addresses stationary flows to the reduced problem with an external force f ¼ div F , that is, time-periodic flows to the original one. Generalizing previous results of G. P. Galdi [19] we show the existence of a unique solution ð'u; pÞ in the class L 3=2; y when both F A L 3=2; y and o are small enough; here L 3=2; y is the weak-L 3=2 space.

show abstract

Section: Navier-stokes Problemmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stationary Navier-Stokes Flow Around a Rotating Obstacle

Farwig

Hishida

2007

Self Cite

View full text Add to dashboard Cite

show abstract

“…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%

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

Maekawa

2017

Arch Rational Mech Anal

View full text Add to dashboard Cite

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

show abstract

“…Later Hishida [11] constructed local mild solutions to the Navier-Stokes problem in the exterior of a rotating obstacle in the context of L 2 by using semigroup techniques (see also [12]). This existence result was extended to the general L p -theory by Geissert et al [8] and Hishida and Shibata [14] showed that this solution is even a global one, provided the data are small enough. However, there are only a few partial results for the case when the fluid flow is subject to an additional outflow condition at infinity (hereby we mean a prescribed velocity of fluid at infinity).…”

Section: Introductionmentioning

confidence: 98%

On the Navier–Stokes equations with rotating effect and prescribed outflow velocity

Hansel

2011

J. Math. Fluid Mech.

View full text Add to dashboard Cite

We consider the equations of Navier-Stokes modeling viscous fluid flow past a moving or rotating obstacle in R d subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use L p -techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in R d is solved by an evolution system on L p σ (R d ) for 1 < p < ∞. For this we use results about time-dependent Ornstein-Uhlenbeck operators. Finally, we prove, for p ≥ d and initial data u 0 ∈ L p σ (R d ), the existence of a unique mild solution to the full Navier-Stokes system. (2000). Primary 35Q30; Secondary 76D03, 76D05. Mathematics Subject Classification

show abstract

L p -L q Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle

Cited by 83 publications

References 45 publications

Stationary Navier-Stokes Flow Around a Rotating Obstacle

Stationary Navier-Stokes Flow Around a Rotating Obstacle

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

On the Navier–Stokes equations with rotating effect and prescribed outflow velocity

Contact Info

Product

Resources

About