On the existence of vanishing at infinity symmetric solutions to the plane stationary exterior Navier–Stokes problem

Pileckas, Konstantin; Russo, Roberto

doi:10.1007/s00208-011-0653-4

Cited by 42 publications

(45 citation statements)

References 20 publications

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“…For the case

u_{\infty} = 0

with external force, A. Russo obtained the existence of weak solutions, if

| α |

is sufficiently small, where α is the total flux from the obstacle defined in (2.14). In particular, if the following outflow condition is satisfied:

\oint_{normalΓ} n (x) \cdot a (x) d s (x) = 0 .

Next, Galdi [, Section ], A. Russo and Pileckas and R. Russo posed the assumption

\begin{matrix} true \\ left The set normalΩ is invariant under the mappings \\ left π_{1} : (x_{1}, x_{2}) \mapsto (- x_{1}, x_{2}), π_{2} : (x_{1}, x_{2}) \mapsto (x_{1}, - x_{2}) \end{matrix}

with a specific coordinate variables

(x_{1}, x_{2})

, and the external force

f (x) = (f_{1} (x), f_{2} (x))

is assumed to satisfy the condition

({, \begin{matrix} f_{1} (- x_{1}, x_{2}) = - f_{1} (x_{1}, x_{2}), & f_{2} (- x_{1}, x_{2}) = f_{2} (x_{1}, x_{2}), \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

Two‐dimensional stationary Navier–Stokes equations with 4‐cyclic symmetry

Yamazaki

2016

Mathematische Nachrichten

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This paper is concerned with the stationary Navier–Stokes equation in the whole plane and in the two–dimensional exterior domain invariant under the action of the cyclic group of order 4, and gives a condition on the potentials yielding the external force, and on the boundary value, sufficient for the unique existence of a small solution equivariant with respect to the aforementioned cyclic group.

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“…For the case

u_{\infty} = 0

with external force, A. Russo obtained the existence of weak solutions, if

| α |

is sufficiently small, where α is the total flux from the obstacle defined in (2.14). In particular, if the following outflow condition is satisfied:

\oint_{normalΓ} n (x) \cdot a (x) d s (x) = 0 .

Next, Galdi [, Section ], A. Russo and Pileckas and R. Russo posed the assumption

\begin{matrix} true \\ left The set normalΩ is invariant under the mappings \\ left π_{1} : (x_{1}, x_{2}) \mapsto (- x_{1}, x_{2}), π_{2} : (x_{1}, x_{2}) \mapsto (x_{1}, - x_{2}) \end{matrix}

with a specific coordinate variables

(x_{1}, x_{2})

, and the external force

f (x) = (f_{1} (x), f_{2} (x))

is assumed to satisfy the condition

({, \begin{matrix} f_{1} (- x_{1}, x_{2}) = - f_{1} (x_{1}, x_{2}), & f_{2} (- x_{1}, x_{2}) = f_{2} (x_{1}, x_{2}), \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

Two‐dimensional stationary Navier–Stokes equations with 4‐cyclic symmetry

Yamazaki

2016

Mathematische Nachrichten

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“…Contrary to the higher dimensional cases, less is known so far for the case of two-dimensional exterior domains. Indeed, for the two-dimensional exterior problem the unique existence of stationary flows decaying at spatial infinity has been achieved mainly under some symmetry conditions on both the domain and given data; see Galdi [14], Russo [41], Yamazaki [46], and Pileckas and Russo [40], and see also Nakatsuka [38] for a recent uniqueness result. In particular, Yamazaki [46] obtained the decay order O(|x| −1 ) for the stationary solutions constructed there.…”

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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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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“…in [27] it is showed that a symmetric D-solution (1.13) to (1.2), uniformly vanishing at infinity, exists under the only natural assumption that a satisfies (1.13) and natural regularity conditions. Note that (1.13) meets the mean property (1.12) with u 0 = 0.…”

Section: Introductionmentioning

confidence: 99%