The 3D Time-Dependent Oseen System: Link Between $$L^p$$-Integrability in Time and Pointwise Decay in Space of the Velocity

Deuring, Paul

doi:10.1007/s00021-021-00562-6

Cited by 6 publications

(37 citation statements)

References 38 publications

(95 reference statements)

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“…where the right-hand side is understood as C|x| −2 for all x ∈ R [7], then we see that his estimate for |x| > 2R is also valid for our integral. But it would be better to give a brief sketch of his proof for convenience of readers.…”

Section: Assume Further Thatsupporting

confidence: 61%

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Spatial pointwise behavior of time-periodic Navier-Stokes flow induced by oscillation of a moving obstacle

Hishida

2022

Preprint

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We study the spatial decay of time-periodic Navier-Stokes flow at the rate |x| −1 with/without wake structure in 3D exterior domains when a rigid body moves periodically in time. In this regime the existence of time-periodic solutions was established first in the 2006 paper by Galdi and Silvestre, however, with little information about spatial behavior at infinity so that uniqueness of solutions was not available. This latter issue has been addressed by Galdi, who has recently succeeded in construction of a unique time-periodic solution with spatial behavior mentioned above if translational and angular velocities of the body fulfill, besides smallness and regularity, either of the following assumptions: (i) translation or rotation is absent; (ii) both velocities are parallel to the same constant vector. This paper shows the existence of a unique time-periodic Navier-Stokes flow in the small with values in the weak-L 3 space and then deduces the desired pointwise decay of the solution under some condition on the rigid motion of the body, that covers the cases (i), (ii) mentioned above.

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Section: Assume Further Thatsupporting

confidence: 61%

“…We begin with some auxiliary estimates, which are essentially known as estimates of the Oseen fundamental solution E(x + ζt, t) and some related potentials, see [7,9,12,15,24,25,47,53,55]. for all x ∈ R 3 with some constant C = C(|ζ|) > 0.…”

Section: Reconstruction Proceduresmentioning

confidence: 99%

Spatial pointwise behavior of time-periodic Navier-Stokes flow induced by oscillation of a moving obstacle

Hishida

2022

Preprint

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“…A link between the rate of spatial decay of ∂ α x u on the one hand and L pintegrability of f with respect to time on the other already appears in [7,Theorem 6.1] and [8,Theorem 5.2], but with certain L p -norms in time and in space of u additionally influencing the spatial asymptotics of u. The former reference specifies the spatial decay of regular solutions to (1.1), and the latter one extends these results to L q -weak solutions of (1.1).…”

mentioning

confidence: 95%

“…The former reference specifies the spatial decay of regular solutions to (1.1), and the latter one extends these results to L q -weak solutions of (1.1). No specific boundary conditions are imposed in [7,8]. A simplified version of [8,Theorem 5.2] is stated below as Theorem 2.7, which is the starting point of the work at hand.…”

mentioning

confidence: 99%

“…A simplified version of [8,Theorem 5.2] is stated below as Theorem 2.7, which is the starting point of the work at hand. In [7,8], we applied [7,Theorem 6.1] and [8,Theorem 5.2], respectively, to some solutions which are known to exist; see [7,Theorem 6.2,6.3] and [8,Theorem 6.1,6.2,6.3]. However, these examples are either restricted to an L 2 -framework ([7, Theorem 6.2] and [8, Theorem 6.1, 6.2, 6.3]), or they deal with solutions whose lifespan T 0 ∈ (0, ∞) is finite, which are much more regular than required for our theory and are associated with a right-hand side f ∈ L p 0, T 0 , L q (Ω c ) 3 required to satisfy the condition p = q ([7, Theorem 6.3]).…”

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Spatial asymptotics of mild solutions to the time-dependent Oseen system

Deuring¹

2021

CPAA

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We consider mild solutions to the 3D time-dependent Oseen system with homogeneous Dirichlet boundary conditions, under weak assumptions on the data. Such solutions are defined via the semigroup generated by the Oseen operator in L q . They turn out to be also L q -weak solutions to the Oseen system. On the basis of known results about spatial asymptotics of the latter type of solutions, we then derive pointwise estimates of the spatial decay of mild solutions. The rate of decay depends in particular on L p -integrability in time of the external force.This latter system is usually considered as a model for the flow of a viscous incompressible fluid around a rigid body moving with constant velocity and without rotation, with the set Ω corresponding to the rigid body. The functions u : Ω c × (0, ∞) → R 3 (velocity) and π : Ω c × (0, ∞) → R (pressure) are the unknowns of problem (1.1), (1.2), whereas the functions f : Ω c × (0, ∞) → R 3 (volume force) and U 0 : Ω c → R 3 (initial velocity), as well as the number τ ∈ (0, ∞) (Reynolds number), are given quantities. Mild solutions to (1.1), (1.2), which only involve the velocity among the two unknowns velocity and pressure, are introduced via the semigroup generated by the Oseen operator. We refer to (3.

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Lq$L^q$‐weak solutions to the time‐dependent 3D Oseen system: Decay estimates

Deuring

2022

Mathematische Nachrichten

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This article deals with 𝐿 𝑞 -weak solutions to the 3D time-dependent Oseen system. This type of solution is defined in terms of the velocity only. It is shown that the velocity may be represented by a sum of integrals none of which involves the pressure and without a surface integral of the spatial gradient of the velocity. On the basis of this representation formula, an estimate of the spatial decay of the velocity and its spatial gradient is derived. No boundary conditions have to be imposed for these results.

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The 3D Time-Dependent Oseen System: Link Between $$L^p$$-Integrability in Time and Pointwise Decay in Space of the Velocity

Cited by 6 publications

References 38 publications

Spatial pointwise behavior of time-periodic Navier-Stokes flow induced by oscillation of a moving obstacle

Spatial pointwise behavior of time-periodic Navier-Stokes flow induced by oscillation of a moving obstacle

Spatial asymptotics of mild solutions to the time-dependent Oseen system

Lq$L^q$‐weak solutions to the time‐dependent 3D Oseen system: Decay estimates

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