Pointwise Spatial Decay of Weak Solutions to the Navier–Stokes System in 3D Exterior Domains

Deuring, Paul

doi:10.1007/s00021-014-0198-x

Cited by 6 publications

(14 citation statements)

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“…Here, A 2 := P 2 ∆ is the Stokes operator, and D(A We point out that key ideas, the technique and the proofs in [5] and in the present paper are completely different.…”

Section: Introductionmentioning

confidence: 92%

On the spatial asymptotic decay of a suitable weak solution to the Navier–Stokes Cauchy problem

Crispo

Maremonti

2016

Nonlinearity

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-We prove space-time decay estimates of suitable weak solutions to the NavierStokes Cauchy problem, corresponding to a given asymptotic behavior of the initial data of the same order of decay. We use two main tools. The first is a result obtained in [4] on the behavior of the solution in a neighborhood of t = 0 in the L ∞ loc -norm, which enables us to furnish a representation formula for a suitable weak solution. The second is the asymptotic behavior of ||u(t)|| L 2 (R 3 \B R ) for R → ∞. Following a Leray's point of view, roughly speaking our result proves that a possible space-time turbulence does not perturb the asymptotic spatial behavior of the initial data of a suitable weak solution.

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“…Here, A 2 := P 2 ∆ is the Stokes operator, and D(A We point out that key ideas, the technique and the proofs in [5] and in the present paper are completely different.…”

Section: Introductionmentioning

confidence: 92%

On the spatial asymptotic decay of a suitable weak solution to the Navier–Stokes Cauchy problem

Crispo

Maremonti

2016

Nonlinearity

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“…A similar representation formula was derived in [15] for solutions to the time-dependent Stokes system; see [15,Theorem 4.3]. However, this formula only leads to pointwise decay estimates of the velocity itself, but not of its spatial gradient, and is valid only if f vanishes and homogeneous Dirichlet boundary conditions are satisfied.…”

Section: Introductionmentioning

confidence: 92%

“…However, this formula only leads to pointwise decay estimates of the velocity itself, but not of its spatial gradient, and is valid only if f vanishes and homogeneous Dirichlet boundary conditions are satisfied. Moreover the theory in [15] is essentially restricted to an L 2 -framework and is based on maximal regularity of solutions to the time-dependent Stokes system. Maximal regularity cannot be expected to hold for solutions to the Oseen system (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…For x ∈ A, t ∈ J, we will write v(t)(x) or v(x, t) depending on whether v is considered as a function on J with values in L q (A) n , or as a function on A × J. If v ∈ W 1,1 loc J, L q (A) n , we write v for the weak derivative of v : J → L q (A) n , and ∂ t v for the weak partial derivative of v as a funtion on A × J; compare [15,Lemma 2.3]. For a function v : J → W 1,1 loc (A) 3 , the notation ∇ x v stands for the gradient of v with respect to x ∈ A, in the sense that…”

Section: Recall That In Section 1 We Introduced the Functionmentioning

confidence: 99%

“…In particular, we indicate why a measurable function f : J → L q (U ) is measurable as a function on J × U , where J ⊂ R n is open, for some n ∈ N. This feature is often used in the work at hand with respect to functions from L 1 loc J, L q (U ) when J is an interval in R. In [15,Lemma 3.5], we already considered the case q = 2, without treating measurability. In the proof of the ensuing lemma, we simplify the argument from [15] by applying Theorem 2.5. 3 integrable as a Bochner integral in L q (U ) 3 .…”

Section: Recall That In Section 1 We Introduced the Functionmentioning

confidence: 99%

See 2 more Smart Citations

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

Deuring

2021

J. Math. Fluid Mech.

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A representation formula without pressure term is derived for regular solutions to the 3D time-dependent Oseen system in exterior Lipschitz domains. This formula is valid even if no boundary conditions are imposed. It is used in order to exhibit how the velocity decays pointwise in space. It turns out that the rate of this decay depends on L p -integrability in time of the velocity. In addition, this work is the basis for successor papers dealing with spatial decay of L q -weak solutions and mild solutions to the time-dependent Oseen system, and with L 2 -strong solutions to the stability problem related to the Navier-Stokes system with Oseen term.

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Stability of Stationary Viscous Incompressible Flow Around a Rigid Body Performing a Translation

Deuring

2017

J. Math. Fluid Mech.

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form-as well as (1.4). The boundary condition at infinity imposed by (1.4) is taken account of by the relations U ∈ L 6 (Ω c) 3 , ∇U ∈ L 2 (Ω c) 9. (We refer to Section 2 for the definition of our function spaces and for other notation.) Leray solutions to (1.3), (1.4) exist under suitable assumptions on F and ∂Ω. More details may be found in [23, Theorem VII.2.1, II.5.1]. If F decays sufficiently fast, it may be shown that U ∈ L p (Ω c) 3 for p ∈ (2, ∞] and ∇U ∈ L p (Ω c) 9 for p ∈ (4/3, ∞] ([24, Theorem X.4.1]). In the work at hand, it is sufficient to suppose U ∈ L p (Ω c) 3 for p ∈ (2, ∞) and ∇U ∈ L p (Ω c) 9 for p ∈ (4/3, 3+ 0 ], with some 0 > 0. Then U ∈ W 1, 3+ 0 (Ω c) 3 , so U ∈ L ∞ (Ω c) 3 by a Sobolev inequality, and also |U (x)| → 0 for |x| → ∞, as explained in [12, p. 209]. Therefore the assumptions U ∈ L p (Ω c) 3 for p ∈ (2, ∞) and ∇U ∈ L p (Ω c) 9 for p ∈ (4/3, 3 + 0 ] mean in particular that U ∈ L p (Ω c) 3 for p ∈ (2, ∞], |U (x)| → 0 for |x| → ∞, (1.4) ∇U ∈ L p (Ω c) 9 for p ∈ (4/3, 3]. These are the properties of U we will actually use in what follows. We are interested in conditions which guarantee asymptotic stability of U , that is, if U ∈ W 1,2 0 (Ω c) 3 ∩ H 2 (Ω c), and if the quantity U − U 2 + ∇(U − U 2 is sufficiently small, then the velocity part v of a solution (v, π) of the initial-boundary value problem Proof: We proceed as in [23, p. 177-178]. Take R and V as in the lemma. First suppose that q < 3. Then Hölder's inequality yields V |B 2R \B R q ≤ C R V |B c R 3q/(3−q). On the other hand, a scaling argument and Theorem 3.4 with A = B R 0 imply V |B c R 3q/(3−q) = (R/R 0) (3−q)/q V (R/R 0) • |B R 0 c 3q/(3−q)

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Pointwise Spatial Decay of Weak Solutions to the Navier–Stokes System in 3D Exterior Domains

Cited by 6 publications

References 50 publications

On the spatial asymptotic decay of a suitable weak solution to the Navier–Stokes Cauchy problem

On the spatial asymptotic decay of a suitable weak solution to the Navier–Stokes Cauchy problem

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

Stability of Stationary Viscous Incompressible Flow Around a Rigid Body Performing a Translation

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