Existence of strong solutions and decay of turbulent solutions of Navier–Stokes flow with nonzero Dirichlet boundary data

Farwig, Reinhard; Kozono, Hideo; Wegmann, David

doi:10.1016/j.jmaa.2017.03.086

Cited by 4 publications

(1 citation statement)

References 19 publications

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“…For the stability of time‐dependent flows in the whole space

\mathbb {R}^3

and exterior domains, the reader is referred to a chapter by Brandolese and Schonbek in [2] for a comprehensive survey of the area. We also refer to Hishida and Schonbek [12] and Farwig, Kozono, and Wegmann [4] for recent work. Although the equations (NS) are imposed on the three‐dimensional domain

\Omega \times \mathbb {T}

, we still cannot apply the Hardy inequality since the stationary flow U does not decay in space along the vertical direction.…”

Section: Introductionmentioning

confidence: 99%

Stability of a two‐dimensional stationary rotating flow in an exterior cylinder

Higaki

2022

Mathematische Nachrichten

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We investigate the stability of an exact stationary flow in an exterior cylinder. The horizontal velocity is the two-dimensional rotating flow in an exterior disk with a critical spatial decay, for which the 𝐿 2 stability is known under smallness conditions. We prove its stability property for three-dimensional perturbations although the Hardy type inequalities are absent as in the twodimensional case. The proof uses a large time estimate for the linearized equations exhibiting different behaviors in the Fourier modes, namely, the standard 𝐿 2 -𝐿 𝑞 decay of the two-dimensional mode and an exponential decay of the three-dimensional modes.

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“…For the stability of time‐dependent flows in the whole space

\mathbb {R}^3

\Omega \times \mathbb {T}

, we still cannot apply the Hardy inequality since the stationary flow U does not decay in space along the vertical direction.…”

Section: Introductionmentioning

confidence: 99%

Stability of a two‐dimensional stationary rotating flow in an exterior cylinder

Higaki

2022

Mathematische Nachrichten

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Maximal regularity of the Stokes operator in an exterior domain with moving boundary and application to the Navier–Stokes equations

2018

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Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system

Winkler

2024

Math. Ann.

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The Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{l} u_t + (u\cdot \nabla ) u =\Delta u+\nabla P + f(x,t), \\ \nabla \cdot u=0, \end{array} \right. \end{aligned}$$ u t + ( u · ∇ ) u = Δ u + ∇ P + f ( x , t ) , ∇ · u = 0 , is considered along with homogeneous Dirichlet boundary conditions in a smoothly bounded planar domain $$\Omega $$ Ω . It is firstly, inter alia, observed that if $$T>0$$ T > 0 and $$\begin{aligned} \int _0^T \bigg \{ \int _\Omega |f(x,t)| \cdot \ln ^\frac{1}{2} \big (|f(x,t)|+1\big ) dx \bigg \}^2 dt <\infty , \end{aligned}$$ ∫ 0 T { ∫ Ω | f ( x , t ) | · ln 1 2 ( | f ( x , t ) | + 1 ) d x } 2 d t < ∞ , then for all divergence-free $$u_0\in L^2(\Omega ;{\mathbb {R}}^2)$$ u 0 ∈ L 2 ( Ω ; R 2 ) , a corresponding initial-boundary value problem admits a weak solution u with $$u|_{t=0}=u_0$$ u | t = 0 = u 0 . For any positive and nondecreasing $$L\in C^0([0,\infty ))$$ L ∈ C 0 ( [ 0 , ∞ ) ) such that $$\begin{aligned} \frac{L(\xi )}{\ln ^\frac{1}{2} \xi } \rightarrow 0 \qquad \text{ as } \xi \rightarrow \infty , \end{aligned}$$ L ( ξ ) ln 1 2 ξ → 0 as ξ → ∞ , this is complemented by a statement on nonexistence of such a solution in the presence of smooth initial data and a suitably constructed $$f:\Omega \times (0,T)\rightarrow {\mathbb {R}}^2$$ f : Ω × ( 0 , T ) → R 2 fulfilling $$\begin{aligned} \int _0^T \bigg \{ \int _\Omega |f(x,t)| \cdot L\big (|f(x,t)|\big ) dx \bigg \}^2 dt < \infty . \end{aligned}$$ ∫ 0 T { ∫ Ω | f ( x , t ) | · L ( | f ( x , t ) | ) d x } 2 d t < ∞ . This resolves a fine structure in the borderline case $$p=1$$ p = 1 and $$q=2$$ q = 2 appearing in results on existence of weak solutions for sources in $$L^q((0,T);L^p(\Omega ;{\mathbb {R}}^2))$$ L q ( ( 0 , T ) ; L p ( Ω ; R 2 ) ) when $$p\in (1,\infty ]$$ p ∈ ( 1 , ∞ ] and $$q\in [1,\infty ]$$ q ∈ [ 1 , ∞ ] satisfy $$\frac{1}{p}+\frac{1}{q}\le \frac{3}{2}$$ 1 p + 1 q ≤ 3 2 , and on nonexistence if here $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) and $$q\in [1,\infty )$$ q ∈ [ 1 , ∞ ) are such that $$\frac{1}{p}+\frac{1}{q}>\frac{3}{2}$$ 1 p + 1 q > 3 2 .

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Existence of strong solutions and decay of turbulent solutions of Navier–Stokes flow with nonzero Dirichlet boundary data

Cited by 4 publications

References 19 publications

Stability of a two‐dimensional stationary rotating flow in an exterior cylinder

Stability of a two‐dimensional stationary rotating flow in an exterior cylinder

Maximal regularity of the Stokes operator in an exterior domain with moving boundary and application to the Navier–Stokes equations

Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system

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