Existence of Strong Mild Solution of the Navier-Stokes Equations in the Half Space With Nondecaying Initial Data

Bae, Hyeong–Ohk; Jin, Bum Ja

doi:10.4134/jkms.2012.49.1.113

Cited by 35 publications

(41 citation statements)

References 15 publications

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“…The above blow‐up estimate was first proved by Leray for

Ω = {double-struckR}^{3}

. See for

n \geq 3

and (, ) for a half space. The statement of Theorem is valid also for a half space and improves regularity properties of mild solutions on

L_{σ}^{\infty}

proved in , . ( D ‐solutions) In , Leray proved the existence of D ‐solutions u satisfying

u - u_{\infty} \in L^{6} false(normalΩ false)

for

u_{\infty} \in {double-struckR}^{3}

in the exterior domain

Ω \subset {double-struckR}^{3}

.…”

Section: Introductionmentioning

confidence: 75%

“…Moreover, the space

L_{σ}^{\infty}

includes vector fields rotating at space infinity; see Remarks 1.2 (iv). When Ω is the whole space or a half space , , the existence of mild solutions of (1.1) on

L_{σ}^{\infty}

is proved by explicit formulas of the Stokes semigroup. In this paper, we prove unique existence of mild solutions on

L_{σ}^{\infty}

for exterior domains based on

L^{\infty}

‐estimates of the Stokes semigroup , .…”

Section: Introductionmentioning

confidence: 99%

“…The solution u is bounded in

Ω \times (0, \infty)

and non‐decaying as

| x | \to \infty

. (Associated pressure) We invoke that the associated pressure of mild solutions on

L^{p}

(

p \geq n

) is determined by the projection operator

Q = I - P

and

\begin{matrix} true \\ right \nabla p = double-struckQ normalΔ u - double-struckQ false(u \cdot \nabla u false) . \end{matrix}

Since the projection

double-struckQ

may not be bounded on

L^{\infty}

, this representation is no longer available for mild solutions on

L_{σ}^{\infty}

. When

Ω = {double-struckR}^{n}

R_{+}^{n}

, the projection

double-struckQ

has explicit kernels and we are able to find associated pressure of mild solutions on

L^{\infty}

; see for

Ω = {double-struckR}^{n}

and , , for

Ω = {double-struckR}_{+}^{n}

. Although explicit kernels are not available for exterior domains, we are able to find the associated pressure of mild solutions on

L^{\infty}

.…”

Section: Introductionmentioning

confidence: 99%

“…See for

n \geq 3

and (, ) for a half space. The statement of Theorem is valid also for a half space and improves regularity properties of mild solutions on

L_{σ}^{\infty}

proved in , .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Exterior Navier–Stokes flows for bounded data

Abe

2016

Mathematische Nachrichten

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show abstract

“…The above blow‐up estimate was first proved by Leray for

Ω = {double-struckR}^{3}

. See for

n \geq 3

and (, ) for a half space. The statement of Theorem is valid also for a half space and improves regularity properties of mild solutions on

L_{σ}^{\infty}

proved in , . ( D ‐solutions) In , Leray proved the existence of D ‐solutions u satisfying

u - u_{\infty} \in L^{6} false(normalΩ false)

for

u_{\infty} \in {double-struckR}^{3}

in the exterior domain

Ω \subset {double-struckR}^{3}

.…”

Section: Introductionmentioning

confidence: 75%

“…Moreover, the space

L_{σ}^{\infty}

includes vector fields rotating at space infinity; see Remarks 1.2 (iv). When Ω is the whole space or a half space , , the existence of mild solutions of (1.1) on

L_{σ}^{\infty}

is proved by explicit formulas of the Stokes semigroup. In this paper, we prove unique existence of mild solutions on

L_{σ}^{\infty}

for exterior domains based on

L^{\infty}

‐estimates of the Stokes semigroup , .…”

Section: Introductionmentioning

confidence: 99%

“…The solution u is bounded in

Ω \times (0, \infty)

and non‐decaying as

| x | \to \infty

. (Associated pressure) We invoke that the associated pressure of mild solutions on

L^{p}

(

p \geq n

) is determined by the projection operator

Q = I - P

and

\begin{matrix} true \\ right \nabla p = double-struckQ normalΔ u - double-struckQ false(u \cdot \nabla u false) . \end{matrix}

Since the projection

double-struckQ

may not be bounded on

L^{\infty}

, this representation is no longer available for mild solutions on

L_{σ}^{\infty}

. When

Ω = {double-struckR}^{n}

R_{+}^{n}

, the projection

double-struckQ

has explicit kernels and we are able to find associated pressure of mild solutions on

L^{\infty}

; see for

Ω = {double-struckR}^{n}

and , , for

Ω = {double-struckR}_{+}^{n}

. Although explicit kernels are not available for exterior domains, we are able to find the associated pressure of mild solutions on

L^{\infty}

.…”

Section: Introductionmentioning

confidence: 99%

“…See for

n \geq 3

and (, ) for a half space. The statement of Theorem is valid also for a half space and improves regularity properties of mild solutions on

L_{σ}^{\infty}

proved in , .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exterior Navier–Stokes flows for bounded data

Abe

2016

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…Bae and Choe [4], Bae and Jin [5][6][7][8][9], Brandolese [10,11], Brandolese and Vigneron [12], Fujigaki and Miyakawa [14] and Schonbek [20][21][22] considered the asymptotic behaviour for weak and strong solutions of (1.1), and many important and interesting results were obtained. Bae and Choe [4], Bae and Jin [5][6][7][8][9], Brandolese [10,11], Brandolese and Vigneron [12], Fujigaki and Miyakawa [14] and Schonbek [20][21][22] considered the asymptotic behaviour for weak and strong solutions of (1.1), and many important and interesting results were obtained.…”

Section: P Hanmentioning

confidence: 99%