Initial-boundary value problem of the Navier–Stokes equations in the half space with nonhomogeneous data

Chang, Tongkeun; Jin, Bum Ja

doi:10.1007/s11565-018-0312-8

Cited by 8 publications

(6 citation statements)

References 54 publications

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“…Because the proofs are similar, we only prove (1) of Theorem 1.3. We will follow the proof of Theorem 1.2 in [17].…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

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Global existence of non-Newtonian incompressible fluid in half space with nonhomogeneous initial-boundary data

Chang,

Jin

2023

Preprint

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In this study, we investigate the global existence of weak solution of non-Newtonian incompressible fluid governed by \eqref{maineq2}. When $u_0 \in \dot B^{\alpha-\frac{2}{p}}_{p,q}({\mathbb R}^{n}_+) \, \cap \,\dot B^{ 1 - \frac4{n+2}}_{\frac{n+2}2,\frac{n+2}2}({\mathbb R}^{n}_+) \,\cap \, \dot B^{ 1 +\frac{n}p}_{p,1} (\R_+)$ is given, we will find the weak solution for the equations \eqref{maineq2} in the function space $C_b ([ 0, \infty; \dot B^{\alpha -\frac2p}_{p,q} ({\mathbb R}^n_+))$, $ n+2 < p < \infty, \,\, 1 \leq q \leq \infty, \,\, 1 + \frac{n+2}p < \alpha < 2$. To show it, we will show the exisence of weak solution in the anisotropic Besov spaces $\dot B^{\alpha, \frac{\alpha}2}_{p,q} (\R_+ \times (0, \infty))$ (see Theorem \ref{thm-navier}) and we will show the embedding $\dot B^{\alpha, \frac{\alpha}2}_{p,q} (\R_+ \times (0, \infty) \subset C_b ([ 0, \infty; \dot B^{\alpha -\frac2p}_{p,q} ({\mathbb R}^n_+))$ (see Lemma \ref{lemma1208}). For the global existence of solution, we assume that the extra stress tensor $S$ is represented by $S({\mathbb A}) = {\mathbb F} ( {\mathbb A}) {\mathbb A}$, where ${\mathbb F}(0) $ is uniformly elliptic matrix and \begin{align*} |\big( {\mathbb F}({\mathbb A}) -{\mathbb F}(0) \big) {\mathbb A} - \big( {\mathbb F}({\mathbb B})- {\mathbb F}(0) \big){\mathbb B}| \leq o(\max ( |{\mathbb A}|, |{\mathbb B}|) ) |{\mathbb A} -{\mathbb B}| \quad \mbox{at zero}. \end{align*} Note that $S_1$, $S_2$ and $S_3$ introduced in \eqref{0207-1} satisfy our assumption.

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“…Because the proofs are similar, we only prove (1) of Theorem 1.3. We will follow the proof of Theorem 1.2 in [17].…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…In this study, we investigage the nonhomogeneous boundary problem for non-Newtonian imcompressible fluid ( u| x n =0 = 0). Over the past decade many mathematicians have studied the Newtonian imcompressible fluid based on nonhomogeneous boundary data (See [2,3,4,14,15,16,25,26,27,28,31,32,33,34,40,46,57] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Global existence of non-Newtonian incompressible fluid in half space with nonhomogeneous initial-boundary data

Chang,

Jin

2023

Preprint

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“…Du and Zhang [6] obtained local and global existence of strong solutions for the SNS in the critical Besov space Ḃ d p −1 p,r . Chang and Yang [3] studied the initial-boundary value problem of the SNS in the half space.…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

Yin

Shen

2022

J Theor Probab

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The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space $$\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$$ B ˙ p , r 4 - 2 α - 3 p . Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.

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“…In particular, if p = q, then the velocity u is contained inḂ α, α 2 pp (R n + ×(0, ∞)) = L p (0, ∞;Ḃ α pp )∩ L p (R n + :Ḃ α 2 pp (0, ∞)) (see [10]…”

Section: Introductionmentioning

confidence: 99%

Global in time solvability of the Navier-Stokes equations in the half-space

Chang

Jin

2019

Journal of Differential Equations

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In this paper, we study the initial value problem of the Navier-Stokes equations in the half-space. Let a solenoidal initial velocity be given in the function spaceḂ α− 2 2 pq,0 (R n + ) for α + 1 = n p + 2 q and 0 < α < 2. We prove the global in time existence of weak solution u ∈ L q (0, ∞;Ḃ α pq (R n + )) ∩ L q 0 (0, ∞; L p 0 (R n + )) for some 1 < p0, q0 < ∞ with n p 0 + 2 q 0 = 1, when the given initial velocity has small norm in function spaceḂ − 2 q 0 p 0 q 0 ,0 (R n + ). The solution is unique in the class L q 0 (0, ∞; L p 0 (R n + )). Pressure estimates are also given.2000 Mathematics Subject Classification: primary 35K61, secondary 76D07.

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Initial-boundary value problem of the Navier–Stokes equations in the half space with nonhomogeneous data

Cited by 8 publications

References 54 publications

Global existence of non-Newtonian incompressible fluid in half space with nonhomogeneous initial-boundary data

Global existence of non-Newtonian incompressible fluid in half space with nonhomogeneous initial-boundary data

Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

Global in time solvability of the Navier-Stokes equations in the half-space

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