On the existence of unique global-in-time solutions and temporal decay rates of solutions to some non-Newtonian incompressible fluids

Bae, Hantaek; Kang, Kyungkeun

doi:10.1007/s00033-021-01489-8

Cited by 2 publications

(1 citation statement)

References 21 publications

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“…Ladyženskaja ([37, 38]) was the first to study the existence of weak solution of (1.1) in bounded domain with no-slip boundary condition (g = 0), and Nečas et al also studied this problem intensively. After them, non-Newtonian incompressible fluid was studied by many mathematicians ( see [10,11,23,36,45] pertaining to the whole space and see [5,7,12,29,35,44,54,55,56,58] for bounded domain with no-slip boundary condition).…”

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

Chang,

Jin

2024

Journal of Mathematical Physics

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In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When u0∈Ḃp,qα−2p(R+n)∩Ḃn+22,n+221−4n+2(R+n)∩Ḃp,11+np(R+n) is given, we will find the weak solutions for the Eq. (1.1) in the function space Cb[0,∞;Ḃp,qα−2p(R+n)∩Cb(0,∞;Ḃn+221−4n+2(R+n))∩L∞(0,∞;Ẇ∞1(R+n)), n+2<p<∞,1≤q≤∞,1+n+2p<α<2. We show the existence of weak solutions in the anisotropic Besov spaces Ḃp,qα,α2(R+n×(0,∞)) (see Theorem 1.2) and we show the embedding Ḃp,qα,α2R+n×(0,∞)⊂Cb[0,∞;Ḃp,qα−2p(R+n) (see Lemma 2.8). For the global existence of solutions, we assume that the extra stress tensor S is represented by S(A)=F(A)A, where F satisfies the assumption (A). Note that S1, S2 and S3 introduced in (1.2) satisfy our assumptions.

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On the existence of unique global-in-time solutions and temporal decay rates of solutions to some non-Newtonian incompressible fluids

Cited by 2 publications

References 21 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

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

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