Existence of regular solutions for a certain type of non-Newtonian fluids

Kang, Kyungkeun; Kim, Hwa Kil; Kim, Jae Myoung

doi:10.1007/s00033-019-1167-2

Cited by 7 publications

(2 citation statements)

References 14 publications

(24 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that the restriction α < 2 (regularity of solution) is based on the assumption of S(Du). With an improvement in the regularity of , the regularity of the solution in Theorem 1.2 will be also improved (see [35]).…”

Section: Theorem 12 Let S Satisfy the Assumptionmentioning

confidence: 99%

“…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%

See 1 more Smart Citation

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

Chang,

Jin

2023

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Theorem 12 Let S Satisfy the Assumptionmentioning

confidence: 99%