Canadian mathematical bulletin | |
Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $mathcal{D}subsetmathbb{R}^3$ | |
关键词: formal Brauer groups; $K3$ surfaces in positive; characteristic; weighted Delsarte surfaces; | |
DOI : 10.4153/CMB-2004-005-3 | |
学科分类:数学(综合) | |
来源: University of Toronto Press * Journals Division | |
【 摘 要 】
Leray's self-similar solution of the Navier-Stokes equations isdefined by $$u(x,t) = U(y)/sqrt{2sigma (t^*-t)},$$where $y = x/sqrt{2sigma (t^*-t)}$, $sigma>0$. Consider theequation for $U(y)$ in a smooth bounded domain $mathcal{D}$ of$mathbb{R}^3$ with non-zero boundary condition: egin{gather*}-u igtriangleup U + sigma U +sigma y cdot abla U + Ucdotabla U + abla P = 0,quad y in mathcal{D}, \abla cdot U = 0, quad y in mathcal{D}, \U = mathcal{G}(y), quad y in partial mathcal{D}.end{gather*}We prove an existence theorem for the Dirichlet problem in Sobolevspace $W^{1,2} (mathcal{D})$. This implies the local existence ofa self-similar solution of the Navier-Stokes equations which blowsup at $t=t^*$ with $t^* < +infty$, provided the function$mathcal{G}(y)$ is permissible.
【 授权许可】
Unknown
【 预 览 】
Files | Size | Format | View |
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RO201912050576335ZK.pdf | 36KB | download |