Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $mathcal{D}subsetmathbb{R}^3$

Canadian mathematical bulletin

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