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

[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.

[学科分类] 数学（综合）

[关键词] formal Brauer groups;$K3$ surfaces in positive;characteristic;weighted Delsarte surfaces