Geodesic
Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $\mathcal{D}\,\subset \,{{\mathbb{R}}^{3}}$
Canadian mathematical bulletin, Tome 47 (2004) no. 1, pp. 30-37

Voir la notice de l'article provenant de la source Cambridge

DOI

Leray's self-similar solution of the Navier-Stokes equations is defined by $$u(x,\,t)\,=\,U(y)/\sqrt{2\sigma ({{t}^{*}}\,-\,t)},$$ where $y\,=\,x/\sqrt{2\sigma ({{t}^{*}}\,-\,t)},\,\sigma \,>\,0$ . Consider the equation for $U(y)$ in a smooth bounded domain $\mathcal{D}$ of ${{\mathbb{R}}^{3}}$ with non-zero boundary condition: $$-v\,\Delta \,U\,+\,\sigma U\,+\,\sigma y\,\cdot \,\nabla U\,+\,U\,\cdot \,\nabla U\,+\,\nabla P\,=\,0,\,\,\,y\,\in \,\mathcal{D}\,$$ $$\nabla \,\cdot \,U\,=\,0,\,\,\,y\,\in \,\mathcal{D},$$ $$U\,=\,\mathcal{G}(y),\,\,\,y\,\in \,\partial \mathcal{D}.$$ We prove an existence theorem for the Dirichlet problem in Sobolev space ${{W}^{1,2}}(\mathcal{D})$ . This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at $t\,=\,{{t}^{*}}$ with ${{t}^{*}}\,<\,+\infty $ , provided the function $\mathcal{G}(y)$ is permissible.
DOI : 10.4153/CMB-2004-005-3
Mots-clés : 76D05, 76B03 
He, Xinyu. Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $\mathcal{D}\,\subset \,{{\mathbb{R}}^{3}}$. Canadian mathematical bulletin, Tome 47 (2004) no. 1, pp. 30-37. doi: 10.4153/CMB-2004-005-3
@article{10_4153_CMB_2004_005_3,
     author = {He, Xinyu},
     title = {Existence of {Leray's} {Self-Similar} {Solutions} of the {Navier-Stokes} {Equations} {In} $\mathcal{D}\,\subset \,{{\mathbb{R}}^{3}}$},
     journal = {Canadian mathematical bulletin},
     pages = {30--37},
     year = {2004},
     volume = {47},
     number = {1},
     doi = {10.4153/CMB-2004-005-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2004-005-3/}
}
TY  - JOUR
AU  - He, Xinyu
TI  - Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $\mathcal{D}\,\subset \,{{\mathbb{R}}^{3}}$
JO  - Canadian mathematical bulletin
PY  - 2004
SP  - 30
EP  - 37
VL  - 47
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2004-005-3/
DO  - 10.4153/CMB-2004-005-3
ID  - 10_4153_CMB_2004_005_3
ER  - 
%0 Journal Article
%A He, Xinyu
%T Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In $\mathcal{D}\,\subset \,{{\mathbb{R}}^{3}}$
%J Canadian mathematical bulletin
%D 2004
%P 30-37
%V 47
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2004-005-3/
%R 10.4153/CMB-2004-005-3
%F 10_4153_CMB_2004_005_3

Cité par Sources :