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

He, Xinyu

Canadian mathematical bulletin, Tome 47 (2004) no. 1, pp. 30-37

Résumé

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.

Détail

DOI : 10.4153/CMB-2004-005-3

Mots-clés : 76D05, 76B03

Bibliographie

[BP] [BP] Boratav, O. N. and Pelz, R. B., Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids 6 (1994), 2757–2784. Google Scholar

[CFM] [CFM] Constantin, P., Fefferman, C. and Majda, A., Geometric constraints on potentially singular solutions for the 3d Euler equations. Comm. Partial Differential Equations 21 (1996), 559–571. Google Scholar

[CKN] [CKN] Caffarelli, L., Kohn, R. and Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), 771–831. Google Scholar

[CP] [CP] Cannone, M. and Planchon, F., Self-similar solutions for Navier-Stokes equations in R3. Comm. Partial Differential Equations 21 (1996), 179–193. Google Scholar

[G] [G] Galdi, G. P., An introduction to the mathematical theory of the Navier-Stokes equations. Vol I and II, Springer Tracts Nat. Philos. 39, Springer, 1994. Google Scholar

[GM] [GM] Giga, Y. and Miyakawa, T., Navier-Stokes flow in R3 with measures as initial vorticity and Morrey spaces. Comm. Partial Differential Equations 14 (1989), 577–618. Google Scholar

[K] [K] Kerr, R. M., The outer regions in singular Euler. In: Fundamental Problematic Issues in Turbulence, (eds., Tsinober and Gyr), Birkh¨auser, 1998. Google Scholar

[L] [L] Leray, J., Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934), 193–248. Google Scholar

[LS] [LS] Leray, J. and Schauder, J., Topologie et équations fonctionnelles. Ann. Sci. École Norm. Sup. 51 (1934), 45–78. Google Scholar

[M] [M] Moffatt, H. K., The interaction of skewed vortex pairs: a model for blow-up of the Navier-Stokes equations. J. Fluid Mech. 409 (2000), 51–68. Google Scholar

[NOZ] [NOZ] Nagayama, M., Okamoto, H. and Zhu, J., On the blow-up of some similarity solutions of the Navier-Stokes equations. Quad. Mat. (2002), to appear. Google Scholar

[NRŠ] [NRŠ] Nečas, J., Råužička, M. and Šver´ak, V., On Leray's self-similar solutions of the Navier-Stokes equations. Acta Math. 176 (1996), 283–294. Google Scholar

[OG] [OG] Ohkitani, K. and Gibbon, J. D., Numerical study of singularity formation in a class of Euler and Navier-Stokes flows. Phys. Fluids 12 (2000), 3181–3194. Google Scholar

[P] [P] Pelz, R. B., Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55 (1997), 1617–1626. Google Scholar

[SS] [SS] Sulem, C. and Sulem, P.-L., The Nonlinear Schródinger Equation. Appl. Math. Sci. 139, Springer, 1999. Google Scholar

[T] [T] Tsai, T.-P., On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Rational Mech. Anal. 143 (1998), 29–51. Google Scholar