zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global attractors for small samples and germs of 3D Navier-Stokes equations. (English) Zbl 1075.35037
The author considers the Navier-Stokes equations $$u_t= \nu\Delta u- (u\nabla)u+ \nabla p+ f,\ \text{div}(u)= 0,\ u= 0\quad\text{on }\partial\Omega,\tag1$$ with $\Omega$ smooth and bounded, and $f$ a constant exterior force. One imposes on (1) a standard Hilbert space setting: $$|u|^2= (u,u)= \sum \int u^2_j\, dx.$$ $H$ is the $L^2$-closure of $V_0= \{u/u\in C_0(\Omega)^3,\text{div}(u)= 0\}$, while $V$ is the closure of $V_0$ with respect to the norm $$\Vert u\Vert^2= \sum\int(\partial_j u)^2\, dx,\quad \partial_j= \partial_{x_j}.$$ A weak solution of (1) is an element $u$ in $L^\infty(0,\infty; H)\cap L^2(0, T;V)$ (for all $T$) subject to three classical conditions, one requiring that $u$ satisfies a standard variational form of (1). The set $W$ of weak solutions is considered as a metric space under the norm $$[u]^2= \int^\infty_0 |u(t)|^2 e^{-t}\,dt.$$ It was proved by {\it G. R. Sell} [J. Dyn. Differ. Eq. 8, No. 1, 1--33 (1996; Zbl 0855.35100)] that the semiflow $S_t$ on $W$ given by $$(S_t u)(x)= u(t+ s),\quad s\ge 0$$ admits a global attractor $A$. The author now takes a generalization of this situation given by {\it J. M. Ball} [J. Nonlinear Sci. 7, No. 5, 475--502 (1997; Zbl 0903.58020)] as starting point who introduced the notion of generalized semiflow $G$ on a metric space $X$. Since this notion is not directly applicable to the space $W$, the author defines a notion of $\varepsilon$-samples as follows. With $u|\varepsilon$ the restriction of $u\in L^\infty(0,\infty; H)$ to $[0,\varepsilon]$ one sets: $$W_\varepsilon= \{u|\varepsilon\mid u\in W\}$$ and with $u\in W$ one associates $\varphi^\varepsilon_u: [0,\infty]\to W$ as follows: $$\varphi^\varepsilon_u(t)= u^t|\varepsilon,\quad\text{where }u^t(s)= u(t+ s),\quad s\ge 0.$$ One then sets: $$G_\varepsilon= \{\varphi^\varepsilon_u\mid u\in W\}.$$ The main results then are: (I) $G_\varepsilon$ is a generalized semiflow on $W_\varepsilon$, (II) with $A$ the attractor for $S_t$ and $A_\varepsilon= \{u\mid \varepsilon\mid u\in A\}$, $A_\varepsilon$ is a global attractor for $G_\varepsilon$. The author extends the above theory to a familiy of objects called “germs”, which are a kind of infinitesimal $\varepsilon$-samples $(\varepsilon= 0)$.

35Q30Stokes and Navier-Stokes equations
35B41Attractors (PDE)
37L30Attractors and their dimensions, Lyapunov exponents
76D05Navier-Stokes equations (fluid dynamics)
Full Text: DOI
[1] Aubin, J. -P.: Un théorème de compacité. CR acad. Sci. Paris sér. I math. 256, 5042-5044 (1963) · Zbl 0195.13002
[2] J.M. Ball, Continuity properties and global attractors of generalised semiflows and the Navier -- Stokes equations, J. Nonlinear Sci. 7 (1997) 475 -- 502. Erratum ibid. 8 (1998) 233. Corrected version appears in Mechanics: from Theory to Computation, Springer, Berlin, 2000, pp. 447 -- 474.
[3] Capiński, M.; Cutland, N. J.: A simple proof of existence of weak and statistical solutions of Navier -- Stokes equations. Proc. roy. Soc. London A 436, 1-11 (1992) · Zbl 0746.35059
[4] M. Capiński, N.J. Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, Series on Advances in Mathematics for Applied Sciences, vol. 27, World Scientific, Singapore, 1995. · Zbl 0824.76003
[5] Capiński, M.; Cutland, N. J.: Attractors for three-dimensional Navier -- Stokes equations. Proc. roy. Soc. London ser. A 453, 2413-2426 (1997) · Zbl 0990.35118
[6] N.J. Cutland, H.J. Keisler, Attractors and neo-attractors for 3D stochastic Navier -- Stokes equations, Stoch. Dynam., to appear. · Zbl 1081.60048
[7] C. Foias, R. Temam, The connection between the Navier -- Stokes equations, dynamical systems, and turbulence theory, in: Directions in Partial Differential Equations (Madison, WI, 1985), Publications of the Mathematics Research Center University of Wisconsin, vol. 54, Academic Press, Boston, 1987, pp. 55 -- 73.
[8] Málek, J.; Pražák, D.: Large time behavior via the method of l-trajectories. J. differential equations 181, 243-279 (2002) · Zbl 1187.37113
[9] Marín-Rubio, P.; Robinson, J. C.: Attractors for the stochastic 3D Navier -- Stokes equations. Stochastics dynamics 3, 279-297 (2003) · Zbl 1059.35100
[10] Sell, G. R.: Global attractors for the three-dimensional Navier -- Stokes equations. J. dynamics differential equations 8, 1-33 (1996) · Zbl 0855.35100
[11] Temam, R.: Navier -- Stokes equations and nonlinear functional analysis. (1983) · Zbl 0522.35002