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2013 | Buch

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Topics in Mathematical Fluid Mechanics

Cetraro, Italy 2010, Editors: Hugo Beirão da Veiga, Franco Flandoli

verfasst von: Peter Constantin, Arnaud Debussche, Giovanni P. Galdi, Michael Růžička, Gregory Seregin

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume brings together five contributions to mathematical fluid mechanics, a classical but still very active research field which overlaps with physics and engineering. The contributions cover not only the classical Navier-Stokes equations for an incompressible Newtonian fluid, but also generalized Newtonian fluids, fluids interacting with particles and with solids, and stochastic models. The questions addressed in the lectures range from the basic problems of existence of weak and more regular solutions, the local regularity theory and analysis of potential singularities, qualitative and quantitative results about the behavior in special cases, asymptotic behavior, statistical properties and ergodicity.

Inhaltsverzeichnis

Frontmatter

Complex Fluids and Lagrangian Particles

Abstract

We discuss complex fluids that are comprised of a solvent, which is an incompressible Newtonian fluid, and particulate matter in it. The complex fluid occupies a region in physical space \({\mathbb{R}}^{d}\). The particles are described using a finite dimensional manifold M, which serves as configuration space. Simple models [15, 29] represent complicated objects by retaining very few degrees of freedom, and in those cases \(M = {\mathbb{R}}^{d}\) or \(M = {\mathbb{S}}^{d-1}\). In general, there is no reason why the number of degrees of freedom of the particles should equal, or be related to the number of degrees of freedom of ambient physical space. We will consider as starting point kinetic descriptions of the particles.

Peter Constantin

Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction

Abstract

The theory of the stochastic Navier–Stokes equations (SNSE) has known a lot of important advances those last 20 years. Existence and uniqueness have been studied in various articles (see for instance [1, 35, 911, 13, 15, 21, 28, 30, 49, 51, 52, 66, 67]) and this part of the theory is well understood. Most of the deterministic results have been generalized to the stochastic context and it is now known that as in the deterministic case the SNSE has unique global strong solutions in dimension two. In dimension three, there exist global weak solutions and uniqueness is also a completely open problem in the stochastic case. The solutions in dimension three are weak in the sense of the theory of partial differential equations and in the sense of stochastic equations: the solutions are not smooth in space and they satisfy the SNSE only in the sense of the martingale problem. In Sect. 2 of these notes, we recall briefly these results and give the ideas of the proof.

Arnaud Debussche

Steady-State Navier–Stokes Problem Past a Rotating Body: Geometric-Functional Properties and Related Questions

Abstract

As is well known, the three-dimensional steady motion of a viscous, incompressible (Navier–Stokes) liquid around a rigid body, \(\mathcal{B}\), is among the fundamental and most studied questions in fluid dynamics; see e.g. [4].

Giovanni P. Galdi

Analysis of Generalized Newtonian Fluids

Abstract

In this paper we want to present an optimal existence result for the steady motion of generalized Newtonian fluids. Moreover, we present an optimal error estimate for a FEM approximation of the corresponding steady p-Stokes system. The presented results are based on long lasting cooperations with L. Berselli, L. Diening, J. Málek, A. Prohl and J. Wolf.

Michael Růžička

Selected Topics of Local Regularity Theory for Navier–Stokes Equations

Abstract

\(\mathbb{R}_{+} =\{ t \in \mathbb{R} :\,\, t > 0\}\), \(\mathbb{R}_{-} =\{ t \in \mathbb{R} :\,\, t < 0\}\); \(\mathbb{R}_{+}^{d} =\{ x = (x^\prime ,x_{d}) :\,\, x^\prime = (x_{i}),\,\,i = 1,2,\ldots ,d - 1,\,\,x_{d} > 0\}\); \(Q_{-} = {\mathbb{R}}^{d} \times R_{-}\), \(Q_{+} = {\mathbb{R}}^{d} \times R_{+}\); \(Q_{\delta ,T} = \Omega \times ]\delta ,T[\), \(Q_{T} = \Omega \times ]0,T[\), \(\Omega \subset {\mathbb{R}}^{d}\); B(x, r) is the ball in \({\mathbb{R}}^{d}\) of radius r centered at the point \(x \in {\mathbb{R}}^{d}\), B(r) = B(0, r), B = B(1); \(B_{+}(x,r) =\{ y = (y^\prime ,y_{d}) \in B(x,r) :\,\, y_{d} > x_{d}\}\) is a half ball, \(B_{+}(r) = B_{+}(0,r)\), \(B_{+} = B_{+}(1)\); Q(z, r) = B(x, r) ×]t − r ², t[ is the parabolic ball in \({\mathbb{R}}^{d} \times \mathbb{R}\) of radius r centered at the point \(z = (x,t) \in {\mathbb{R}}^{d} \times \mathbb{R}\), Q(r) = Q(0, r), Q = Q(1); \(Q_{+}(r) = Q_{+}(0,r) = B_{+}(r)\times ] - {r}^{2},0[\); \(L_{s}(\Omega )\) and \(W_{s}^{1}(\Omega )\) are the usual Lebesgue and Sobolev spaces, respectively; \(L_{s,l}(Q_{T}) = L_{l}(0,T;L_{s}(\Omega ))\), \(L_{s}(Q_{T}) = L_{s,s}(Q_{T})\); \(W_{s,l}^{1,0}(Q_{T}) =\{ \vert v\vert + \vert \nabla v\vert \in L_{s,l}(Q_{T})\}\) and \(W_{s,l}^{1,0}(Q_{T}) =\{ \vert v\vert + \vert \nabla v\vert + \vert {\nabla }^{2}v\vert + \vert \partial v\vert \in L_{s,l}(Q_{T})\}\) are parabolic Sobolev spaces; \(C_{0,0}^{\infty }(\Omega ) =\{ v \in C_{0}^{\infty }(\Omega ) :\,\, \mathrm{div}\,v = 0\}\); \({ \circ \atop J} (\Omega )\) is the closure of the set \(C_{0,0}^{\infty }(\Omega )\) in the space \(L_{2}(\Omega )\), \({ \circ \atop J} _{2}^{1}(\Omega )\) is the closure of the same set with respect to the Dirichlet integral; BMO is the space of functions having bounded mean oscillation;

Gregory Seregin

Backmatter

Titel: Topics in Mathematical Fluid Mechanics
verfasst von: Peter Constantin
Arnaud Debussche
Giovanni P. Galdi
Michael Růžička
Gregory Seregin
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-36297-2
Print ISBN: 978-3-642-36296-5
DOI: https://doi.org/10.1007/978-3-642-36297-2

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