Partial Differential Equations Seminar
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Mon, 02/05/2011 17:00 |
Michael Ruzicka (Universitaet Freiburg) |
Partial Differential Equations Seminar  |
Gibson 1st Floor SR |
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Fluids that are not adequately described by a linear constitutive relation are usually referred to as "non-Newtonian fluids". In the last 15 years we have seen a significant progress in the mathematical theory of generalized Newtonian fluids, which is an important subclass of non-Newtonian fluids. We present some recent results in the existence theory and in the error analysis for approximate solutions. We will also indicate how these techniques can be generalized to more general constitutive relations. |
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Mon, 09/05/2011 17:00 |
Giovanni Alberti (Universita di Pisa) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
| We give a characterization of divergence-free vector fields on the plane such that the Cauchy problem for the associated continuity (or transport) equation has a unique bounded solution (in the sense of distribution). Unlike previous results in this directions (Di Perna-Lions, Ambrosio), the proof relies on a dimension-reduction argument, which can be regarded as a variant of the method of characteristics. Note that our characterization is not stated in terms of function spaces, but is based on a suitable weak formulation of the Sard property for the potential associated to the vector-field. This is a joint work with S. Bianchini (SISSA, Trieste) and Gianluca Crippa (Parma). | |||
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Mon, 16/05/2011 17:00 |
Mariarosaria Padula (Universita di Ferrara) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
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Given a film of viscous heavy liquid with upper free boundary over an inclined plane, a steady laminar motion develops parallel to the flat bottom ofthe layer. We name this motion Poiseuille Free Boundary PFBflow because of its (half) parabolic velocity profile. In flowsover an inclined plane the free surface introduces additionalinteresting effects of surface tension and gravity. These effectschange the character of the instability in a parallel flow, see{Smith} [1]. \parBenjamin [2], and Yih [3], have solved the linear stabilityproblem of a uniform film on a inclined plane. Instability takesplace in the form of an infinitely long wave, howeversurface waves of finite wavelengths are observed, see e.g.Yih [3]. Up to date direct nonlinear methods for the study ofstability seem to be still lacking.Aim of this talk is the investigation of nonlinear stability ofPFB providing a rigorous formulation of the problem by theclassical direct Lyapunov method assuming periodicity in theplane, when above the liquid there is a uniform pressure due tothe air at rest, and the liquid is moving with respect to the air.Sufficient conditions on the non dimensional Reynolds, Webernumbers, on the periodicity along the line of maximum slope, onthe depth of the layer and on the inclination angle are computedensuring Kelvin-Helmholtz nonlinear stability. We usea modified energy method, cf. [4],[5], which providesphysically meaningful sufficient conditions ensuring nonlinearexponential stability. The result is achieved in the class ofregular solutions occurring in simply connected domains havingcone property.\parNotice that the linear equations, obtained by linearization of ourscheme around the basic Poiseuille flow, do coincide with theusual linear equations, cf. {Yih} [3]. References [1] M.K. Smith, The mechanism for the long-waveinstability in thin liquid films J. Fluid Mech., 217,1990, pp.469-485. [2] Benjamin T.B., Wave formation in laminar flow down aninclined plane, J. Fluid Mech. 2, 1957, 554-574. [3] Yih Chia-Shun, Stability of liquid flow down aninclined plane, Phys. Fluids, 6, 1963, pp.321-334. [4] Padula M., On nonlinear stability of MHD equilibriumfigures, Advances in Math. Fluid Mech., 2009, 301-331. [5] Padula M., On nonlinear stability of linear pinch,Appl. Anal. 90 (1), 2011, pp. 159-192. |
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Mon, 23/05/2011 17:00 |
Didier Bresch (Savoie University) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
| In this talk, we will present some recent mathematical features around two-fluid models. Such systems may be encountoured for instance to model internal waves, violent aerated flows, oil-and-gas mixtures. Depending on the context, the models used for simulation may greatly differ. However averaged models share the same structure. Here, we address the question whether available mathematical results in the case of a single fluid governed by the compressible barotropic equations for single flow may be extended to two phase model and discuss derivations of well-known multi-fluid models from single fluid systems by homogeneization (assuming for instance highly oscillating density). We focus on existence of local existence of strong solutions, loss of hyperbolicity, global existence of weak solutions, invariant regions, Young measure characterization. | |||
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Mon, 30/05/2011 17:00 |
Sergiu Kleinerman (Princeton University) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
| Please note that this seminar has been cancelled due to unforeseen circumstances. | |||
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Mon, 06/06/2011 17:00 |
Jesenko Vukadinovic (City University of New York) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
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The talk will address two recent results concerning the Doi-Smoluchowski equation and the Onsager model for nematic liquid crystals. The first result concerns the existence of inertial manifolds for the Smloluchowski equation both in the presence and in the absence of external flows. While the Doi-Smoluchowski equation as a PDE is an infinite-dimensional dynamical system, it reduces to a system of ODEs on a set coined inertial manifold, to which all other solutions converge exponentially fast. The proof uses a non-standard method, which consists in circumventing the restrictive spectral-gap condition, which the original equation fails to satisfy by transforming the equation into a form that does. The second result concerns the isotropic-nematic phase transition for the Onsager model on the circle using more complicated potentials than the Maier-Saupe potential. Exact multiplicity of steady-states on the circle is proven for the two-mode truncation of the Onsager potential. |
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Mon, 13/06/2011 17:00 |
Adriana Garroni (Universita di Roma) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
| The main mechanism for crystal plasticity is the formation and motion of a special class of defects, the dislocations. These are topological defects in the crystalline structure that can be identify with lines on which energy concentrates. In recent years there has been a considerable effort for the mathematical derivation of models that describe these objects at different scales (from an energetic and a dynamical point of view). The results obtained mainly concern special geometries, as one dimensional models, reduction to straight dislocations, the activation of only one slip system, etc. The description of the problem is indeed extremely complex in its generality. In the presentation will be given an overview of the variational models for dislocations that can be obtained through an asymptotic analysis of systems of discrete dislocations. Under suitable scales we study the “variational limit” (by means of Gamma-convergence) of a three dimensional (static) discrete model and deduce a line tension anisotropic energy. The characterization of the line tension energy density requires a relaxation result for energies defined on curves. | |||
