**Summer Seminars on Differential Equations**

1ª palestra: 10h – 10h40

**Higher-order models in fluid film lubrication **

*Igor Pažanin (University of Zagreb – Croácia) *

**Resumo:** The aim of this talk is to present recent results on new asymptotic models for ﬂuid ﬁlm lubrication. We consider the situation in which two rigid surfaces being in relative motion are separated by a thin layer of ﬂuid acting as a lubricant. Lower surface is assumed to be perfectly smooth, while the upper is rough with roughness described by some function *h*. Such situation appears naturally in many engineering applications consisting of moving machine parts, e.g. in journal bearings or computer disk drives. The ﬁrst part of the talk will be devoted to the isothermal setting. The lubricant is ﬁrst assumed to be classical Newtonian ﬂuid, and then we extend our results to incompressible micropolar ﬂuid taking into account the ﬂuid microstructure as well. Using asymptotic analysis with respect to the ﬁlm thickness, we derive the higherorder corrections of the standard Reynolds approximation. The eﬀective equations are similar to the Brinkman model for porous medium ﬂow. In the second part of the talk we investigate the ﬂow and heat transfer inside a thin layer of a ﬂuid. We assume that the problem is described by the Stokes equation coupled with the heat equation involving the nonlinear viscous dissipation term. A second-order eﬀective model is proposed in the form of the explicit formulae for the velocity and temperature. Justiﬁcation via error estimate is also provided.

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2ª palestra: 11h – 11h40

**Locally Periodic Unfolding Operator for Highly Oscillating Rough Domains **

*Ravi Prakash (Universidad de Concepción – Chile)*

**Resumo:** In this talk, we try to understand the locally-periodic oscillating domain via unfolding operators. A three dimensional rough domain Ωε, ε > 0 a small parameter, will be considered for the study where the boundary is rapidly oscillating with high amplitude. There are few articles with locally-periodic boundary oscillations with small amplitude. But, we do not ﬁnd any literature with high-amplitude (O(1)) locally-periodic oscillating domains. This is an attempt to study asymptotic behavior of partial diﬀerential equations in locally periodic rough domains with an eye towards the general oscillating domains without periodicity. We will see the development of locally-periodic unfolding operators to study few model problems. We recall the work of M. Ptashnyk where he has developed an unfolding operator in [Multiscale Model. Simul. (2015)] to study PDEs with locally-periodic coeﬃcients. The development of these unfolding operators motivated the deﬁnition and analysis of unfolding operators for locally-periodic oscillating domains with high amplitude. We consider a non-linear inhomogeneous Robin boundary value problem posed on this domain to demonstrate the utility of this new operator.

**(Café às 10h40 - ao lado do auditório)**