Applied Mathematics and Numerical Analysis Seminar 

The Navier-Stokes equations are known since the 19th century. They were derived under the assumption that the fluid is a continuous medium, under Newton's law, and moreover under an a priori assumption that velocity and pressure have a certain smoothness. The existence of solutions with this smoothness in a 3D-case still remains an open mathematical problem!

Comment: There exist mechanisms in real fluids which do not enable the speed of motion to increase above all limits. It is highly desirable to know whether the Navier-Stokes model (a good model?) also involves such mechanisms or whether it admits solutions with singularities?

There are many real fluids such as water, which the Newtonianconstitutive equation describes extremely well in almost all flowsituations. Small amounts of polymer additive can have a dramaticeffect on the behavior of these liquids. For example, B. A. Tomsstudied flows in straight pipes and discovered that in theturbulent regime small amounts of polymer additive couldsignificantly reduce the pressure drop necessary to attain a givenflow rate. These changes in behavior are attributed to theviscoelastic nature of the polymeric solution and numerousconstitutive equations have been developed to model these fluids.In contrast, in curved pipes polymer additives were foundexperimentally both to alter the relationship between pressure dropand flow rate in the laminar regime and to alter the criticalReynolds number for transition to turbulence. In this talk, we willdiscuss results for steady, fully developed flows of viscoelasticfluids in curved pipes and contrast this behavior with flows ofNewtonian fluids. Following the approach of W. R. Dean and otherauthors, we have used regular perturbation methods to study flowsof viscoelastic fluids in curved pipes. We have obtained explicitsolutions to the perturbation equations at first order for secondorder fluids and a modified Oldroyd-B fluid. In the absence ofinertial effects, flows of Newtonian fluids in curved pipes do notdisplay a secondary flow, rather a uniaxial flow exists whichdiffers only slightly from the straight pipe solution. In contrast,even in the absence of inertial effects, the class of viscoelasticfluids studied display a secondary motion (see, e.g. Thomas 1963,Bowen et al. 1991, Robertson and Muller 1996). Significantly, for acountable number of combinations of material parameters andReynolds numbers, there is a loss of uniqueness of the solution tothe perturbation equations. For other values of material parametersand Reynolds number, a solution does not even exist. There is aregion in parameter space which is free of such singularities. Thislack of existence to the perturbation equations regardless of themagnitude of the curvature ratio, implies a lack of existence of asolution which is a steady, fully developed perturbation of thestraight pipe solution. The implications of this result are underinvestigation.

In this talk, we review some recent results on fractionalevolution equations of type $D^\alpha_t u+ Au = f$, $t\geq 0$,where $\alpha\in (0,2)$, $\alpha\neq 1$.

First, as a typical example, we consider the fractional Burgersequation. Next, we formulate a general result with $A$$m$-accretive. Some comments on maximal regularity follow; finallywe present an unsolved problem concerning fractional nonlinearhyperbolic equations.

Os problemas de optimização não linear aparecem frequentementeassociados ao controlo em simulação, ao projecto em engenharia e àidentificação de parâmetros e ao ajuste de dados em experimentação.A função objectivo e as restrições apresentam, com regularidade,uma estrutura e um escalonamento próprios, associados àdiscretização de equações diferenciais.

Métodos numéricos modernos em optimização não linear, como sãopor exemplo os casos dos métodos SQP e dos métodos de regiões deconfiança, podem e devem ser adaptados para tirar partido dascaracterísticas da aplicação. Outras situações, como adegenerescência das restrições, por um lado, e a dificuldadecomputacional ou experimental em obter o valor das funções e dassuas derivadas, por outro, colocam sérios desafios aooptimizador.

We compute bound on effective properties of homogenizedmaterials in conduction and linear elasticity. We use $H$-measures,following the original ideas and exposition of Luc Tartar. After having recovered classical Hashin-Shtrikman simple bounds wepropose to deal with coupled bounds in conduction.