Sumários
The Transport Equation. The Wave Equation.
16 Fevereiro 2017, 11:00 • Hugo Tavares
The Transport Equation [E, 17-19]: solution of the initial-value problem and the nonhomogeneous case; characteristics and the propagation of a discontinuity [S, 180-185]. The Wave Equation [E, 67-69]: d’Alembert solution for the Cauchy problem in the line and in the half-line; [S, 273-83] well-posedness, in Hadamard sense, in the uniform norm and in energy; generalised solutions of the 1-D wave equations; reference to the Dirichlet problem in 1-D, by separation of variables [S, 267-272] and the energy method in general [E, 82-84] — the cone of dependence and the range of influence.
Presentation. A brief historical overview of the classical PDEs.
14 Fevereiro 2017, 14:00 • Hugo Tavares
Presentation of the Syllabus and of the professors. The menu of PDEs in [E, 3-6]. A brief historical overview of the classical PDEs with some key facts: Begin of Calculus with the Leibniz ODE (1684) for the exponencial and the Taylor ODE (1713) for linear oscillations; d’Alembert solution for the wave equation (1746), via characteristics, Daniel Bernoulli solution (1753) in harmonic series and the first controversy about admissible solutions between Euler and d’Alembert; The heat equation, via separation of variables, and its radiation boundary condition by Fourier (1807); the Dirichlet principle used by Riemann (1851) associated with the Laplace equation (1784) and related to the Cauchy-Riemann conditions, already used by Euler and d’Alembert in the 1750’s; its role in the development of the Calculus of Variations and Analysis with Hilbert (1900) an the Gottingen school and B. Levi and Tonelli and the Italian school (1921); the appearance of generalised and variational solutions in 1934 with Sobolev, for the wave equation, and with Leray, for the Navier-Stokes system; the development of Sobolev spaces and the Distribution theory, in the books of Sobolev and of L Schwarz in 1950, exploring the technique of integration by parts, already used by Lagrange in 1761 and Poincaré in 1890, the concept of duality in functional spaces and a priori estimates; the explosion of modern methods, monotonicity, compactness, iterative and constructive (numerical) methods to linear and nonlinear problems after the middle of the 1960’s, with pioneers such that Ladyzhenskaya, Oleinik in Russia, J-L Lions in France, Courant and Nirenberg in the USA; reference to free boundary problems with the modern treatment of the Stefan problem, the Signorini problems and some other contemporary developments.