Partial Differential Equations

The area of Analysis and Partial Differential Equations is one of those with highest growth rate and spreads itself throughout the whole national territory. This area represents an essential basis for several other areas of mathematics itself and, mainly, applications. There are important groups in the axis Rio-São Paulo, notably at IMPA, LNCC, UFRJ, UNICAMP, USP and UFSCar, as well as in Pernambuco, in UFPE, in Brasilia, in UNB, and also in emergent and less traditional or even isolated centers in Florianópolis, UFSC, Maringá, UEM, Londrina, UEL, João Pessoa and Campina Grande, in UFPB, Fortaleza, in UFCE, Goiania, in UFGO. A presentation of the more expressive lines of research in Analysis and Partial Differential Equations in the national landscape is made below.

Team and Interlocutors

D. Figueiredo and his group at UNICAMP and UnB, H. Frid, J. P. Zubelli, J. Hounie, P. Cordaro, R. Iório, C. Tomei, A. Loula, D. Marchesin, A. Nachbin

J. Bona, A. Grunbaum, P. Markowich, F. Treves

Equations of Fluid Dynamics, Continuum Mechanics, Thermodynamics and Conservation Laws

Therein we find celebrated systems such as the Euler and Navier-Stokes equations for inviscid, viscous, compressible and incompressible fluids, of central importance in mechanical, oceanographic, aeronautic and aerospace engineering, the geostrophic equations, which model the displacement of fluids in the atmosphere, among others, with applications in meteorology and climatology. The equations of linear and nonlinear elasticity, magnetohydrodynamics, chromatography, kinetic theory of gases, visco and thermoelasticity. Among those working in this line of research, we cite H. Frid, D. Marchesin (IMPA), G. Perla, J. Rivera (LNCC), R. Rosa (UFRJ), H. N. Lopes, M. Lopes, J. L. Boldrini, M. A. Rojas-Medar, M. Santos (UNICAMP), J. Hounie and C. Kondo (UFSCar), C. G. Ragazzo (USP). In the context of South America, we may cite interchanges with researchers in Chile, notably C. Conca, and Colombia, notably Y. Lu and L. Rendon. There is an intense articulation with several researchers of the highest level in the international plane, among whom we mention C. Dafermos (Brown University), G-Q. Chen (Northwestern University), and P.-L. Lions (University of Paris XIII).

Linear and Nonlinear Dispersive Equations and Inverse Problems

Therein we find equations such as KdV, Benjamim-Ono, linear and nonlinear Schrödinger, K-P, etc., which constitute models for propagation of water-waves in channels, of plasma-waves, models in atomic physics, quantum mechanics, etc.. The inverse problems in Scattering Theory arise in important applications such as computerized tomography, non-invasive methods in magnetocardiography, seismic and oil prospection, etc.. Among those working in this line of research, we cite R. Iório, F. Linares, J. Zubelli (IMPA), C. Tomei (PUC), G. Perla (LNCC), H. Biagioni, M. Scialom, J. Angulo, O. Lopes (UNICAMP), E. Alarcon (UFGO), F. Montenegro (UFCE), E. Lueders (UEL). In the context of South America, we may cite interchanges with researchers of Colombia, notably G. Rodriguez and F. Soriano, and of Argentina, notably D. Rial. There is an intense cooperation with researchers of the highest level in the international plane, among whom we mention J. Bona (University of Texas at Austin), F. A. Grunbaum (University of California at Berkeley), P. Markowich, (University of Vienna).

Linear and Nonlinear Elliptic Equations

Therein we find equations of geometry e.g., minimal surfaces, equations arising in variational problems of diverse types, stationary fluids, elasticity, plasma, filtration through porous media, theory of potential, etc.. Among those working in this line of research, we cite H. Frid, R. Iório (IMPA), D. G. Figueiredo, O. Lopes, H. N. Lopes (UNICAMP), J. V. Gonçalves, E. Alves (UNB), A. M. Bertone, C. Alves, D. C. Moraes Filho, J. M. Bezerra do Ó (UFPB). There is an intense articulation with researchers of the highest level in the international plane, among whom we mention L. Nirenberg (Courant Institute), L. Caffarelli (University of Texas at Austin), P. Rabinovitch (University of Wisconsin).

Pseudo-Differential Operators, Microlocal Analysis, Harmonic Analysis, Linear and Nonlinear Hyperbolic Equations

Therein we find the linear and nonlinear hyperbolic equations, present everywhere in Mathematical Physics, where feature beside others already cited, the Einstein equations, of general relativity, the Yang-Mills equations, of relativistic quantum mechanics, among others. Among those working in this line of research, we cite H. Frid, F. Linares, J. Zubelli (IMPA), J. Hounie, A. Bergamasco, J. R. Santos Filho (UFSCar), P. Cordaro, P. P. Schirmer, S. Toscano R. de Melo (USP), H. N. Lopes (UNICAMP), F. Cardoso, M. L. Leite, J. Tavares (UFPE). There is an intense cooperation with researchers of the highest level in the international plane, among whom we mention, besides those already cited above, F. Treves (University of Rutgers).

Nonlinear Differential Equations

We detail research objectives pursued by the group of D. Figueiredo at UNICAMP and UnB: to study the properties of solutions of ordinary and partial differential equations, namely, existence, uniqueness, multiplicity, symmetries, asymptotic behavior, qualitative properties, controlability, implementation and convergence of numerical approximations.

This group includes O. Lopes, M. A. Teixeira, J. L. Boldrini, J. V. Gonçalves, E. Silva, H. Biagioni, M. Lopes, H. N. Lopes, Y. Jianfu, M. Scialom, K. Rezende, J. Ângulo, M. Santos, M. Casarin, M. Rojas.

Geometric Theory of Partial Differential Equations and Several Complex Variables

Some of the most relevant questions in the field of systems of first order linear PDEs, to be pursued by P. Cordaro, J. Hounie and A. Bergamasco are:

Study of the general properties of the solutions to the homogeneous systems (e.g. approximation properties, propagation of singularities and supports, hypoanalyticity, etc.); Local solvability for the differential complex associated to a given involutive system; Involutive systems of first order non-linear PDE; Global problems and normal forms.

It is important to remark that classical results that are known for complex and Cauchy-Riemann structures find a natural context within the theory of involutive systems. The analysis under this new point of view has given new and important results, which certainly have clarified the understanding of the theory. problems, and are the approaches of main interest in this project.