Analysis of the Einstein Equations

FCT Project PTDC/MAT/108921/2008 (2010-12)

Centro de Matemática (CMAT), Universidade do Minho, Braga, Portugal
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos (CAMGSD), Instituto Superior Técnico, Lisboa, Portugal


Team Members

Filipe Mena (CMAT, Project Coordinator)
José Natário (CAMGSD, Coordinator of the IST team)
Simone Calogero (Granada, Spain, formerly at CMAT)
Pedro Girão (CAMGSD)
Leonor Godinho (CAMGSD)
João Lopes Costa (CAMGSD)
Jorge Silva (CAMGSD)
Artur Alho (CMAT, Ph.D. student)


Project Summmary

This project is aimed to study a number of problems which involve nonlinear partial differential equations in General Relativity. In particular, we aim to investigate (i) The existence of global solutions to the spatially homogeneous Einstein-Boltzmann system (ii) The cosmic censorship conjecture of Penrose in spacetimes with a cosmological constant (iii) Initial boundary value problems arising in spacetime matchings.

Publications of team members in 2010-12

29 - S. Calogero, J. M. Heinzle, Oscillations toward the singularity of LRS Bianchi type IX cosmological models with Vlasov matter. SIAM J. Appl. Dyn. Syst. 9, 1244-1262 (2010) http://arxiv.org/abs/1011.3982
28 - L. Godinho and M. E. Sousa-Dias, The fundamental group of S^1-manifolds, Canad. J. Math. 62 (2010), 1082-1098. http://www.math.ist.utl.pt/~lgodin/ Page1.html
27 -T. Harmark, J. Natário and R. Schiappa, Greybody Factors for d-Dimensional Black Holes, Adv. Theor. Math. Phys. 14 (2010) 727-794. http://arxiv.org/abs/0708.0017
26 - I. Brito, J. Carot, F. C. Mena and E. L. R. Vaz, Cylindrically symmetric static solutions of the Einstein Field Equations for elastic matter, Journal of Mathematical Physics, vol. 53 (2012) 122504 (16 pages) web: http://scitation.aip.org/content/aip/journal/jmp/53/12/10.1063/1.4769223
25 - J. A. Alcántara, S. Calogero: Newtonian limit and trend to equilibrium for the relativistic Fokker-Planck equation, accepted in J. Math. Phys, 2012, web: http://arxiv.org/abs/1107.5140
24 - S. Calogero, Cosmological models with fluid matter undergoing velocity diffusion. J. Geom. Phys. 62 (2012), no. 11, 2208 -2213. web: http://arxiv.org/abs/1202.4888
23 - S. Calogero: Exponential convergence to equilibrium for kinetic Fokker- Planck equations. Comm. Partial Differential Equations 37 (2012), no. 8, 1357–1390 web: http://arxiv.org/abs/1009.5086
22 - J. Costa, A. Alho, J. Natário, "Spherical linear waves in de Sitter spacetime", J. Math. Phys. 53 (2012) 052501 (9 pages) web: http://arxiv.org/abs/1107.0802
21 - J. Natario, Optimal time travel in the Gödel universe, Gen. Rel. Grav. 44 (2012) 855-874. web: http://arxiv.org/abs/1105.6197
20 - J. Natario, P. G. Henriques, The rocket problem in general relativity, J. Optim. Theory Appl. 154 (2012) 500-524. web: http://arxiv.org/abs/1105.5235
19 - F. Costa, C. Herdeiro, J. Natario, M. Zilhão, Mathisson's helical motions for a spinning particle - are they unphysical?, Phys. Rev. D 85 (2012) 024001 (11 pages). web: http://arxiv.org/abs/1109.1019
18 - P. T. Chrusciel, J. L. Costa, Markus Heusler, Stationary Black Holes: Uniqueness and Beyond, Living Rev. Relativity 15, (2012), 7 web: http://arxiv.org/abs/1205.6112
17 - Girão, P. (2011). Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue. Nonlinear Anal. 74 (1), 94-113.
16 - S. Calogero: A kinetic theory of diffusion in general relativity with cosmological scalar field. J. Cosm. Astrop. Phys. 11 (2011), 016 web: http://arxiv.org/abs/1107.4973
15 - J. A. Alcántara, S. Calogero: On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models 4 , 401-426 (2011) web: http://arxiv.org/abs/1011.5429
14 - S. Calogero, J. M. Heinzle, Bianchi Cosmologies with Anisotropic Matter: Locally Rotationally Symmetric Models. Physica D: Nonlin. Phen. 240 , 636-669 (2011) web: http://arxiv.org/abs/0911.0667
13 - A. Alho and F. C. Mena , Pre-inflationary homogeneization of scalar field cosmologies, Physics Letters B, vol. 703 (2011) 537-542
12 - M. Le Delliou, F. C. Mena and J. Mimoso, The role of shell crossing on the existence and stability of trapped matter shells in spherical inhomogeneous Lambda-CDM models, Physical Review D, vol. 83 (2011) 103528 (16 pages)
11. Girão P and Tehrani H, On the Fucik spectrum of the wave operator and an asymptotically linear problem, J. Math. Anal. Appl. 366 (2010), 55–66
10. Calogero S, Calvo J, Sánchez O and Soler J, Virial inequalities for steady states in relativistic galactic dynamics, Nonlinearity, 23 (2010) 1851-1871
9. Calogero S and Heinzle J M, Asymptotics of LRS Bianchi type I cosmological models with elastic matter, General Relativity and Gravitation, 42 (2010) 1491-1512
8. Calogero S, Calvo J, Sánchez O and Soler J, Dispersive behavior in Galactic Dynamics, Discr. Cont. Dyn. Syst., series B 14 (2010) 1-16
7. Calogero, S and Heinzle J M, Closed cosmological models that satisfy the strong energy condition but do not recollapse, Physical Review D, 81 (2010) 023520
6. Alho A, On the linear stability of a scalar-field with exponential potential in a flat Robertson-Walker background, Proceedings of the Spanish Relativity Meeting 2009, Journal of Physics: Conference Series, 229 (2010) 012015 (4 pages)
5. Alho A, Mena F C and Valiente-Kroon J, The Einstein-Klein-Gordon-Friedrich system and the non-linear stability of scalar field cosmologies, Submitted of publication, 2010 online
4. Mena F C, Natario J and Tod P, Formation of Higher Dimensional Topological Black Holes, Annales Henri Poincaré, vol. 10 (2010) 1359-1376
3. Mena F C, On the asymptotic stability of De-Sitter spacetime: a non-linear perturbative approach, Journal of Physics A: Mathematical and Theoretical, vol. 43 (2010) 165401 (8 pages)
2. Mimoso J, Le Delliou M and Mena F C, Spherically symmetric models with pressure: separating expansion from contraction and generalizing the TOV condition, Physical Review D, vol. 81 (2010) 123514 (17 pages)
1. Natario J, An elementary derivation of the Montgomery phase formula for the Euler top, Journal of Geometric Mechanics, 2 (2010) 113-118


Communications of team members in 2010-11

29- S. Calogero, Relativistic Diffusion, ”Third Minho Meeting on Mathematical Physics”, Univ. Minho, Novembro 4, 2011.
28- J. Costa, Towards the Einstein-scalar field system with positive cosmological constant, ”Third Minho Meeting on Mathematical Physics”, Univ. Minho, Novembro 4, 2011.
27- J. Natario, Cosmic censorship in spherical symmetry, ”Third Minho Meeting on Mathematical Physics”, Univ. Minho, Novembro 4, 2011.
26 - A. Alho, "Spherical linear waves in de Sitter spacetime", poster na "XX Fall workshop on Geometry and Physics", Madrid, Espanha (2011)
25 - A. Alho, "Spherical linear waves in de Sitter spacetime", IV Black Holes Workshop, Universidade de Aveiro (2011)
24- F. Mena, Junction conditions and formation of black holes in higher dimensions", poster na "XX Fall workshop on Geometry and Physics", Madrid, Espanha (2011)
23 - F. Mena, On the asymptotic stability of de-Sitter: a non-linear perturbative approach", poster no "Spanish Relativity Meeting 2011", Madrid, Espanha (2011)
22- F. Mena, Junction conditions and formation of black holes in higher dimensions", poster na "XX Fall workshop on Geometry and Physics", Madrid, Espanha (2011)
21 - F. Mena, Radiative Gravitational Collapse to Black Holes in 5 Dimensions", IV Black Holes Workshop, Universidade de Aveiro (2011)
20 - F. Mena, Gravitational collapse to higher dimensional black holes", Spanish Relativity Meeting 2011, Madrid, Espanha (2011)
19 - J. Silva, "Local and Global Well-posedness for the Critical Schrödinger- Debye System", Problems in the Dynamics of Nonlinear Dispersive Equations Meeting, Wolfgang Pauli Institute, Viena, Áustria, 28/4/2011.
18- A. Alho, "The Einstein-Klein-Gordon-Friedrich system and the non-linear stability of scalar field cosmologies", PDEs, Relativity and non-linear waves, 5-9/April/2010, Granada, Spain
17 - S. Calogero, Second Minho meeting in mathematical physics, University of Minho, Guimarães (Portugal) TITLE: Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations on Riemannian Manifolds, November 5, 2010
16 - S. Calogero, Fluid-Kinetic Modelling in Biology, Physics and Engineering, Isaac Newton Institute, University of Cambridge (UK), Trend to Equilibrium for Kinetic Fokker-Planck Equations on Riemannian Manifolds, September 6--10, 2010
15 - F. Mena, "Spacetime Matching and Black Holes", apresentada na sessão de "String Theory and Mathematical Physics", 3rd Iberian Mathematical Meeting, Universidade do Minho, Novembro 2010.
14 - F. Mena, "First order perturbations of the Einstein-Straus model", 19th International Meeting on General Relativity and Gravitation, 5-9/July/ 2010, Mexico City, Mexico
13 - A. Alho, "The Einstein-Klein-Gordon-Friedrich system and the non-linear stability of scalar field cosmologies", 19th International Meeting on General Relativity and Gravitation, 5-9/July/ 2010, Mexico City, Mexico
12 - J. Natário, Godel universe and the isoperimetric problem, III Workshop on Black Holes, Universidade do Minho, December 2010.
11 - J. Natário, Formation of higher-dimensional topological black holes, PDEs, Relativity & Nonlinear Waves, Granada, 5-9 April 2010.
10 - J. Silva, "Global Well-Posedness for a Critical Perturbation of the Nonlinear Schrödinger Equation", 2011 ICMC Summer Meeting on Differential Equations, USP-São Carlos, Brasil, 7/2/2011.
9 - S. Calogero, Trend to equilibrium for kinetic Fokker-Planck equations, Probability and Geometry (ProbaGeo), Strasbourg (France), May 30--June 1, 2011
8 - S. Calogero, Dynamics of spatially homogeneous solutions to the Einstein- Vlasov system, Vlasov models in kinetic theory, ICERM (Brown University), Providence (USA), September 19--23, 2011
7 - J. Silva, "Global Well-Posedness for a Critical Perturbation of the Nonlinear Schrödinger Equation", 2011 ICMC Summer Meeting on Differential Equations, USP-São Carlos, Brasil, 7/2/2011.
6 - J. Natário, An elementary derivation of the Montgomery phase formula for the Euler top, Geometria em Lisboa, April 2010
5 - S. Calogero, March 21, 2011, Chalmers University, Gothenburg (Sweden), TITLE: Hypocoercivity of kinetic Fokker-Planck equations on Riemannian manifolds
4 - S. Calogero, April 1, 2011, Center of Mathematics for the Applications, Oslo (Norway), TITLE: Hypocoercivity of kinetic Fokker-Planck equations on Riemannian manifolds
3- J. Silva, 5/5/2010 "Local well-posedness for generalized KP-II type equations on cylinders", Centro de Matemática e Aplicações, FCT - Universidade Nova de Lisboa
2- J. Silva, 29/7/2010 "Local well-posedness for generalized KP-II type equations on cylinders", Centro de Matemática, Universidade do Minho.
1- J. Silva, 18/2/2011 "Boa colocação local para equações generalizadas, de tipo KP-II, em cilindros", Instituto de Matemática - Universidade Federal de Alagoas, Maceió, Brasil.



Contact Details

Filipe Mena, Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal.
Tel: +351 253 604087 (direct), Fax: +351 253 604369 (dept.)
Email: fmena@math.uminho.pt