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The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant
Get PDFIn this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in 1+3 spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state p=c[superscript 2]ρ,0 < c[superscript 2] < 1/3, the background metric + fluid solutions are globally future-stable under small irrotational perturbations of their initial data. In particular, we prove that the perturbed spacetime solutions, which have the topological structure [0,∞)XT[superscript 3], are future causally geodesically complete. Our analysis is based on a combination of energy estimates and pointwise decay estimates for quasilinear wave equations featuring dissipative inhomogeneous terms. Our main new contribution is showing that when 0 < c[superscript 2] < 1/3, exponential spacetime expansion is strong enough to suppress the formation of fluid shocks. This contrasts against a well-known result of Christodoulou, who showed that in Minkowski spacetime, the corresponding constant-state irrotational fluid solutions are unstable
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A fully anisotropic mechanism for formation of trapped surfaces in vacuum
Get PDFWe present a new, fully anisotropic, criterion for formation of trapped surfaces in vacuum. More precisely we provide local conditions on null data, concentrated in a neighborhood of a short null geodesic segment (possibly flat in all other directions) whose future development contains a trapped surface. This extends considerably the previous result of Christodoulou (Monographs in Mathematics. European Mathematical Society, Switzerland, 2009) which required instead a uniform condition along all null geodesic generators. To obtain our result we combine Christodoulou’s mechanism for the formation of a trapped surface with a new deformation process which takes place along incoming null hypersurfaces.National Science Foundation (U.S.) (Grant DMS-1001500)National Science Foundation (U.S.). Focused Research Group (Grant DMS-1065710
Linear and nonlinear wave equations on black hole spacetimes
No full textIn this thesis, I study three problems related to the linear and nonlinear wave equations on black hole spacetimes. These problems are motivated by the nonlinear stability of Kerr spacetime.
First, I prove that sufficiently regular solutions to the wave equation on the exterior of the Schwarzschild black hole obey the estimates and on a compact region of , including inside the black hole region.
Second, I prove that sufficiently regular solutions to the wave equation on the exterior of the sufficiently slowly rotating Kerr black hole also obey the estimates . The first two results are proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite and along the event horizon.
Third, I study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr
spacetimes. I prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method and in particular makes use of the improved decay rates obtained in the first and second results
Nonlinear wave equations on time dependent inhomogeneous backgrounds
No full textIn this thesis, I study the nonlinear wave equations on a class of asymptotically flat Lorentzian manifolds (R3+1, g) with time dependent inhomogeneous metric g.
Based on a new approach for proving the decay of solutions of linear wave equations, I give several applications to nonlinear problems. In particular, I show the small data global existence result for quasilinear wave equations satisfying the null condition on a class of time dependent inhomogeneous backgrounds which do not settle to any
particular stationary metric
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
The wave equation on the exterior of a Schwarzschild black hole
No full textThis thesis investigates a wave equation on the Minkowski
and Schwarzschild spacetimes. We rely on the vector field method and
the rp-weighted inequalities discovered by Dafermos and Rodnianski to
show the decay of energy and solutions in the cases studied
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