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Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

Juan Luis Vázquez

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199202973
eISBN:
9780191707919
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199202973.001.0001
Subject:
Mathematics, Applied Mathematics

This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as ... More


The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence

Robert W. Batterman

Published in print:
2001
Published Online:
November 2003
ISBN:
9780195146479
eISBN:
9780199833078
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/0195146476.001.0001
Subject:
Philosophy, Philosophy of Science

This book focuses on a form of reasoning in science that I call “asymptotic reasoning.” At base, this type of reasoning involves methods that eliminate details and, in some sense, precision. ... More


Asymptotic Differential Algebra and Model Theory of Transseries

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691175423
eISBN:
9781400885411
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175423.001.0001
Subject:
Mathematics, Computational Mathematics / Optimization

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a ... More


Asymptotic Explanation

Robert W. Batterman

in The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence

Published in print:
2001
Published Online:
November 2003
ISBN:
9780195146479
eISBN:
9780199833078
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0195146476.003.0004
Subject:
Philosophy, Philosophy of Science

This chapter provides a fairly detailed discussion of the renormalization group account of the universality of critical phenomena. This discussion allows one to determine the distinctive features of ... More


Criteria ensuring global existence

Hans Ringström

in On the Topology and Future Stability of the Universe

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199680290
eISBN:
9780191760235
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199680290.003.0026
Subject:
Mathematics, Mathematical Physics, Geometry / Topology

In Chapter 26, we formulate criteria ensuring future global existence of solutions (in the spatially homogeneous setting). Moreover, we provide some rough conclusions concerning the asymptotics. ... More


A potential with a positive non-degenerate local minimum

Hans Ringström

in On the Topology and Future Stability of the Universe

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199680290
eISBN:
9780191760235
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199680290.003.0027
Subject:
Mathematics, Mathematical Physics, Geometry / Topology

In Chapter 27, we focus on the case that the scalar field converges to a positive non-degenerate minimum of the potential. In the spatially homogeneous setting, it is then possible to derive detailed ... More


Global existence

Hans Ringström

in On the Topology and Future Stability of the Universe

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199680290
eISBN:
9780191760235
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199680290.003.0031
Subject:
Mathematics, Mathematical Physics, Geometry / Topology

In Chapter 31, we prove future global existence in the case that the initial data are specified on the 3-torus, and assuming the initial data are close to those of de Sitter space. The proof is a ... More


Asymptotics

Hans Ringström

in On the Topology and Future Stability of the Universe

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199680290
eISBN:
9780191760235
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199680290.003.0032
Subject:
Mathematics, Mathematical Physics, Geometry / Topology

In Chapter 32, we derive detailed asymptotics of the solutions constructed in Chapter 31. We separate between the two cases that the scalar field is zero and non-zero. The case of zero scalar field ... More


The Semi-group on

Charles L. Epstein and Rafe1 Mazzeo

in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157122
eISBN:
9781400846108
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157122.003.0012
Subject:
Mathematics, Probability / Statistics

This chapter deals with the semi-group on the space Β‎⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components ... More


The sharp Weyl formula

Christopher D. Sogge

in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.003.0003
Subject:
Mathematics, Numerical Analysis

This chapter considers the sharp Weyl formula using the tools provided in the previous chapter. It attempts to prove the sharp Weyl formula which says that there is a constant c, depending on (M,g) ... More


Improved spectral asymptotics and periodic geodesics

Christopher D. Sogge

in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.003.0005
Subject:
Mathematics, Numerical Analysis

This chapter proves an improved Weyl formula under the assumption that the set of periodic geodesics for (M,g) has measure zero. It then shows trace estimates associated with shrinking spectral ... More


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