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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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