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2026: 2742025: 12472024: 18512023: 168

Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle

2025 · Book · Koleksi Perpustakaan

The goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, small and smooth enough initial data, is almost globally defined in time on Sobolev spaces, provided the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, a normal forms-based procedure is used, eliminating those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations form a quasi-linear system, the usual normal forms approaches would face the well-known problem of losses of derivatives in the unbounded transformations. To overcome this, after a paralinearization of the capillary-gravity water waves equations, we perform several paradifferential reductions to obtain a diagonal system with constant coefficient symbols, up to smoothing remainders. Then we start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we seek solutions even in space, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.

The State of Food and Agriculture 2020

2025 · Book · Koleksi Perpustakaan

Intensifying water constraints threaten food security and nutrition. Thus, urgent action is needed to make water use in agriculture more sustainable and equitable. Irrigated agriculture remains by far the largest user of freshwater, but scarcity of freshwater is a growing problem owing to increasing demand and competition for freshwater resources. At the same time, rainfed agriculture is facing increasing precipitation variability driven by climate change. These trends will exacerbate disputes among water users and inequality in access to water, especially for small-scale farmers, the rural poor and other vulnerable populations. The State of Food and Agriculture 2020 presents new estimates on the pervasiveness of water scarcity in irrigated agriculture and of water shortages in rainfed agriculture, as well as on the number of people affected. It finds major differences across countries, and also substantial spatial variation within countries. This evidence informs a discussion of how countries may determine appropriate policies and interventions, depending on the nature and magnitude of the problem, but also on other factors such as the type of agricultural production system and countries’ level of development and their political structures. Based on this, the publication provides guidance on how countries can prioritize policies and interventions to overcome water constraints in agriculture, while ensuring efficient, sustainable and equitable access to water.

The Breadth and Depth of Continuum Mechanics

2025 · Book · Koleksi Perpustakaan

This volume collects papers dedicated to Jerry Ericksen on his sixtieth birthday, December 20, 1984. They first appeared in Volumes 82-90 (1983-1985) of the Archive for Rational Mechanics and Analysis. At the request of the Editors the list of authors to be invited was drawn up by C. M. Dafermos, D. D. Joseph, and F. M. Leslie. The breadth and depth of the works here reprinted reflect the corresponding qualities in Jerry Ericksen's research, teaching, scholarship, and inspiration. His interests and expertness center upon the mechanics of materials and extend to everything that may contribute to it: pure analysis, algebra, geometry, through all aspects of theoretical mechanics to fundamental experiment, all of these illumi nated by an intimate and deep familiarity with the sources, even very old ones. He is independent of school and contemptuous of party spirit; his generosity in giving away his ideas is renowned, but not everyone is capable of accepting what is offered. His writings are totally free of broad claims and attributions beyond his own study. Some are decisive, some are prophetic, and all are forthright. His work has served as a beacon of insight and simple honesty in an age of ever more trivial and corrupt science. The authors of the memoirs in this volume are his students, colleagues, admirers, and (above all) his friends.