Yoshikazu Giga – författare
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This book addresses the issue of uniqueness of a solution to a problem – a very important topic in science and technology, particularly in the field of partial differential equations, where uniqueness guarantees that certain partial differential equations are sufficient to model a given phenomenon.
This book is intended to be a short introduction to uniqueness questions for initial value problems. One often weakens the notion of a solution to include non-differentiable solutions. Such a solution is called a weak solution. It is easier to find a weak solution, but it is more difficult to establish its uniqueness. This book examines three very fundamental equations: ordinary differential equations, scalar conservation laws, and Hamilton-Jacobi equations. Starting from the standard Gronwall inequality, this book discusses less regular ordinary differential equations. It includes an introduction of advanced topics like the theory of maximal monotone operators as wellas what is called DiPerna-Lions theory, which is still an active research area. For conservation laws, the uniqueness of entropy solution, a special (discontinuous) weak solution is explained. For Hamilton-Jacobi equations, several uniqueness results are established for a viscosity solution, a kind of a non-differentiable weak solution. The uniqueness of discontinuous viscosity solution is also discussed. A detailed proof is given for each uniqueness statement.
The reader is expected to learn various fundamental ideas and techniques in mathematical analysis for partial differential equations by establishing uniqueness. No prerequisite other than simple calculus and linear algebra is necessary. For the reader’s convenience, a list of basic terminology is given at the end of this book.
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The aim of this proceeding is addressed to present recent developments of the mathematical research on the Navier-Stokes equations, the Euler equations and other related equations. In particular, we are interested in such problems as:1) existence, uniqueness and regularity of weak solutions2) stability and its asymptotic behavior of the rest motion and the steady state3) singularity and blow-up of weak and strong solutions4) vorticity and energy conservation5) fluid motions around the rotating axis or outside of the rotating body6) free boundary problems7) maximal regularity theorem and other abstract theorems for mathematical fluid mechanics.
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Japan is a tiny country that occupies only 0.25% of the world’s total land area. However, this small country is the world’s third largest in economy: the Japanese GDP is roughly equivalent to the sum of any two major countries in Europe as of 2012.This book is a first attempt to ask leaders of top Japanese companies, such as Toyota, about their thoughts on mathematics. The topics range from mathematical problems in specific areas (e.g., exploration of natural resources, communication networks, finance) to mathematical strategy that helps a leader who has to weigh many different issues and make decisions in a timely manner, and even to mathematical literacy that ensures quality control. The reader may notice that every article reflects the authors’ way of life and thinking, which can be evident in even one sentence.This book is an enlarged English edition of the Japanese book What Mathematics Can Do for You: Essays and Tips from Japanese Industry Leaders. In this edition we have invited the contributions of three mathematicians who have been working to expand and strengthen the interaction between mathematics and industry.The role of mathematics is usually invisible when it is applied effectively and smoothly in science and technology, and mathematical strategy is often hidden when it is used properly and successfully. The business leaders in successful top Japanese companies are well aware of this invisible feature of mathematics in applications aside from the intrinsic depth of mathematics. What Mathematics Can Do forYou ultimately provides the reader an opportunity to notice what is hidden but key to business strategy.
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Role Of Metrics In The Theory Of Partial Differential, The - Proceedings Of The 11th Mathematical Society Of Japan, Seasonal Institute (Msj-si)
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