European Mathematical Society - 35 Partial differential equations
https://euro-math-soc.eu/msc/35-partial-differential-equations
enHot molecules, cold electrons
https://euro-math-soc.eu/review/hot-molecules-cold-electrons
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Paul Nahin is well known as a popular science writer. Some twenty books he has published since he started at the end of the previous century with a biography of Oliver Heaviside. Most of his books are dealing with topics involving physics, but there is always keen attention given to mathematics. For example he authored books on explicit mathematical topics like <em>An imaginary tale. The story of √-1</em> (1998) and <em>Dr. Euler's fabulous formula</em> (2006).</p>
<p>The present book is like a mathematical textbook for engineering or science students in which all the derivations are given. Nahin uses an historical approach to introduce Fourier analysis, derive the heat equation, and solve it for different geometries and boundary conditions. When applied to a cooling sphere, this illustrates how William Thomson (Lord Kelvin) estimated the age of our planet by computing how a molten sphere cools down to a sphere with a solid crust (that explains the hot molecules of the title). When the equation is solved for a long cable, it explains how electrons travel through the transatlantic submarine telegraph cable (hence the cold electrons).</p>
<p>So there are a lot of formulas and derivations, but it is not a course as it would be written in modern times. It is taken out of a regular university curriculum and it assumes only the basic calculus from a course at a first year science, engineering, or mathematics level. Fourier series and the Fourier transform are developed from basic principles. Nowadays, the heat equation can be solved efficiently using for example Laplace transforms, but Nahin prefers to use essentially the mathematics available to Fourier who solved it in the time domain. Every step is explained to the smallest details. Sometimes the approach is using an engineering style of mathematics. This means that Nahin is just using an insight from the underlying physics to propose a certain method or to justify a certain solution. Infinite sums and integrals are interchanged, postponing to when the eventual result is obtained whether this makes sense or not. The square root of minus 1 is however denoted by the mathematical standard i, and not by j as is customized by the engineering community to distinguish it from electric current which is also indicated by i or I.</p>
<p>This "engineering mathematics" is also what Fourier applied. His original report on the solution of the heat equation in 1807 was criticized by Lagrange and Laplace because he used his formally obtained infinite sums as if they were ordinary functions. It is not until his "new mathematics" was better understood, ten years later that he was taken seriously and was accepted as a member of the French Academy of Science. Chapter 1 is an eye opener to the sort of mathematics that Fourier introduced. It is for example shown how Fourier obtained $\frac{\pi}{4}=\sum_{k=0}^\infty (−1)^k\frac{\cos(2k+1)x}{2k+1}$. This is well known for $x=0$ (Leibniz formula), but there are many other values of $x$ for which this is also true, much to the surprise of Fourier's contemporaries.</p>
<p>In Chapter 2, the Fourier series are derived and it is shown that they are optimal approximations in a least squares sense. Convergence is not proved. Nahin asks the reader to "accept that our mathematician colleagues have, indeed, established its truth". In this way Fourier series, the Parseval identity, Dirichlet's integral, and the Fourier transform are introduced.</p>
<p>Chapter 3 derives the heat equation $\frac{\partial u}{\partial t}=k(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2})$ from first principles. When the medium is a long radiating cable, it is essentially one dimensional and a simple solution is found as a decaying exponential assuming a constant energy loss per unit length, not depending on time. The solution of the equation for different geometries and different physical boundary conditions is discussed in the next chapter. It starts with a cooling problem of an infinite slab with finite thickness ($0\le x \le L$) using a separation of variables ($u(t,x)=f(t)g(x)$) as Johann Bernoulli did. This results in an infinite series with terms of the form $\exp(−ak^2t)\sin(bkx)$ which has to satisfy the boundary conditions. Next, the spherical problem is solved. Assuming isotropy for a sphere, it becomes one-dimensional in the radius $r$. This problem was solved by Lord Kelvin when he applied it to a cooling Earth, which however drastically underestimated its existence to 98 million years because he did not know about radioactive decay or tectonic plates. Next is the solution in a semi-infinite medium with infinite thickness. This is the first case of the slab where the thickness $L$ goes to $\infty$. This is an occasion to show how the Fourier series used for finite $L$ migrates into the Fourier transform when $L\to\infty$. The heat equation is also solved for other cases like a circular ring and an insulated sphere These were also discussed by Fourier in his <em>Théorie analytique de la chaleur</em> (1822), although the last one did not result in a Fourier series.</p>
<p>Chapter 5 starts with a crash course on electrical circuits: resistors, capacitors, inductors and Kirchoff's laws and describing the behaviour of electrons in an electrical field. And lo and behold, the electrons in a one-dimensional semi-infinite induction-free telegraph cable behave according to the heat equation, again an ingenious insight of Lord Kelvin. Solving that equation was a theoretical achievement, producing the cable and letting it sink to the bottom of the ocean was a risky and adventurous enterprise. In this book, that technological adventure is only lingering in the background. A nice account of this adventure can be found for example in the book <em><a target="_blank" href="/review/mind-play-how-claude-shannon-invented-information-age">A Mind at Play: How Claude Shannon Invented the Information Age</a></em> by J. Soni and R. Goodman (2017).</p>
<p>Heaviside also features in the last chapter discussing the evolution after the 1866 Atlantic cable was realized. He added the inductance to the heat equation which turns it into a wave equation (actually the telegrapher's equation describing traveling waves in transmission lines, smartly solved by d'Alembert). That removes the assumed instantaneous action at a distance in the heat equation, which was causing a diffusion of the signal. The parameters of the cable can be controlled to remove that effect and this improved the usefulness of the cable considerably. Nahin ends by discussing the computation of how an arbitrary signal is transmitted. The diffusion however destroys the information during the transmission. This is illustrated by a matlab program that computes this deformation. The short code is given so that you can try it out yourself. The example shows that the signal is unrecognizable, it can still work though for a binary signal since the only information that one needs to detect is whether or not a bit is zero or one. We can also read how Heaviside explained the asymmetry of the transmission time: a message sent from England took longer than a message sent to England.</p>
<p>The sources used by Nahin, and some additional historical notes are listed at the end of the book, organized per chapter. There is no separate bibliography but there is an index that includes references to these notes. He has also one appendix about Leibniz's formula, i.e., how to compute the derivative of an integral if the boundaries of the integral are varying.</p>
<p>The book confirms what is already known from his previous books: Nahin knows how to write a book mixing physics and (a lot of) mathematics and (still) make it readable for a (relatively) broad public (with only some basic mathematical knowledge). The mathematics in this book certainly take the leading role like it does in lecture notes about the solution of differential equations. Nahin takes his time to explain everything and derive things from the very basics. When the mathematics become too involved or advanced, he uses intuition and asks the reader to accept and believe the result. The hard core mathematical mind may have some problems with his "engineering approach", but it works perfectly well for a first introduction. Anyway, from the historical perspective, this approach was used by the people who originally developed the theory.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Nahin introduces us through an historical approach to Fourier series, Fourier transforms, and how Fourier used this to solve the heat equation. Lord Kelvin used the heat equation to model the cooling of the Earth and hence estimate its age and he, and others, solved essentially the same equation to model the flow of electrons in the transatlantic telegraph cable.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691191720 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 24.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">232</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Partial Differential Equations</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691191720/hot-molecules-cold-electrons" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691191720/hot-molecules-cold-electrons</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/35k05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35K05</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42a16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/35s30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35S30</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/35l05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35L05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/35k57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35K57</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/94c05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">94C05</a></li></ul></span>Tue, 26 May 2020 16:18:52 +0000Adhemar Bultheel50806 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/hot-molecules-cold-electrons#commentsHYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS AND GEOMETRIC OPTICS
https://euro-math-soc.eu/review/hyperbolic-partial-differential-equations-and-geometric-optics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The interested reader has access to a vast literature on hyperbolic Partial Differential Equations. The ubiquitous presence of hyperbolic PDEs (from pure Mathematics to applied Engineering) has made these equations the centre of many important references. Probably because of this fact, those books covering more than the fundamentals of the theory are almost forced to concentrate efforts to specialized topics of the field. The book under review has chosen short wavelength asymptotics as the main theme of its exposition. The author, Prof. J. Rauch is a renowned researcher on PDEs with interesting contributions on non-linear microlocal analysis, control of waves and non-linear Geometric Optics. He is also the author of a monograph on PDEs of which this second book can be considered a sequel for deeper and specialized study.<br />
The book contains some general facts about PDEs (characteristics, the Cauchy problem, dispersion,…) but it rapidly focus its attention to linear and non-linear geometric optics, that is, the realm of solutions of hyperbolic PDEs with short wavelength in the sense of the Fourier expansion. In this situation, the existence of solutions, interference, microlocal analysys, resonance and other interesting issues are tackled. It is important to note that short wave length solutions are not only important in Optics, but also in other relevant situations as Fluids. These other situations are also contemplated in the book which, in addition, is illustrated with many examples and exercises that help the reading of this dense work. The result is a book certainly intended for researchers and graduate students interested in the topic.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Marco Castrillon Lopez</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book introduces graduate students and researchers to the partial differential equations of hyperbolic type with special emphasis on propagation of waves and singularities for short wavelenght solutions.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/j-raouch" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">J. Raouch</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematica-society" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">American Mathematica Society</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-8218-7291-8</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">363</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li><li class="vocabulary-links field-item odd"><a href="/imu/partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Partial Differential Equations</a></li></ul></span><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/35a18" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35A18</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/35a21" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35A21</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/35a27" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35A27</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/78a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">78A05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/93b07" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">93B07</a></li></ul></span>Mon, 23 Feb 2015 11:20:12 +0000Marco Castrillon Lopez46053 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/hyperbolic-partial-differential-equations-and-geometric-optics#commentsSmoothing and Decay Estimates for Nonlinear Diffusion Equations - Equations of Porous Medium Type
https://euro-math-soc.eu/review/smoothing-and-decay-estimates-nonlinear-diffusion-equations-equations-porous-medium-type
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The central object of this book is the nonlinear partial differential equation, ut – div (|u|m-1grad u); x Rn, t > 0, equipped with the initial value condition u = u0; x Rn, t = 0. The author is concerned with the smoothing effect of the equation and the time decay of positive solutions, i.e. whether the fact that u0 belongs to some function space X implies that the solution u(t) in time t > 0 is a member of some "better" function space Y and if it is possible to get estimates of the form |u(t)|Y < C(t, X, n, m,|u0|X). Well-posedness of the problem and some other substantial results such as the comparison theorem are mentioned in the preliminary part of the book and references are given for the proofs. Smoothing is carefully studied for all n N, m R (if m = 1 the classical results for the heat equation are reconstructed), X and Y being Lebesgue or weak Lebesgue spaces, which naturally appear as the correct spaces for studies of smoothing. It is very interesting that depending on m, n, X, Y, the solutions of the equation exhibit qualitatively very different properties, which are sometimes very surprising. The last chapter is devoted to the question of whether the results for the equation introduced at the beginning of the review also remain valid for the p-Laplacian equation. The book is very nicely written, well ordered and gives a rather complete overview of known results in the chosen field. At the beginning of each chapter there is a summary of the whole chapter with remarks of which sections of the chapter are essential for the following sections. The text is equipped with historical notes, remarks and a number of exercises. These properties make the book useful as a graduate textbook or a source of information for graduate students and researchers.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">kapl</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/j-l-v%C3%A1zquez" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">j. l. vázquez</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press-oxford" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press, oxford</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-19-920297-3 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 45</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span>Sun, 23 Oct 2011 18:33:23 +0000Anonymous40089 at https://euro-math-soc.euMulti-dimensional Hyperbolic Partial Differential Equations - First-order Systems and Applications
https://euro-math-soc.eu/review/multi-dimensional-hyperbolic-partial-differential-equations-first-order-systems-and
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This carefully written and well-thought-out book presents a comprehensive view on mathematical theory concerning multi-dimensional hyperbolic partial differential equations. The main body of the book consists of four parts. The first part deals with the theory of linear Cauchy problems that involve both constant and variable coefficients, the latter illustrating the power of pseudo-differential and para-differential calculus (presented separately in the appendix). The second part is devoted to linear initial boundary value problems, proceeding from simpler to more complicated systems (symmetric, constant coefficients, variable coefficients). The third part is devoted to the theory of nonlinear problems, focusing on the notion of a smooth solution and a piecewise smooth solution suitable for analysis of shock waves (as is carefully proved). The last part investigates problems in gas dynamics and it includes a discussion of appropriate boundary conditions and shock-wave analysis. The text is completed with an extensive bibliography including classical and recent papers both in partial differential equation analysis and applications (mainly in gas dynamics).</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jmal</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/s-benzoni-gavage" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">s. benzoni-gavage</a></li><li class="vocabulary-links field-item odd"><a href="/author/d-serre" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">d. serre</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press-oxford-oxford-mathematical-monographs" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press, oxford: oxford mathematical monographs</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">0-19-921123-X </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 60</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span>Sun, 23 Oct 2011 14:31:15 +0000Anonymous40058 at https://euro-math-soc.euNonlinear Partial Differential Equations with Applications
https://euro-math-soc.eu/review/nonlinear-partial-differential-equations-applications
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book concerns the mathematical analysis of quasilinear partial differential equations (PDEs) where the leading operator is nonlinear, elliptic and in divergence form. Techniques such as monotone operators, pseudomonotone operators, accretive operators, potential operators, variational inequalities and set-valued mappings form the cornerstone basis for the presented analysis. Special attention is also devoted to penalty methods. The author treats both steady-state problems in part I and corresponding evolutionary problems in part II. For time-dependent problems, the Rothe and Faedo-Galerkin methods are incorporated in detail. </p>
<p>Each section has the same structure: a general abstract framework is accompanied by applications of theoretical results to carefully selected examples starting from sample cases up to the cases that have their origin in the physical sciences (thermofluid mechanics, thermoelasticity, reaction-diffusion problems, material science). Frequently the author shows that dealing with a specific (system of) PDE(s) can strengthen the results obtained by abstract methods. Each section is completed with exercises and a representative list of relevant literature. This carefully written book, addressed to graduate and PhD students and researchers in PDEs, applied analysis and mathematical modelling contains on one hand several mathematical approaches developed to understand the basic mathematical properties of nonlinear PDEs and on the other hand many interesting examples. Some are solved, others are complemented by hints and the rest are left for the reader to complete.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jomal</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/t-roub%C3%AD%C4%8Dek" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">t. roubíček</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel-international-series-numerical-mathematics-vol-153" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser, basel: international series of numerical mathematics, vol. 153</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">3-7643-7293-1</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 108</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span>Sun, 23 Oct 2011 12:20:23 +0000Anonymous40048 at https://euro-math-soc.euAnalytical Methods for Markov Semigroups
https://euro-math-soc.eu/review/analytical-methods-markov-semigroups
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book describes past and present results on Markov semigroups. It begins with the existence of solutions for elliptic and parabolic equations with unbounded coefficients on the whole of Rn . Then it continues with uniqueness and nonuniqueness results and regularity properties (e.g. compactness, uniform and pointwise estimates of the derivatives) of the associated semigroup, both on the space of bounded continuous functions and on Lp with invariant measure. One chapter is devoted to the Ornstein-Uhlenbeck operator as a prototype of an elliptic operator with unbounded coefficients. The second part of the book is devoted to elliptic and parabolic problems on open unbounded domains in Rn with Dirichlet and Neumann boundary conditions. The third part deals with degenerate problems. The monograph contains a very well-arranged collection of the results on Markov semigroups. It will be mainly appreciated by experts on Markov semigroups as well as researchers working in related topics. But not only by them, since the results are presented in a way suitable for applications. The book does not contain many examples but they are not necessary since the text is suitably understandable.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">tba</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/l-lorenzi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">l. lorenzi</a></li><li class="vocabulary-links field-item odd"><a href="/author/m-bertoldi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">m. bertoldi</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc-boca-raton-pure-and-applied-mathematics-vol-283" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman & hall/crc, boca raton: pure and applied mathematics, vol. 283</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">1-58488-659-5 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 99,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span>Sun, 23 Oct 2011 12:14:43 +0000Anonymous40043 at https://euro-math-soc.euStrichartz Estimates for Schrödinger Equations with Variable Coefficients
https://euro-math-soc.eu/review/strichartz-estimates-schr%C3%B6dinger-equations-variable-coefficients
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is devoted to the proof of regularity results for the solution of the Schrödinger equation i ∂t u - A u + V(x) = 0. Coefficients of the second order differential operator A are allowed to depend on the space variable x but A is assumed to be asymptotically (as |x| ‒> +∞) a perturbation of the Laplacian. For solutions of the equation the authors show the Strichartz estimate. The main step of the proof is to express the solution for an auxiliary problem via a Fourier integral operator with complex phase, which is described in chapter 6. It relies on a careful study of phase and transport equations, presented in chapter 4 and 5. When the authors have the explicit formula for the solution, they then deduce the dispersion estimate and use a general result [M. Kell, T. Tao: End point Strichartz estimate, Amer. J. Math. 120 (1998), p. 955-980] to conclude the proof in chapter 7. The proof of the theorem is quite technically demanding but the book is well ordered and carefully written. It is very interesting and experts in the field will surely appreciate it.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">pkap</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/l-robbiano" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">l. robbiano</a></li><li class="vocabulary-links field-item odd"><a href="/author/c-zuily" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">c. zuily</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/soci%C3%A9t%C3%A9-math%C3%A9matique-de-france-paris-m%C3%A9moires-de-la-soci%C3%A9t%C3%A9-math%C3%A9matique-de-france-no-101" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">société mathématique de france, paris: mémoires de la société mathématique de france, no. 101-102</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">2-85629-180-5 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 41</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span>Sat, 22 Oct 2011 18:51:55 +0000Anonymous40011 at https://euro-math-soc.euSystemes différentiels involutifs
https://euro-math-soc.eu/review/systemes-diff%C3%A9rentiels-involutifs
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The origin of the theory of involutive systems of partial differential equations goes back to Élie Cartan, who studied them in terms of exterior differential systems. During the last fifty years, methods of homological algebra were applied with success to study such systems. In the first part of the book, the author reviews the theory of involutive systems from the point of view of partial differential equations (a relation to the approach by exterior differential systems is explained in appendix B). In the main part of the book (chapters 4 and 5) the author introduces a notion of D-analytic spaces, proves the finiteness theorem and proves the involutiveness of the system for a generic case. In such a way, the author gives a precise sense to the old statement of Cartan that “by a prolongation, a differential system becomes eventually involutive”. This is a small booklet with a very interesting content related to many questions studied recently in the realm of partial differential equations with algebraic or analytic coefficients.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">vs</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/b-malgrange" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">b. malgrange</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/soci%C3%A9t%C3%A9-math%C3%A9matique-de-france-paris-panoramas-et-synth%C3%A8ses-no-19" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">société mathématique de france, paris: panoramas et synthèses, no. 19</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">2-85629-178-3 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 26</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span>Sat, 22 Oct 2011 18:38:32 +0000Anonymous40002 at https://euro-math-soc.euRandomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions
https://euro-math-soc.eu/review/randomly-forced-nonlinear-pdes-and-statistical-hydrodynamics-2-space-dimensions
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This booklet is based on the author's lecture-course at ETH Zürich and it addresses the incompressible isothermal Navier-Stokes equation in a two-dimensional domain with periodic boundary conditions forced randomly by the right-hand side. The topic is related to turbulence theory but heuristic considerations are suppressed only to discussion while the focus is on rigorously proved mathematical results. The author shows, for example, that statistical characteristics of a turbulent flow stabilize with time growing to characteristics independent of an initial velocity profile, that the time average of any characteristic of a turbulent flow equals the ensemble average, and that the turbulent flow is a Gaussian process at large time scales. A short introductory section on function spaces and partial differential equations (focused on the Navier-Stokes equation) is included but the reader’s probability background is assumed. The book is primarily intended for experts in probability theory and partial differential equations with a focus on theoretical fluid dynamics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">trou</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/s-b-kuksin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">s. b. kuksin</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/european-mathematical-society-z%C3%BCrich-zurich-lectures-advanced-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">european mathematical society, zürich: zurich lectures in advanced mathematics</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">3-03719-021-3</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 28</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span>Sat, 22 Oct 2011 18:34:00 +0000Anonymous39999 at https://euro-math-soc.euEnnio De Giorgi - Selected Papers
https://euro-math-soc.eu/review/ennio-de-giorgi-selected-papers
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This volume contains a well-balanced choice of papers written by Ennio De Giorgi, an outstanding Italian mathematician of the 20th century. Many important papers were written by Ennio De Giorgi in Italian, hence the main motivation behind the project was to make them available to a larger public. In addition to their English translations, some of papers are also included in their Italian versions in order to give the reader a feeling of De Giorgi's original style (e.g. this is the case with the celebrated article, "Sulla differenziabilita e l'analiticita delle estremali degli integrali multipli regolari", one of the most important contributions to regularity theory). After a short biography, there is a description of the scientific work of Ennio De Giorgi, ending with contributions by Louis Nirenberg (remarks on some analytic works of Ennio De Giorgi) and Louis Caffarelli (De Giorgi's contribution to the regularity theory of elliptic equations). Then there is a collection of 43 papers (out of 152). In my opinion the volume, which covers all the main directions of the author's activities (measure theory, the 19th Hilbert problem, minimal hypersurfaces, G- and Γ-convergence, and foundations of mathematics) should be on the shelves of every mathematical library.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">oj</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/l-ambrosio-et-al" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">l. ambrosio et al.</a></li><li class="vocabulary-links field-item odd"><a href="/author/eds" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">eds.</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-berlin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer, berlin</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">3-540-26169-9 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 149,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/35-partial-differential-equations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35 Partial differential equations</a></li></ul></span>Sat, 22 Oct 2011 16:59:48 +0000Anonymous39975 at https://euro-math-soc.eu