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"title": "Book review",
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"Stochastically Forced Compressible Fluid Flows ",
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"by Dominic Breit, Eduard Feireisl and Martina Hofmanová"
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"Reviewed by Donatella Donatelli\n"
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"content": [
"The book is focused on systematically developing a consistent mathematical\ntheory of compressible fluids driven by random initial data and stochastic\nexternal forces in the context of classical continuum fluid mechanics."
]
},
{
"type": "Paragraph",
"id": "Sx1.p4",
"content": [
"The theory of continuum fluid mechanics is derived from basic physical\nprinciples under the assumption that all quantities – fields – are smooth,\nand the Navier–Stokes system became a well-established model working as a\nreliable basis of investigation for both theoretical and applied aspects. Built\non the foundation of conservation laws, fluid mechanics helps to describe the\nflow and interactions of gases, liquids and/or plasmas, as well as the forces\nacting on them. Until fairly recently, these forces have largely been\nconsidered to be deterministic. This means that they are functions of\nmicroscopic space and time parameters, so that at any given instant of time the\nfluid position in space is expected to be known. There are still many important\nopen problems, but the literature concerning the deterministic case is very\nwell-established and extensive; see for example the monographs [E. Feireisl,\n",
{
"type": "Emphasis",
"content": [
"Dynamics of Viscous Compressible Fluids"
]
},
", Oxford Lecture Series in Mathematics\nand its Applications, vol. 26, Oxford University Press, Oxford, 2004] or\n[P. L. Lions, ",
{
"type": "Emphasis",
"content": [
"Mathematical Topics in Fluid Mechanics"
]
},
", Vol. 2: ",
{
"type": "Emphasis",
"content": [
"Compressible\nModels"
]
},
", Oxford Lecture Series in Mathematics and its Applications, vol. 10, The\nClarendon Press, Oxford Science Publications, Oxford University Press, New\nYork, 1998]."
]
},
{
"type": "Paragraph",
"id": "Sx1.p5",
"content": [
"However, this description is a fairly weak idealisation, which is obvious\nalready from the fact that we are still unable to model extreme fluid mechanic\nevents like turbulence to a sufficient level of accuracy. In fact, the\nmodelling of turbulence can be considered as the prime motivation for the\nintroduction of stochasticity in the study of fluids. Turbulence is frequently\nassociated with an intrinsic element of randomness, and furthermore,\nexperimental studies of turbulence lead more to a statistical approach than to\na deterministic one. Moreover, the addition of stochastic terms to the basic\ngoverning equations is often used to account for other numerical, empirical or\nphysical uncertainties. Therefore it becomes important, in the framework of\npartial differential equations, to set up a stochastic PDE theory for fluid\nflow."
]
},
{
"type": "Paragraph",
"id": "Sx1.p6",
"content": [
"Nowadays there exists a large amount of literature concerning the dynamics of\nincompressible fluids driven by stochastic forcing. The first results can be\nfound in the pioneering work by Bensoussan–Temam (1973). See also the lecture\nnotes [A. Debussche, Ergodicity results for the stochastic Navier–Stokes\nequations: An introduction, In ",
{
"type": "Emphasis",
"content": [
"Topics in Mathematical Fluid Mechanics"
]
},
",\nvolume 2073 of Lecture Notes in Math., pages 23–108, Springer, Heidelberg,\n2013], [Flandoli, An introduction to 3D stochastic fluid dynamics, In\n",
{
"type": "Emphasis",
"content": [
"SPDE in Hydrodynamic: Recent Progress and Prospects"
]
},
", volume 1942 of\nLecture Notes in Math., pages 51–150, Springer, Berlin, 2008]. Nevertheless,\nfar less is known in the case of compressible fluids. Important questions of\nwell-posedness and even mere existence of solutions to problems dealing with\nstochastic perturbations of compressible fluids are largely open, with only a\nfew rigorous results available. This monograph is an exhaustive and up-to-date\noverview of the most recent results by different authors on stochastic\ncompressible fluids."
]
},
{
"type": "Paragraph",
"id": "Sx1.p7",
"content": [
"The book contains eight chapters and is divided into three parts. It starts\nwith Part I, a very didactic introduction providing the necessary background.\nIn a very clear manner, Part I provides the non-expert readers in the field\nwith all the basic results of the theory and, at the same time, a description\nof more advanced tools in the theory of stochastic PDEs. Part II is the core of\nthe book, containing all that is really new and original compared to the\nexisting literature. The most recent existence results on compressible\nstochastic fluids are described. This part consists of five chapters, which\nguide the reader step by step towards the proof of the existence of solutions.\nEach chapter is devoted to one of the main aspects of the existence theory: the\nsetup of the model, approximation schemes and their convergence, energy\ninequalities, relative energy inequality, and weak strong uniqueness. In\nparticular, it starts with the existence of local strong solutions defined on a\nmaximal time interval bounded above by a positive stopping time that may depend\non the size of the initial data; then, because all real world problems require\nsolutions defined globally in time, one has to switch to the notion of weak\nsolutions. This approach is based on the idea of including some form of the\nenergy/entropy balance as an integral part of a weak formulation, and goes back\nto Dafermos (1979) concerning conservation laws and to Germain (2011) who\nintroduced a similar concept in the context of the deterministic compressible\nNavier–Stokes system. Therefore, the solutions constructed in this part of the\nbook are the so-called dissipative martingale solutions, which are weak\nmartingale solutions also satisfying a variant of the energy balance."
]
},
{
"type": "Paragraph",
"id": "Sx1.p8",
"content": [
"Finally, Part III of the book is focused on applications such as singular\nlimits. Indeed, by scaling the equations by means of appropriately chosen\nreference units, the parameters determining the behaviour of the system become\nevident. Asymptotic analysis and/or singular limits provide a useful tool in\nsituations where these parameters vanish or become infinite. In this part, the\nauthors describe a rigorous mathematical approach to asymptotic analysis in the\ncase of incompressible and inviscid–incompressible limits for the compressible\nNavier–Stokes system with stochastic perturbations."
]
},
{
"type": "Paragraph",
"id": "Sx1.p9",
"content": [
"To conclude, this is the first book in which one can find a complete\ndescription of the available theory on compressible stochastic fluid equations.\nCompared to the previous literature, this is a new point of view that makes the\nbook original and of very high quality. It is a really valuable and much-needed\ncontribution to the literature in the domain. This monograph is built in a\nmasterly manner, in such a way as to provide not only a complete and up-to-date\noverview of the problems under consideration, but also a detailed introduction\nto the topic for the uninitiated reader. The book is very well and rigorously\nstructured, having the excellent attribute of being valuable to both\nexperienced researchers in the domain and to graduate students who wish to\nexplore the different topics in this challenging area of research. Overall, it\nconstitutes an ideal book for researchers (in the broadest sense) who want to\nenlarge their mathematical knowledge of fluid mechanics."
]
},
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"id": "Sx1.p10",
"content": [
"Dominic Breit, Eduard Feireisl and Martina Hofmanová,\n",
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"content": [
"Stochastically Forced Compressible Fluid Flows"
]
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".\nDe Gruyter Series in Applied and Numerical\nMathematics 3. De Gruyter,\n2018, 330 pagesISBN 978-3-11-049050-3.\neBook ISBN 978-3-11-049255-2.\n"
]
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