Mathematical Theory of a Fluid Flow Around a Rotating and Translating Body

by Sárka Necasová, Stanislav Kracmar, Jirí Neustupa, Patrick Penel

Overview

The book deals with qualitative analysis of the mathematical model of flow of a viscous incompressible fluid around a translating and rotating body. The considered mathematical model, which represents the description of the flow in a coordinate system attached to the body, is derived from the Navier–Stokes equations by means of an appropriate transformation. The core of the book is the mathematical theory of the transformed equations. Most of the text is devoted to the theory of the linearized versions of these equations (i.e. the Stokes- and Oseen-type equations), because they play a fundamental role in the theory of the complete nonlinear system.

Considering strong, weak, and very weak solutions, we present the L2 and Lq theories and the weighted space theory (with Muckenhaupt's weights) in the whole space and in an exterior domain. The book also contains the spectral analysis of the associated linear Stokes-Oseen-type operators and the information on semigroups generated by these operators, and related resolvent estimates.

Moreover, the book describes the asymptotic behavior of solutions and leading profiles of solutions for linear and as well as nonlinear systems.

Further, the book contains studies of the problem with artificial boundary (important in numerical analysis), an introduction to the theory of the corresponding complete nonlinear system in both steady and nonsteady cases, a brief description of the situation when the rotation is not parallel to the velocity at infinity and necessary estimates of the related Oseen kernels.

Product Details

ISBN-13:	9783031841194
Publisher:	Springer Nature Switzerland
Publication date:	08/02/2025
Series:	Springer Monographs in Mathematics
Pages:	624
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Šárka Nečasová was born in 1965. She studied mathematics at the Faculty of Mathematics and Physics (Charles University, Prague), graduated in 1988, and obtained her Ph.D. in 1991 (at the Czech Technical University in Prague). She defended her habilitation thesis at the University of Pau (France) in 2010, and she also achieved the Doctor of Sciences degree at the Czech Academy of Sciences in 2013. She worked as an assistant professor at the Czech Technical University in Prague in 1991–1995. In 1995, she became a researcher at the Mathematical Institute of the Academy of Sciences of the Czech Republic, Prague. She has been the head of the Department of Evolution Differential Equations since 2010. She obtained the following awards: the Wichterle prize (the prize of the Academy of Sciences of the Czech Republic for young researchers) in 2003; member of the Learned Society of the Czech Republic (since 2018, head of the section Mathematics and Physics 2020–2024, now the scientific secretary); and Praemium Academiae (awarded by the Czech Academy of Sciences) in 2021. She gave mini-courses at Zagreb (2017), Tata Institute Bangalore (2017), University of Wuerzburg (2018), and University of Nanjing (2020). Her main fields of interest are the mathematical analysis of models of fluids, partial differential equations, and qualitative theory.

Stanislav Kračmar was born in 1955. He obtained his master degree at Kharkov University, Kharkov, Ukraine, in 1980. Then he received the title RNDr. (MSc. equivalent) at the Faculty of Mathematics and Physics of the Charles University in Prague in 1982. He defended his Ph.D. thesis at the Czech Technical University in 1991 and his habilitation in 2001. He is currently Associate Professor at the Czech Technical University. His main fields of interest are mathematical modeling in mechanics of fluids, partial differential equations, and qualitative theory.

Jiří Neustupa was born in 1949. He defended his master degree in 1972 and his Ph.D. degree in 1977, both at the Charles University in Prague. He defended his habilitation in 1982. He became Full Professor in 1999. He worked at the Czech Technical University from 1977 to 2005. In 2005, he moved to the Mathematical Institute of the Czech Academy of Sciences. He gave mini–courses on the theory of the Navier–Stokes equations at the Petroleum Institute Abu Dhabi (2010, 2011), Banach Center Warsaw (2012), Waseda University Tokyo (2012), Tata Institute Bangalore (2014), and Yonsei University Seoul (2019). His main fields of interest are mathematical modeling in mechanics of fluids, partial differential equations, and qualitative theory.

Patrick Penel was born in 1948. He studied mathematics at the University of Paris-XI-Orsay, graduating in 1969. He defended his Ph.D. thesis in 1972 and his State Doctorate ès Science in 1975, both at Paris-XI-Orsay. He was appointed an assistant in Paris XIII (1969), he worked as an assistant professor in 1972–1974. He became Full Professor at the University of Toulon and Var in 1976 and stayed at this position until 2010. After having been appointed Emeritus Professor twice, he is now retired and honorary. His main fields of interest are mathematical fluid mechanics, partial differential equations, and their qualitative theory. He was a responsible manager for international university co-operations (Antananarivo–Madagascar 1980–1993, Prague–Czech Republic 1990–2017), and co-organizer of a series of international conferences in CIRM, Marseille–Luminy (2008, 2011, 2014, 2017). Furthermore, he has been Academician of Var since 2015, elected to Chair No. 15.

Table of Contents

Part I Introduction and preliminaries.- 1 Introduction.- 2 Preliminaries.- Part II Linear theory.- 3 The steady Stokes — type problem.- 4 The Steady Oseen — type problem.- 5 Representation formula and asymptotic behavior.- 6 Artificial boundary conditions for Oseen type problem.- 7 The Oseen-type problem: a weak solution in anisotropical L2 -spaces.- 8 Stokes– and Oseen–type operators: spectral theory and generated semigroups.- Part III Navier-Stokes type equations.- 9 The stationary Navier–Stokes–type problem.- 10 Pointwise decay for the stationary Navier–Stokes type problem.- 11 Asymptotic behavior of the solutions of the Navier–Stokes type problem.- 12 The nonstationary Navier–Stokes–type problem.- Part IV Appendix.- 13 Appendix.