Abstract
There exists a series of other works dealing with flows in time varying domains that concern the motion of one or more bodies in a fluid. The fluid and the bodies are studied as an interconnected system so that the position of the bodies in the fluid is not apriori known. The weak solvability of such a problem, provided the bodies do not touch each other or they do not strike the boundary, was proved by B. Desjardins and M. J. Esteban [4, 5], K. H. Hoffmann and V. N. Starovoitov [13] (the 2D case), C. Conca et al. [2] and M. D. Gunzburger et al. [12].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams R.: Sobolev Spaces. Academic Press, New York, San Francisco, London (1975)
Conca C., San Martín J., Tucsnak M.: Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. Part. Diff. Equat. 25 (5&6), 1019–1042 (2000)
Dautray R., Lions M.J.: Mathematical Analysis and Numerical Methods for Science and Technology II. Springer, Berlin, Heidelberg (2000)
Desjardins B., Esteban M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Rat. Mech. Anal. 146, 59–71 (1999)
Desjardins B., Esteban M.J.: On weak solutions for fluid-rigid structure interaction: compressible and incompressible models. Comm. Part. Diff. Equat. 25 (7&8), 1399–1413 (2000)
Feireisl E.: On the motion of rigid bodies in a viscous incompressible fluid. J. Evol. Equat. 3, 419–441 (2003)
Fujita H., Sauer N.: On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries. J. Fac. Sci. Univ. Tokyo, Sec. 1A, 17, 403–420 (1970)
Galdi G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I, Linear Steady Problems. Springer Tracts in Natural Philosophy 38 (1998)
Galdi G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II, Nonlinear Steady Problems. Springer Tracts in Natural Philosophy 39 (1998)
Galdi G.P.: An Introduction to the Navier–Stokes initial–boundary value problem. In Fundamental Directions in Mathematical Fluid Mechanics, ed. G.P.Galdi, J. Heywood, R. Rannacher, series “Advances in Mathematical Fluid Mechanics”, Vol. 1, Birkhauser–Verlag, Basel, 1–98 (2000)
Galdi G.P.: On the motion of a rigid body in a viscous fluid: a mathematical analysis with applications. In Handbook of Mathematical Fluid Dynamics I, Ed. S. Friedlander and D. Serre, Elsevier (2002)
Gunzburger M.D., Lee H.C., Seregin G.: Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. J. Math. Fluid Mech. 2, 219–266 (2000)
Hoffmann K.H., Starovoitov V.N.: On a motion of a solid body in a viscous fluid. Two–dimensional case. Adv. Math. Sci. Appl. 9, 633–648 (1999)
Hoffmann K.H., Starovoitov V.N.: Zur Bewegung einer Kugel in einen zähen Flüssigkeit. Documenta Mathematica 5, 15–21 (2000)
Hopf E.: Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1950)
Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)
Leray J.: Sur le mouvements d’un liquide visqueux emplissant l’espace. Acta Mathematica 63, 193–248 (1934)
Lions J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Gauthier–Villars, Paris (1969)
Neustupa J.: Existence of a weak solution to the Navier-Stokes equation in a general time–varying domain by the Rothe method. Preprint (2007)
San Martín J., Starovoitov V.N., Tucsnak M.: Global weak solutions for the two–dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Rat. Mech. Anal. 161, 113–147 (2002)
Starovoitov V.N.: Behavior of a rigid body in an incompressible viscous fluid near a boundary. Int. Series of Num. Math. 147, 313–327 (2003)
Temam R.: Navier-Stokes Equations. North-Holland, Amsterdam–New York–Oxford (1977)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Neustupa, J., Penel, P. (2010). A Weak Solvability of the Navier-Stokes Equation with Navier’s Boundary Condition Around a Ball Striking theWall. In: Rannacher, R., Sequeira, A. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04068-9_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-04068-9_24
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04067-2
Online ISBN: 978-3-642-04068-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)