Abstract
We consider approximations of Hopf’s weak form of the Navier-Stokes equation. The initial-boundary value problem of each of these “stabilized” equations has a unique weak solution which is stable on any compact time interval, too. This solution is computable by means of Hopf’s Galerkin method, because in case of any stabilized equation the whole sequence of all Hopf-approximationsis convergent. The formulas of the Hopf-Galerkin method are given explicitly.
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
GOLOVKIN, K.K.: New equations modeling the motion of a viscous fluid, and their unique solvability. Proc. Steklov Inst. Math. 102 (1967), 29–54.
HOPF, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213–231.
KIRCHGXSSNER, K.: Neuere Ergebnisse zur Theorie der Navier-Stokesschen Gleichungen. Z. Angew. Math. Mech. 50, T158–163 (1970).
LADYZENSKAJA, O.A.: The Mathematical Theory of Viscous Incompressible Flow. 2. Ed. Gordon and Breach, New York (1969).
LADYZENSKAJA, O.A.: New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them. Proc. Steklov Inst. Math. 102 (1967), 95–118.
LADYZENSKAJA, O.A.: Example of Nonuniqueness in the Hopf Class of Weak Solutions for the NavierStokes-Equations. Math. USSR-Izvestija 3 (1969), 229–236.
LERAY, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 (1934), 193–248.
LIONS, J.L., PRODI, G.: Un théorème d’existence et unicité dans les équations de Navier-Stokes en dimension 2. C. R. Acad. Sci. Paris 248 (1959), 3519–3521.
PRODI, G.: Un Teorema di unicité per le equazioni di Navier-Stokes. Annali di Mat. 48 (1959), 173–182.
RAUTMANN, R.: Ein Näherungsverfahren für spezielle parabolische Anfangswertaufgaben mit Operatoren, in: Numerische Lösung nichtlinearer partieller Differential-und Integrodifferentialgleichungen, Ed. R. Ansorge, W. Törnig, Springer-Lecture-Notes in Mathematics 267 (1972), 187–231.
RAUTMANN, R.: Lösung einer Anfangswertaufgabe für substantielle Differentialgleichungen, in: Numerische Lösung nichtlinearer partieller Differential-und Integrodifferentialgleichungen, Ed. R. Ansorge, W. Törnig, Springer-Lecture-Notes in Mathematics 395 (1974), 85–108.
RAUTMANN, R.: Die Anfangsrandwertaufgabe einer stabilisierten Navier-Stokesschen Gleichung, to appear.
SERRIN, J.: The initial value problem for the Navier-Stokes equations. in: Nonlinear Problems (ed. R.E. Langer) MRC Madison 1963, 69–98.
SIBAGAKI, W., RIKIMARU, H.: On the Hopf’s weak solution of the initial value problem for the Navier-Stokes equations. Memoirs of the Faculty of Science Kyushu University, Ser. A, 21 (1967), 194–240.
SMIRNOV, W.I.: Lehrgang der höheren Mathematik Teil V. VeB Deutscher Verlag der Wissenschaften Berlin (1962).
WALTER, W.: Differential and Integral Inequalities. Springer Berlin (1970).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1975 Springer Basel AG
About this chapter
Cite this chapter
Rautmann, R. (1975). On the Convergence of a Galerkin Method to Solve the Initial Value Problem of a Stabilized Navier-Stokes Equation. In: Numerische Behandlung von Differentialgleichungen. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 27. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5532-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5532-7_15
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5533-4
Online ISBN: 978-3-0348-5532-7
eBook Packages: Springer Book Archive