Abstract
In this paper we study the vorticity equations for viscous incompressible flows in the half space under the no-slip boundary condition on the velocity field. In particular, the boundary condition for the vorticity field is presented explicitly, and the solution formula for the linearized problem is obtained.
Dedicated to Professor Yoshihiro Shibata on his sixtieth birthday
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
References
H. Fujita, T. Kato, On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. I (Springer, New York, 1994)
Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L r spaces. Math. Z. 178, 297–329 (1981)
Y. Giga, Domains of fractional powers of the Stokes operator in L r spaces. Arch. Ration. Mech. Anal. 89, 251–265 (1985)
Y. Giga, T. Miyakawa, Solutions in L r of the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 89, 267–281 (1985)
Y. Giga, T. Miyakawa, Navier-Stokes flow in R 3 with measures as initial vorticity and Morrey spaces. Commun. Partial Diff. Equ. 14, 577–618 (1989)
M.-H. Giga, Y. Giga, J. Saal, Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-similar Solutions (Birkhäuser, Boston, Basel, Berlin, 2010)
T. Kato, H. Fujita, On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32, 243–260 (1962)
Y. Maekawa, Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit. Adv. Diff. Equ. 18, 101–146 (2013)
Y. Maekawa, On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane. Commun. Pure Appl. Math. 67, 1045–1128 (2014)
A.J. Majda, A.L. Bertozzi, Vorticity and Incompressible Flows (Cambridge University Press, Cambridge, 2002)
L. Quartapelle, Numerical Solution of the Incompressible Navier-Stokes Equations (Birkäuser, Basel, 1993)
R. Rautmann, A direct approach to vorticity transport & diffusion, in Kyoto Conference on the Navier-Stokes Equations and Their Applications, vol. B1 (RIMS Kokyuroku Bessatsu, Kyoto, 2007), pp. 305–329
V. Ruas, A new formulation of the three-dimensional velocity-vorticity system in viscous incompressible flow. Z. Angew. Math. Mech. 79(1), 29–36 (1999)
H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach (Birkhäuser, Basel, 2001)
V.A. Solonnikov, Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations. Am. Math. Soc. Transl. 75(2), 1–116 (1968)
V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math. 8 467–529 (1977)
S. Ukai, A solution formula for the Stokes equation in \(\mathbf{R}_{+}^{n}\). Commun. Pure Appl. Math. 11, 611–621 (1987)
F.B. Weissler, The Navier-Stokes initial value problem in L p. Arch. Ration. Mech. Anal. 74, 219–230 (1980)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Basel
About this chapter
Cite this chapter
Kosaka, H., Maekawa, Y. (2016). On Vorticity Formulation for Viscous Incompressible Flows in R 3+ . In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0939-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0939-9_19
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0938-2
Online ISBN: 978-3-0348-0939-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)