The 3D Time-Dependent Oseen System: Link Between $$L^p$$-Integrability in Time and Pointwise Decay in Space of the Velocity

A representation formula without pressure term is derived for regular solutions to the 3D time-dependent Oseen system in exterior Lipschitz domains. This formula is valid even if no boundary conditions are imposed. It is used in order to exhibit how the velocity decays pointwise in space. It turns out that the rate of this decay depends on $$L^p$$ -integrability in time of the velocity. In addition, this work is the basis for successor papers dealing with spatial decay of $$L^q$$ -weak solutions and mild solutions to the time-dependent Oseen system, and with $$L^2$$ -strong solutions to the stability problem related to the Navier-Stokes system with Oseen term.

[1]  Giovanni P. Galdi,et al.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems , 2011 .

[2]  W. Borchers,et al.  On the equations rot v=g and div u=f with zero boundary conditions , 1990 .

[3]  V. A. Solonnikov,et al.  Estimates for solutions of nonstationary Navier-Stokes equations , 1977 .

[4]  T. Hishida.  Large time behavior of a generalized Oseen evolution operator, with applications to the Navier–Stokes flow past a rotating obstacle , 2017, 1706.03344.

[5]  H. Sohr,et al.  The Navier-Stokes Equations: An Elementary Functional Analytic Approach , 2012 .

[6]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[7]  P. Deuring,et al.  Exterior stationary Navier‐Stokes flows in 3D with non‐zero velocity at infinity: approximation by flows in bounded domains , 2004 .

[8]  E. Hille.  Functional Analysis And Semi-Groups , 1948 .

[9]  P. Deuring.  Pointwise Spatial Decay of Weak Solutions to the Navier–Stokes System in 3D Exterior Domains , 2015 .

[10]  K. Masuda.  On the Stability of Incompressible Viscous Fluid Motions Past Objects (非線形問題の解析 : Analysis of Nonlinear Problems, RIMS, 1974) , 1975 .

[11]  谷内 靖,et al.  書評 Hermann Sohr : The Navier-Stokes Equations : An Elementary Functional Analytic Approach , 2014 .

[12]  J. G. Heywood.  The exterior nonstationary problem for the Navier-Stokes equations , 1972 .

[13]  Takayuki Kobayashi,et al.  On the Oseen equation in the three dimensional exterior domains , 1998 .

[14]  Ryuichi Mizumachi.  On the asymptotic behavior of incompressible viscous fluid motions past bodies , 1984 .

[15]  Lutz Weis,et al.  Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity , 2001 .

[16]  P. Deuring.  On boundary-driven time-dependent Oseen flows , 2008 .

[17]  Y. Shibata,et al.  Local energy decay of solutions to the oseen equation in the exterior domains , 2004 .

[18]  Paul Deuring,et al.  Spatial asymptotics of mild solutions to the time-dependent Oseen system , 2021, Communications on Pure & Applied Analysis.

[19]  P. Deuring.  Pointwise Decay in Space and in Time for Incompressible Viscous Flow Around a Rigid Body Moving with Constant Velocity , 2017, Journal of Mathematical Fluid Mechanics.

[20]  Y. Shibata.  On an exterior initial boundary value problem for Navier-Stokes equations , 1999 .

[21]  Paul Deuring,et al.  Spatial Decay of Time-Dependent Oseen Flows , 2009, SIAM J. Math. Anal..

[22]  Paul Deuring,et al.  On Pointwise Decay of Linearized Stationary Incompressible Viscous Flow around Rotating and Translating Bodies , 2011, SIAM J. Math. Anal..

[23]  Paul Deuring.  Spatial Decay of Time-Dependent Incompressible Navier-Stokes Flows with Nonzero Velocity at Infinity , 2013, SIAM J. Math. Anal..

[24]  P. Deuring,et al.  On Oseen resolvent estimates , 2010, Differential and Integral Equations.

[25]  Tetsuro Miyakawa,et al.  On nonstationary solutions of the Navier-Stokes equations in an exterior domain , 1982 .

[26]  Shuji Takahashi,et al.  A weighted equation approach to decay rate estimates for the Navier-Stokes equations , 1999 .

[27]  Hyeong‐Ohk Bae,et al.  Estimates of the wake for the 3D Oseen equations , 2008 .

[28]  P. Deuring.  Time-dependent incompressible viscous flows around a rigid body: Estimates of spatial decay independent of boundary conditions , 2020, Journal of Differential Equations.

[29]  P. Deuring.  A Representation Formula for the Velocity Part of 3D Time-Dependent Oseen Flows , 2014 .

[30]  A Potential-Theoretic Approach to the Time-Dependent Oseen System , 2010 .

[31]  J. G. Heywood,et al.  On the Global Existence and Convergence to Steady State of Navier-Stokes Flow Past an Obstacle that is Started from Rest , 1997 .

[32]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[33]  T. Hishida.  Decay Estimates of Gradient of a Generalized Oseen Evolution Operator Arising from Time-Dependent Rigid Motions in Exterior Domains , 2019, Archive for Rational Mechanics and Analysis.

[34]  P. Deuring.  Pointwise spatial decay of time-dependent Oseen flows: The case of data with noncompact support , 2013 .

[35]  Y. Shibata,et al.  On the Rate of Decay of the Oseen Semigroup in Exterior Domains and its Application to Navier–Stokes Equation , 2005 .

[36]  Hyeong‐Ohk Bae,et al.  Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains , 2012 .

[37]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[38]  P. Deuring.  The Single-Layer Potential Associated with the Time-dependent Oseen System , 2006 .

[39]  G. Knightly.  Some decay properties of solutions of the Navier-Stokes equations , 1980 .

[40]  R. Farwig.  The stationary exterior 3 D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces , 1992 .

[41]  Zhongwei Shen,et al.  Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders , 1991 .