TY - JOUR

T1 - Weak solutions of the Navier-Stokes equations with non-zero boundary values in an exterior domain satisfying the strong energy inequality

AU - Farwig, Reinhard

AU - Kozono, Hideo

N1 - Funding Information:
The authors are partly supported by the International Research Training Group (IRTG 1529) on Mathematical Fluid Dynamics Darmstadt–Tokyo funded by DFG and JSPS . The first author was also supported by the Center of Smart Interfaces (CSI), TU Darmstadt , the second author by Grant No. 24224003 of the Japan Society for the Promotion of Science .

PY - 2014/4/1

Y1 - 2014/4/1

N2 - In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0, T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

AB - In an exterior domain Ω⊂R3 and a time interval [0, T), 0<T≤∞, consider the instationary Navier-Stokes equations with initial value u0∈Lσ2(Ω) and external force f=divF, F∈L2(0, T;L2(Ω)). As is well-known there exists at least one weak solution in the sense of J. Leray and E. Hopf with vanishing boundary values satisfying the strong energy inequality. In this paper, we extend the class of global in time Leray-Hopf weak solutions to the case when u=g with non-zero time-dependent boundary values g. Although uniqueness for these solutions cannot be proved, we show the existence of at least one weak solution satisfying the strong energy inequality and a related energy estimate.

KW - Exterior domain

KW - Instationary Navier-Stokes equations

KW - Non-zero boundary values

KW - Strong energy inequality

KW - Time-dependent data

KW - Weak solutions

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U2 - 10.1016/j.jde.2014.01.029

DO - 10.1016/j.jde.2014.01.029

M3 - Article

AN - SCOPUS:84895908566

VL - 256

SP - 2633

EP - 2658

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 7

ER -