TY - CHAP
T1 - Global Leray-Hopf weak solutions of the Navier-Stokes equations with Nonzero time-dependent boundary values
AU - Farwig, R.
AU - Kozono, H.
AU - Sohr, H.
PY - 2011/1/1
Y1 - 2011/1/1
N2 - In a bounded smooth domain Ω ⸦ ℝ3 and a time interval [0, T), 0 < T ≤ ∞, consider the instationary Navier-Stokes equations with initial value U0 ∈ L2 σ(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|∂Ω= g with non-zero time-dependent boundary values g. Although there is no uniqueness result for these solutions, they satisfy a strong energy inequality and an energy estimate. In particular, the long-time behavior of energies will be analyzed.
AB - In a bounded smooth domain Ω ⸦ ℝ3 and a time interval [0, T), 0 < T ≤ ∞, consider the instationary Navier-Stokes equations with initial value U0 ∈ L2 σ(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|∂Ω= g with non-zero time-dependent boundary values g. Although there is no uniqueness result for these solutions, they satisfy a strong energy inequality and an energy estimate. In particular, the long-time behavior of energies will be analyzed.
KW - Energy inequality
KW - Instationary Navier-Stokes equations
KW - Long-time behavior
KW - Non-zero boundary values
KW - Time-dependent data
KW - Weak solutions
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U2 - 10.1007/978-3-0348-0075-4_11
DO - 10.1007/978-3-0348-0075-4_11
M3 - Chapter
AN - SCOPUS:84895907203
T3 - Progress in Nonlinear Differential Equations and Their Application
SP - 211
EP - 232
BT - Progress in Nonlinear Differential Equations and Their Application
PB - Springer US
ER -