TY - JOUR
T1 - Navier-stokes equations in thin 3D domains with navier boundary conditions
AU - Iftimie, Dragoş
AU - Raugel, Geneviève
AU - Sell, George R.
PY - 2007
Y1 - 2007
N2 - We consider the Navier-Stokes equations on a thin domain of the form Ωε = {x ∈ ℝ3 | x1, x 2 ∈ (0, 1), 0 < x3 < εg {x1, x2)} supplemented with the following mixed boundary conditions: periodic boundary conditions on the lateral boundary and Navier boundary conditions on the top and the bottom. Under the assumption that ∥u 0∥H1(Ωε ≤ Cε 1/2, par;Mu0i∥L2(Omega;ε) C for i ∈ {1,2} and similar assumptions on the forcing term, we show global existence of strong solutions; here u0i denotes the i-th component of the initial data u0 and M is the average in the vertical direction, that is, Mu0i(x1, x2 = (εg)-1 ∫0εg u0 i(x1, x2, x3) dx3 Moreover, if the initial data, respectively the forcing term, converge to a bidimensional vector field, respectively forcing term, as ε → 0, we prove convergence to a solution of a limiting system which is a Navier-Stokes-like equation where the function g plays an important role. Finally, we compare the attractor of the Navier-Stokes equations with the one of the limiting equation. Indiana University Mathematics Journal
AB - We consider the Navier-Stokes equations on a thin domain of the form Ωε = {x ∈ ℝ3 | x1, x 2 ∈ (0, 1), 0 < x3 < εg {x1, x2)} supplemented with the following mixed boundary conditions: periodic boundary conditions on the lateral boundary and Navier boundary conditions on the top and the bottom. Under the assumption that ∥u 0∥H1(Ωε ≤ Cε 1/2, par;Mu0i∥L2(Omega;ε) C for i ∈ {1,2} and similar assumptions on the forcing term, we show global existence of strong solutions; here u0i denotes the i-th component of the initial data u0 and M is the average in the vertical direction, that is, Mu0i(x1, x2 = (εg)-1 ∫0εg u0 i(x1, x2, x3) dx3 Moreover, if the initial data, respectively the forcing term, converge to a bidimensional vector field, respectively forcing term, as ε → 0, we prove convergence to a solution of a limiting system which is a Navier-Stokes-like equation where the function g plays an important role. Finally, we compare the attractor of the Navier-Stokes equations with the one of the limiting equation. Indiana University Mathematics Journal
KW - Global attractor
KW - Global existence
KW - Global regularity
KW - Navier-stokes equations
KW - Thin domain
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U2 - 10.1512/iumj.2007.56.2834
DO - 10.1512/iumj.2007.56.2834
M3 - Article
AN - SCOPUS:34547452104
SN - 0022-2518
VL - 56
SP - 1083
EP - 1155
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
IS - 3
ER -