By Hajime Koba
A desk bound resolution of the rotating Navier-Stokes equations with a boundary is named an Ekman boundary layer. This ebook constructs desk bound ideas of the rotating Navier-Stokes-Boussinesq equations with stratification results within the case whilst the rotating axis isn't inevitably perpendicular to the horizon. the writer calls such desk bound options Ekman layers. This publication exhibits the lifestyles of a susceptible way to an Ekman perturbed procedure, which satisfies the powerful strength inequality. additionally, the writer discusses the individuality of susceptible ideas and computes the decay fee of vulnerable suggestions with appreciate to time less than a few assumptions at the Ekman layers and the actual parameters. the writer additionally exhibits that there exists a special global-in-time robust answer of the perturbed procedure whilst the preliminary datum is satisfactorily small. evaluating a vulnerable answer gratifying the robust power inequality with the powerful resolution means that the susceptible resolution is soft with admire to time while time is adequately huge
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Extra info for Nonlinear stability of Ekman boundary layers in rotating stratified fluids
0, T ) × R3+ . 29) mh →∞ T 1 wm , φ dt = h lim 0 2 w1 , φ dt = 0 for all φ ∈ L2 ((0, T ) × R3+ ). 0 Let Φ ∈ L2 (s, t; L (R3+ )). Set Φ(τ ), s < τ < t, 0, 0 ≤ τ ≤ s, t ≤ τ ≤ T. 29) to see that t s T 1 wm , φ dt = h 1 wm , Φ dt h 0 T t → w1 , Φ dt = 0 w1 , φ dt. 25). 23), we see that for each k = 1, 2, 3, 4 whk ≤ whk L2 (s,t;H 1 (R3+ )) L2 (0,T ;H 1 (R3+ )) −1 ≤(CE + T )1/2 w0 L2 < ∞. 31) k ∂l wm → vlk weakly in L2 (s, t; L2 (R3+ )) for each l = 1, 2, 3. j We now check vlk = ∂l wk . Let ϕ ∈ C01 (R3+ ).
Set Φ(τ ), s < τ < t, 0, 0 ≤ τ ≤ s, t ≤ τ ≤ T. 29) to see that t s T 1 wm , φ dt = h 1 wm , Φ dt h 0 T t → w1 , Φ dt = 0 w1 , φ dt. 25). 23), we see that for each k = 1, 2, 3, 4 whk ≤ whk L2 (s,t;H 1 (R3+ )) L2 (0,T ;H 1 (R3+ )) −1 ≤(CE + T )1/2 w0 L2 < ∞. 31) k ∂l wm → vlk weakly in L2 (s, t; L2 (R3+ )) for each l = 1, 2, 3. j We now check vlk = ∂l wk . Let ϕ ∈ C01 (R3+ ). 30) to have t mj →∞ t k wm , ∂l ϕ dτ = j lim s wk , ∂l ϕ dτ. 32) lim mj →∞ s t k ∂l wm , ϕ dτ = j ∂l wk , ϕ dτ for all ϕ ∈ C01 (R3+ ).
41) Jm u L∞ ≤C Jm u ≤C Jm u 1/4 3/4 L4 ∇Jm u L4 1/16 3/16 ∇Jm u L2 L2 ∇Jm u 3/4 L4 =: (RHS). 43) L2 ≤ C (A + 1)1/2 Jm u ≤C (A + 1)Jm u L2 1/2 L2 Jm u 1/2 L2 . 44) It suﬃces to show that (A + 1)Jm u (A + 1)Jm u L2 1/4 L2 L2 3/4 L2 . (A + 1)Jm u ≤ C(m) u L2 . 39), we have ≤C( (A + BE + m)Jm u − (BE + m)Jm u ≤C (LE + m)Jm u L2 + C∗∗ BE J m u L2 L2 + u + C(m) u L2 ) L2 , where BE = LE − A. 39), we see C∗∗ BE Jm u L2 ≤C( ∇Jm u L2 + Jm u L2 ) ε0 ∇2 Jm u L2 + C(ε0 ) u ≤ C∗ (2) ≤ε0 (A + 1)Jm u L2 + C(ε0 ) u L2 L2 for each ε0 > 0.
Nonlinear stability of Ekman boundary layers in rotating stratified fluids by Hajime Koba