By Andrei V. Fursikov, Giovanni P. Galdi, Vladislav V. Pukhnachev

The clinical pursuits of Professor A.V. Kazhikhov have been essentially dedicated to Mathematical Fluid Mechanics, the place he completed remarkable effects that had, and now have, an important effect in this field.

This quantity, devoted to the reminiscence of A.V. Kazhikhov, provides the newest contributions from well known global experts in a couple of new vital instructions of Mathematical Physics, typically of Mathematical Fluid Mechanics, and, extra normally, within the box of nonlinear partial differential equations. those effects are quite often regarding boundary price difficulties and to manage difficulties for the Navier-Stokes equations, and for equations of warmth convection. different vital subject matters comprise non-equilibrium tactics, Poisson-Boltzmann equations, dynamics of elastic physique, and similar difficulties of functionality conception and nonlinear analysis.

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Extra resources for New Directions in Mathematical Fluid Mechanics: The Alexander V. Kazhikhov Memorial Volume (Advances in Mathematical Fluid Mechanics)

Example text

2). 4. A priori estimates We proceed by returning to the dimensional variables. 8) depend on δ. We call the electrolyte concave, linear or convex if c− c− c− i qi > 0, i qi = 0, i qi < 0, ± ± ± respectively. We recall that q1 = q+ > 0 and q2 = q− < 0. We write p1 = 4πq1 c− 1 , εf p2 = 4π|q2 |c− 2 , εf r1 = q1 , kT r2 = |q2 | . 11) in the fluid domain becomes ϕzz = p2 er2 [ϕ(z)−ϕ(Hd (z))] − p1 e−r1 [ϕ(z)−ϕ(Hd (z))] . We prove that there is a positive constant B0 such that L 0 ϕ2z (z)dz ≤ B0 , max |ϕ(z)| ≤ B0 .

The case p = ∞. 9) with p = ∞. 7) with the constants m, C being independent of θ. e. t ∈ (0, T ), − ∂t hθ + ∂t hθ = div Gθ on Ω, ∂t hθ Γ = ∂t aθ with Gθ := −vθ ωθ , such that ||Gθ ||L∞ (0,T ;Lq (Ω)) C and C is independent of θ. e. 1) with C = C(q), independent of θ. 7), there exists a subsequence of {ωθ , hθ , vθ }, such that hθ h weakly − ∗ in L∞ (0, T ; Wq2(Ω)), ωθ ω weakly − ∗ in L∞ (ΩT ), vθ → v strongly in L∞ (0, T ; Lq (Ω)) for ∀q < ∞. Using the same argument of Lemma 5, we have also ∂h ∂hθ → (v · n) ≡ strongly in L2 (ΓT ).

Amirat and V. Shelukhin It follows from the identity z w (n) (z) − w (n) wz(n) (s)ds (dn ) = dn that |w(n) (z)| ≤ B3 for any interval an < z < bn . 2) as in the linear electrolyte case. The case of a concave electrolyte (p2 < p1 ) can be considered similarly. 24) where w(z) = ϕ(z) − ϕ(Hd (z)) and g1 (0) = p2 − p1 . 2) that N −1 an+1 wz2 (z)dz ≤ B4 , n=0 a n max |w(z)| ≤ B5 . 0≤z≤L We prove that there is a constant B6 independent of δ such that |g1 (w(z)) − g1 (0)| ≤ B6 δ. 16). By definition, g1 (w(z)) − g1 (0) = g1 (ϕ (z ) − ϕ (Hd (z ))) − g1 (0).

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