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On existence of solutions for the nonstationary Stokes system with boundary slip conditions

100%
Alame W.
Applicationes Mathematicae
|
2005
|
tom 32
|
nr 2
195-223
EN
Existence of solutions for equations of the nonstationary Stokes system in a bounded domain Ω ⊂ ℝ³ is proved in a class such that velocity belongs to $W_p^{2,1}(Ω × (0,T))$, and pressure belongs to $W_p^{1,0}(Ω × (0,T))$ for p > 3. The proof is divided into three steps. First, the existence of solutions with vanishing initial data is proved in a half-space by applying the Marcinkiewicz multiplier theorem. Next, we prove the existence of weak solutions in a bounded domain and then we regularize them. Finally, the problem with nonvanishing initial data is considered.
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On the existence of solutions for the nonstationary Stokes system with slip boundary conditions in general Sobolev-Slobodetskii and Besov spaces

100%
Alame W.
Banach Center Publications
|
2005
|
tom 70
|
nr 1
21-49
EN
We prove the existence of solutions to the evolutionary Stokes system in a bounded domain Ω ⊂ ℝ³. The main result shows that the velocity belongs either to $W_p^{2s+2,s+1}(Ω^T)$ or to $B_{p,q}^{2s+2,s+1}(Ω^T)$ with p > 3 and s ∈ ℝ₊ ∪ 0. The proof is divided into two steps. First the existence in $W_p^{2k+2,k+1}$ for k ∈ ℕ is proved. Next applying interpolation theory the existence in Besov spaces in a half space is shown. Finally the technique of regularizers implies the existence in a bounded domain. The result is generalized to the spaces $W_p^{2s,s}(Ω^T)$ and $B_{p,q}^{2s,s}$ with p > 2 and s ∈ (1/2,1).
3
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Large time regular solutions to the MHD equations in cylindrical domains

63%
Alame W., Zajączkowski W.
Applicationes Mathematicae
|
2011
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tom 38
|
nr 2
117-146
EN
We prove the large time existence of solutions to the magnetohydrodynamics equations with slip boundary conditions in a cylindrical domain. Assuming smallness of the L₂-norms of the derivatives of the initial velocity and of the magnetic field with respect to the variable along the axis of the cylinder, we are able to obtain an estimate for the velocity and the magnetic field in $W₂^{2,1}$ without restriction on their magnitude. Then the existence follows from the Leray-Schauder fixed point theorem.
4
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Global existence of solutions for incompressible magnetohydrodynamic equations

63%
Alame W., Zajączkowski W.
Applicationes Mathematicae
|
2004
|
tom 31
|
nr 2
201-208
EN
Global-in-time existence of solutions for incompressible magnetohydrodynamic fluid equations in a bounded domain Ω ⊂ ℝ³ with the boundary slip conditions is proved. The proof is based on the potential method. The existence is proved in a class of functions such that the velocity and the magnetic field belong to $W_p^{2,1}(Ω×(0,T))$ and the pressure q satisfies $∇q ∈ L_p(Ω×(0,T))$ for p ≥ 7/3.
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