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On the existence for the Cauchy-Neumann problem for the Stokes system in the $L_p$-framework

100%
Mucha P. B., Zajączkowski W.
Studia Mathematica
|
2000
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tom 143
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nr 1
75-101
EN
The existence for the Cauchy-Neumann problem for the Stokes system in a bounded domain $Ω ⊂ ℝ^3$ is proved in a class such that the velocity belongs to $W^{2,1}_r (Ω × (0,T))$, where r > 3. The proof is divided into three steps. First, the existence of solutions is proved in a half-space for vanishing initial data 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.
2
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On the existence for the Dirichlet problem for the compressible linearized Navier-Stokes system in the $L_p$-framework

100%
Mucha P., Zajączkowski W.
Annales Polonici Mathematici
|
2002
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tom 78
|
nr 3
241-260
EN
The existence of solutions to the Dirichlet problem for the compressible linearized Navier-Stokes system is proved in a class such that the velocity vector belongs to $W^{2,1}_r$ with r > 3. The proof is done in two steps. First the existence for local problems with constant coefficients is proved by applying the Fourier transform. Next by applying the regularizer technique the existence in a bounded domain is shown.
3
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On local-in-time existence for the Dirichlet problem for equations of compressible viscous fluids

100%
Mucha P., Zajączkowski W.
Annales Polonici Mathematici
|
2002
|
tom 78
|
nr 3
227-239
EN
The local existence of solutions for the compressible Navier-Stokes equations with the Dirichlet boundary conditions in the $L_p$-framework is proved. Next an almost-global-in-time existence of small solutions is shown. The considerations are made in Lagrangian coordinates. The result is sharp in the $L_p$-approach, because the velocity belongs to $W^{2,1}_r$ with r > 3.
4
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On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion

100%
Mucha P. B., Zajączkowski W.
Applicationes Mathematicae
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2000
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tom 27
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nr 3
319-333
EN
The local-in-time existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion is proved. We show the existence of solutions with lowest possible regularity for this problem such that $u\in W^{2,1}_r(\widetilde{{\mitΩ}}^T)$ with r>3. The existence is proved by the method of successive approximations where the solvability of the Cauchy-Neumann problem for the Stokes system is applied. We have to underline that in the $L_p$-approach the Lagrangian coordinates must be used. We are looking for solutions with lowest possible regularity because this simplifies the proof and decreases the number of compatibility conditions.
5
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Existence and stability theorems for abstract parabolic equations, and some of their applications

100%
Ströhmer G., Zajączkowski W.
Banach Center Publications
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1996
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tom 33
|
nr 1
369-382
EN
For a class of semi-abstract evolution equations for sections on vector bundles on a three-dimensional compact manifold we prove that for initial values with certain symmetries strong solutions exist for all times. In case these solutions become small after some time, strong solutions exist also for small perturbations of these initial values. Many systems from fluid mechanics are included in this class.
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