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Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case

100%
Mucha P. B.
Banach Center Publications
|
2000
|
tom 52
|
nr 1
175-180
2
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On the existence for the Cauchy-Neumann problem for the Stokes system in the $L_p$-framework

63%
Mucha P. B., Zajączkowski W.
Studia Mathematica
|
2000
|
tom 143
|
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.
3
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On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion

63%
Mucha P. B., Zajączkowski W.
Applicationes Mathematicae
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2000
|
tom 27
|
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.
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