# On the existence for the Cauchy-Neumann problem for the Stokes system in the ${L}_{p}$-framework

Studia Mathematica (2000)

• Volume: 143, Issue: 1, page 75-101
• ISSN: 0039-3223

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## Abstract

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The existence for the Cauchy-Neumann problem for the Stokes system in a bounded domain $\Omega \subset {ℝ}^{3}$ is proved in a class such that the velocity belongs to ${W}_{r}^{2,1}\left(\Omega ×\left(0,T\right)\right)$, 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.

## How to cite

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Mucha, Piotr, and Zajączkowski, Wojciech. "On the existence for the Cauchy-Neumann problem for the Stokes system in the $L_p$-framework." Studia Mathematica 143.1 (2000): 75-101. <http://eudml.org/doc/216810>.

@article{Mucha2000,
abstract = {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.},
author = {Mucha, Piotr, Zajączkowski, Wojciech},
journal = {Studia Mathematica},
keywords = {Stokes system; Marcinkiewicz theorem; Cauchy-Neumann initial boundary value problem; the Fourier transform; existence of solutions; initial-boundary value Cauchy-Neumann problem; non-stationary Stokes system; Fourier transforms; Marcinkiewicz multiplier theorem; regulaziation; unique solvability},
language = {eng},
number = {1},
pages = {75-101},
title = {On the existence for the Cauchy-Neumann problem for the Stokes system in the $L_p$-framework},
url = {http://eudml.org/doc/216810},
volume = {143},
year = {2000},
}

TY - JOUR
AU - Mucha, Piotr
AU - Zajączkowski, Wojciech
TI - On the existence for the Cauchy-Neumann problem for the Stokes system in the $L_p$-framework
JO - Studia Mathematica
PY - 2000
VL - 143
IS - 1
SP - 75
EP - 101
AB - 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.
LA - eng
KW - Stokes system; Marcinkiewicz theorem; Cauchy-Neumann initial boundary value problem; the Fourier transform; existence of solutions; initial-boundary value Cauchy-Neumann problem; non-stationary Stokes system; Fourier transforms; Marcinkiewicz multiplier theorem; regulaziation; unique solvability
UR - http://eudml.org/doc/216810
ER -

## References

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1. [1] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow, 1975 (in Russian).
2. [2] J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math. 8 (1939), 78-91.
3. [3] S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965. Zbl0129.07701
4. [4] P. B. Mucha and W. Zajączkowski, On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Appl. Math. (Warsaw) 27 (2000), 319-333. Zbl0996.35050
5. [5] P. B. Mucha and W. Zajączkowski, On stability of equilibrium solutions of the free boundary problem for a viscous self-gravitating incompressible fluid, in preparation. Zbl0996.35050
6. [6] V. A. Solonnikov, Estimates of solutions of the nonstationary linearized Navier-Stokes system, Trudy Mat. Inst. Steklov. 70 (1964), 213-317 (in Russian).
7. [7] V. A. Solonnikov, On the nonstationary motion of an isolated volume of a viscous incompressible fluid, Izv. Akad. Nauk SSSR 51 (1987), 1065-1087 (in Russian).
8. [8] V. A. Solonnikov, On some initial-boundary value problems for the Stokes system, Trudy Mat. Inst. Steklov. 188 (1990), 150-188 (in Russian).
9. [9] V. A. Solonnikov, Estimates of solutions of an initial-boundary value problem for the linear nonstationary Navier-Stokes system, Zap. Nauchn. Sem. LOMI 59 (1976), 178-254 (in Russian). Zbl0357.76026
10. [10] V. A. Solonnikov, On the solvability of the second initial-boundary value problem for the linear nonstationary Navier-Stokes system, ibid. 69 (1977), 200-218 (in Russian). Zbl0348.35079
11. [11] H. Triebel, Spaces of Besov-Hardy-Sobolev Type, Teubner, Leipzig, 1978. Zbl0408.46024
12. [12] W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Dissertationes Math. 324 (1993). Zbl0771.76059

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