On the existence for the Cauchy-Neumann problem for the Stokes system in the L p -framework

Piotr Mucha; Wojciech Zajączkowski

Studia Mathematica (2000)

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

Abstract

top
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 r 2 , 1 ( 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.

How to cite

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.