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

• Autorzy: Piotr Bogusław Mucha , Wojciech Zajączkowski
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Stokes system; Marcinkiewicz theorem; Cauchy-Neumann initial boundary value problem; the Fourier transform; existence of solutions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 143
• Numer: 1
• Strony: 75-101

Daty

wydano

2000

otrzymano

2000-01-31

poprawiono

2000-05-05

poprawiono

2000-06-14

Twórcy

autor

Piotr Bogusław Mucha

• Institute of Applied, Mathematics and Mechanics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

autor

Wojciech Zajączkowski

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland

Bibliografia

• [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] J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math. 8 (1939), 78-91.
• [3] S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965.
• [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.
• [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.
• [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] 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] V. A. Solonnikov, On some initial-boundary value problems for the Stokes system, Trudy Mat. Inst. Steklov. 188 (1990), 150-188 (in Russian).
• [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).
• [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).
• [11] H. Triebel, Spaces of Besov-Hardy-Sobolev Type, Teubner, Leipzig, 1978.
• [12] W. M. Zajączkowski, On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface, Dissertationes Math. 324 (1993).