On the existence for the Cauchy-Neumann problem for the Stokes system in the $L_p$-framework
Volume 143 / 2000
Studia Mathematica 143 (2000), 75-101
DOI: 10.4064/sm-143-1-75-101
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.
