Abstract
In this paper, we investigate Lp-estimates for the solution of the Stokes equation in a half space H where 1 <p <∞. It is shown that the solution of the Stokes equation is governed by an analytic semigroup on B U Cσ(H), C0,σ(H) or Lσ ∞(H). From the operatortheoretical point of view it is a surprising fact that the corresponding result for Lσ 1(H) does not hold true. In fact, there exists an L1 -function f satisfying div f = 0 such that the solution of the corresponding resolvent equation with right hand side f does not belong to L1 . Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the L1 -result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded H∞-calculus on Lp for 1 <p <∞ and obtain as a consequence maximal Lp-Lq-regularity for the solution of the Stokes equation.
Original language | English |
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Pages (from-to) | 115-142 |
Number of pages | 28 |
Journal | Journal of Evolution Equations |
Volume | 1 |
Issue number | 1 |
Publication status | Published - 2001 |
Externally published | Yes |
Keywords
- Analytic semigroups
- H-calculus
- Maximal L-regularity
- R-boundedness
- Resolvent estimates
- Stokes system
ASJC Scopus subject areas
- Ecology, Evolution, Behavior and Systematics