Maximum modulus estimate for the solution of the Stokes equations

Abstract

A maximum modulus estimate for the nonstationary Stokes equations in C 2 domain is found. The singular part and regular part of Poisson kernel are analyzed. The singular part consists of the gradient of single layer potential and the gradient of composite potential defined on only normal component of the boundary data. Furthermore, the normal velocity near the boundary is bounded if the boundary data is bounded. If the normal component of the boundary data is Dinicontinuous and the tangential component of the boundary data is bounded, then the maximum modulus of velocity is bounded in whole domain.

A remark on the non-uniqueness in $$L^\infty $$ of the solutions to the two-dimensional Stokes problem in exterior domains

“…After the articles [25,27,28], in the last decades, the Stokes initial boundary value problem with an initial datum in L ∞ , jointly with L ∞ -estimates of the solutions, has been considered by several authors, both with homogeneous boundary data, see, e.g., [3][4][5][6]14,17], and with non-homogeneous; see, e.g., [7]. In the case of a twodimensional exterior domain, apart from the contributions given in [27,28] related to the non-homogeneous and homogeneous boundary data, respectively, based on the theory of the hydrodynamic potentials, the quoted literature, based on methods of functional analysis, achieves some results in a sequence of different papers [3,4] and [1,2].…”

Quasi-Maximum Modulus Principle for the Stokes Equations

“…In this section, we consider the maximum modulus estimate in the half space. In [1], the authors proved the following proposition.…”

“…For the interior L ∞ bound estimate, the authors in [1] showed the boundedness using the layer potential method in [8]. Let x ∈ Ω.…”

Solvability for Stokes System in Hölder Spaces in Bounded domains and Its Applications

“…The estimate (1.4) was improved by the first author and Choe as following inequality (see [5]) There are various literatures for the solvability of the Stokes system (1.1)-(1.2) with homogeneous boundary data, that is, with φ = 0 (see e.g. [11,17,18,19,22], and references therein).…”