Divergence equation in thin-tube structures

Abstract

Divergence equation is considered in a thin-tube structure, connected finite union of thin finite cylinders (in the case of two dimensions, respectively, thin rectangles) where the ratio of the diameter and the heights of the cylinders is the small parameter . Various norms of the solution of divergence equation are estimated over the norms of the right-hand side. The exact dependence of constants in these estimates on the small parameter is obtained.

Keywords:

• viscous fluid flow
• divergence equation
• thin-tube structure
• estimates

AMS Subject Classifications:

• 35Q30
• 76M45
• 76M55

Notes

1 We apologize not cited authors and refer to the book by Galdi [19].

The present work is partially supported by the Research Federative Structures MODMAD FED 4169 and FR CNRS 3490, by the French-German grant PROCOPE EGIDE 28481WB “Homogenization based optimization for elasticity on the network of beams”, and by LABEX MILYON (ANR-10-LABX-0070) of University of Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The research was funded by the grant number [14-11-00306] of Russian Scientific Foundation and by the Lithuanian-Swiss cooperation programme to reduce economic and social disparities within the enlarged European Union under project agreement No. CH-3- S̆ MM-01/01.