Local Well-Posedness for Volume-Preserving Mean Curvature and Willmore Flows with Line Tension

2014/03/05 by Helmut Abels, Harald Garcke, Abels, Helmut +3
Mathematics · #35K35 #35K55 #53C44 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35K35 #msc:35K55 #msc:53C44

paper · pdf · doi:10.48550/arxiv.1403.1132

37 pages

arxiv created 2014/03/25 · arxiv updated 2014/03/26

Abstract

We show the short-time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by volume-preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non-local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a Hanzawa transformation to write the flows as graphs over a fixed reference hypersurface.

Related

• Long-time existence of mean curvature flow with external force fields
• Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness
• Well-Posedness of a Navier-Stokes/Mean Curvature Flow system
• Local well-posedness of the contact line problem in 2-D Stokes flow
• Well-posedness for a class of compressible non-Newtonian fluids equations