Local Well-posedness of the Free Boundary Incompressible Elastodynamics with Surface Tension

Abstract

In this paper, we consider a free boundary problem of the incompressible elatodynamics, a coupling system of the Euler equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with the surface tension, we establish its well-posedness theory on a short time interval. Our method is the vanishing viscosity limit by establishing a uniform a priori estimates with respect to the viscosity. As a by-product, the inviscid limit of the incompressible viscoelasticity (the system coupling with the Navier-Stokes equations) is also justified. We point out that based on a crucial new observation of the inherent structure of the elastic term on the free boundary, the framework here is established solely in standard Sobolev spaces, but not the co-normal ones used in [33].

Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension

“…Remark. Very recently, the first author and Lei [24] proved the LWP of incompressible elastodynamics with surface tension by proving the inviscid limit of visco-elastodynamics system in standard Sobolev spaces. We also note that the inviscid limit of free-boundary MHD was recently proved by Chen-Ding [7] in co-normal Sobolev spaces.…”

Low Mach number limit of nonisentropic inviscid Hookean elastodynamics

“…More discussions on the physical background and equations of elastodynamics can be found in the book of Dafermos. 1 The mathematical theory of both compressible and incompressible elastodynamics have been studied extensively in the last few decades, for example, previous studies [3][4][5][6][7][8][9][10][11][12] and references therein. In this paper, we will focus mainly on the low Mach number limit of (1.5).…”

Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics

“…Finally, ∂ 5 A 3α in J 1 has the top order contribution ∂ 6 η × ∂η that cannot be directly controlled. To overcome this difficulty, we write A 3α = (∂ 1 η × ∂ 2 η) α and use it to observe a crucial symmetric structure on the boundary, while the 2D version of this symmetric structure was developed in [11] and plays an important role in the local well-posedness of free-surface incompressible elastodynamics.…”