A Regularity Result for the Incompressible Elastodynamics with a Free Interface

Abstract

We consider the incompressible inviscid elastodynamics equations with a free surface and establish the regularity of solutions for elastic system. Compared with the previous result on this free boundary problem [ Gu X and Wang F, Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition, J. Math. Anal. Appl., 2020, 482(1): 123529] in space $H^3,$ we are able to establish the regularity in space $H^{2.5+δ}.$ It is achieved by reformulating the system into the Lagrangian formulation, presenting the uniform estimates for the pressure, the tangential estimates for the system, as well as the divergence and curl estimates.