Asymptotics for incompressible elastodynamics

Source abstract

Abstract The aim of this paper is to clarify the asymptotic behavior of global classical solutions with small initial data for the Cauchy problem of incompressible neo‐Hookean elastodynamics. To this end, we first provide a new and streamlined proof of the global existence result, employing only general derivatives and spatial rotation operators as commuting vector fields. Building upon this new proof, we then show that the global solution scatters in the energy sense; namely, it converges to a solution of the homogeneous linear wave equations as time tends to infinity. We also prove the following rigidity result: if the scattering data vanish, then the global solution vanishes identically. Finally, we show that for compactly supported initial data, the global solution is uniquely determined by the scattering data; in other words, the inverse scattering property holds.