Asymptotic limits for strain-gradient viscoelasticity with nonconvex energy

Abstract We consider the system of viscoelasticity with higher-order gradients and nonconvex energy in several space dimensions. We establish the asymptotic limits when the viscosity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> or when the dispersion coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . For the latter problem, it is worth noting that, for the case of two space dimensions, we also establish a rate of convergence. This result bears analogies to a result of Chemin (1996 Commun. PDE 21 1771–79) on the rate of convergence of the zero-viscosity limit for the two-dimensional Navier–Stokes equations with bounded vorticity.