Abstract
We consider the quasilinear m × m system of partial differential equations that governs the motion of a viscoelastic material of strain-rate type on a bounded domain in \({\mathbb{R}}^n\) . The dependence of the stress on both the strain and strain-rate is nonlinear, and our hypotheses allow for a potential energy which is a nonconvex function of the strain. The critical hypothesis is that the dependence of the stress function on the strain rate is uniformly strictly monotone (in the sense of Minty and Browder). We prove the existence and uniqueness of weak solutions to a natural initial-boundary value problem for a large class of constitutive functions. We then treat the question of H 2-regularity of solutions and show that, while regularity in the initial data is preserved, solutions do not, in general, become more regular than their initial data. This generalizes a result for the semilinear case due to Rybka.
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Tvedt, B. Quasilinear Equations for Viscoelasticity of Strain-Rate Type. Arch Rational Mech Anal 189, 237–281 (2008). https://doi.org/10.1007/s00205-007-0109-x
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DOI: https://doi.org/10.1007/s00205-007-0109-x