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Existence and uniqueness for a viscoelastic Kelvin-Voigt model with nonconvex stored energy

Banff International Research Station for Mathematical Innovation and Discovery

We consider the Kelvin-Voigt model for viscoelasticity and prove propagation of $H^1$-regularity for the deformation gradient of weak solutions in two and three dimensions assuming that the stored energy satisfies the Andrews-Ball condition, in particular allowing for a non-monotone stress. By contrast, a counterexample indicates that for non-monotone stress-strain relations (even in 1-d) initial oscillations of the strain lead to solutions with sustained oscllations. In addition, in two space dimensions, we prove that the weak solutions with deformation gradient in $H^1$ are in fact unique, providing a striking analogy to the 2D Euler equations with bounded vorticity. </p>

(joint work with K. Koumatos (U. of Sussex), C. Lattanzio and S. Spirito (U. of Lâ Aquila)). </p>

Mathematics; Partial differential equations; Fluid mechanics; Fluid dynamics

video/mp4

Author affiliation: KAUST

2021-05-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/78468

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/