Title: Thermo-elastodynamics of nonlinearly viscous solids Show Chinese title

Title: 非线性粘性固体的热弹性动力学

Abstract: In this paper, we study the thermo-elastodynamics of nonlinearly viscous solids in the Kelvin-Voigt rheology where both the elastic and the viscous stress tensors comply with the frame-indifference principle. The system features a force balance including inertia in the frame of nonsimple materials and a heat-transfer equation which is governed by the Fourier law in the deformed configuration. Combining a staggered minimizing movement scheme for quasi-static thermoviscoelasticity with a variational approach to hyperbolic PDEs, our main result consists in establishing the existence of weak solutions in the dynamic case. This is first achieved by including an additional higher-order regularization for the dissipation. Afterwards, this regularization can be removed by passing to a weaker formulation of the heat-transfer equation which complies with a total energy balance. The latter description hinges on regularity theory for the fourth order $p$-Laplacian which induces regularity estimates of the deformation beyond the standard estimates available from energy bounds. Besides being crucial for the proof, these extra regularity properties might be of independent interest and seem to be new in the setting of nonlinear viscoelasticity, also in the static or quasi-static case. Show Chinese abstract

Abstract: 在本文中，我们研究了具有Kelvin-Voigt流变学的非线性粘性固体的热弹性动力学，其中弹性应力张量和粘性应力张量都符合参考系无关性原理。 该系统包括非简单材料框架中的惯性力平衡以及由变形构型中的傅里叶定律控制的热传导方程。 通过将用于准静态热粘弹性问题的交错最小化运动方案与双曲型偏微分方程的变分方法相结合，我们的主要结果是在动态情况下建立了弱解的存在性。 这是通过为耗散增加一个额外的高阶正则化项首次实现的。 随后，可以通过转向与总能量平衡一致的热传导方程的较弱形式来消除这种正则化。 后一种描述依赖于四阶$p$-拉普拉斯算子的正则性理论，该理论在标准能量界限之外提供了变形的正则性估计。 除了对证明至关重要外，这些额外的正则性性质可能具有独立的重要性，并且似乎在非线性粘弹性设置中是新的，即使在静态或准静态情况下也是如此。