双重非线性波动方程的加权惯性-耗散-能量方法
Weighted Inertia-Dissipation-Energy approach to doubly nonlinear wave equations
AI总结：
本文提出加权惯性-耗散-能量（WIDE）变分方法研究双重非线性波动方程，证明WIDE泛函极小元存在且其欧拉-拉格朗日系统强可解，并证明参数趋于零时极小元收敛到原方程解，同时分析了粘性极限。
摘要：
我们讨论了一类形如 $ρu_{tt} + g (u_t) - Δu + f (u)=0$ 的双重非线性波动方程的变分方法。该方法的核心在于对整个轨迹上的一族依赖参数的均匀凸泛函进行极小化，即所谓的加权惯性-耗散-能量（WIDE）泛函。我们证明了 WIDE 泛函存在极小元，并且相应的欧拉-拉格朗日系统在强意义下可解。此外，我们验证了当参数趋于 $0$ 时，依赖参数的极小元在子列意义下收敛到目标双重非线性波动方程的解。该分析依赖于对 WIDE 极小元的特定估计、对 WIDE 泛函次微分的分解，以及在极限过程中对非线性的识别。最后，我们在泛函层面和方程层面研究了粘性极限 $ρ\ o 0$。
英文摘要：
We discuss a variational approach to doubly nonlinear wave equations of the form $ρu_{tt} + g (u_t) - Δu + f (u)=0$. This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WIDE) functionals. We prove that the WIDE functionals admit minimizers and that the corresponding Euler-Lagrange system is solvable in the strong sense. Moreover, we check that the parameter-dependent minimizers converge, up to subsequences, to a solution of the target doubly nonlinear wave equation as the parameter goes to $0$. The analysis relies on specific estimates on the WIDE minimizers, on the decomposition of the subdifferential of the WIDE functional, and on the identification of the nonlinearities in the limit. Eventually, we investigate the viscous limit $ρ\to 0$, both at the functional level and on that of the equation.
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