速度-焓形式下若干非线性声学模型的适定性、长时间行为与离散化
Well-posedness, long-time behavior, and discretization of some models of nonlinear acoustics in velocity-enthalpy formulation
AI总结：
本文基于速度-焓弱形式研究 Westervelt、Kuznetsov 及 Rasmussen 非线性声学模型，证明小数据下解全局存在且指数收敛到平衡态，并利用端口哈密顿结构实现保结构离散化。
摘要：
我们研究一类非线性声学模型，包括著名的 Westervelt 方程和 Kuznetsov 方程，以及一个可视为后者热力学一致修正的 Rasmussen 模型。利用线性化、能量估计和不动点论证，我们建立了解的存在性与唯一性；对于充分小的数据，这些解在时间上全局存在并以指数速度收敛到平衡态。与以往工作不同，我们的分析基于问题的速度-焓形式，其弱形式揭示了底层的端口哈密顿结构。此外，该问题的弱形式特别适合进行保结构离散化。数值试验验证了这一点，同时也展示了所考虑模型的典型特征。
英文摘要：
We study a class of models for nonlinear acoustics, including the well-known Westervelt and Kuznetsov equations, as well as a model of Rasmussen that can be seen as a thermodynamically consistent modification of the latter. Using linearization, energy estimates, and fixed-point arguments, we establish the existence and uniqueness of solutions that, for sufficiently small data, are global in time and converge exponentially fast to equilibrium. In contrast to previous work, our analysis is based on a velocity-enthalpy formulation of the problem, whose weak form reveals the underlying port-Hamiltonian structure. Moreover, the weak form of the problem is particularly well-suited for a structure-preserving discretization. This is demonstrated in numerical tests, which also highlight typical characteristics of the models under consideration.
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