An algorithmic construction of entropies in higher-order nonlinear PDEs
An algorithmic construction of entropies in higher-order nonlinear PDEs
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DOI：
10.1088/0951-7715/19/3/006
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发表时间：
2006-03
期刊：
影响因子：
1.7
通讯作者：
A. Jüngel;D. Matthes
中科院分区：
文献类型：
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作者：
A. Jüngel;D. Matthes
提出了一种构造一维偶数阶非线性演化偏微分方程熵和熵产生的新方法。证明熵耗散的任务被重新表述为多项式系统的决策问题。该方法成功地应用于多孔介质方程、薄膜方程和量子漂移扩散模型。在所有的情况下，一个无限数量的熵泛函连同相关的熵生产。我们的技术可以扩展到高阶熵，包含衍生物的解决方案，并在几个空间维度。此外，还可以得到对数Sobolev不等式.
A new approach to the construction of entropies and entropy productions for a large class of nonlinear evolutionary PDEs of even order in one space dimension is presented. The task of proving entropy dissipation is reformulated as a decision problem for polynomial systems. The method is successfully applied to the porous medium equation, the thin film equation and the quantum drift–diffusion model. In all cases, an infinite number of entropy functionals together with the associated entropy productions is derived. Our technique can be extended to higher-order entropies, containing derivatives of the solution, and to several space dimensions. Furthermore, logarithmic Sobolev inequalities can be obtained.