A Note on Weak Solutions of Conservation Laws and Energy/Entropy Conservation
A Note on Weak Solutions of Conservation Laws and Energy/Entropy Conservation
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DOI：
10.1007/s00205-018-1238-0
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发表时间：
2017-06
影响因子：
2.5
通讯作者：
P. Gwiazda;M. Michálek;Agnieszka Świerczewska-Gwiazda
中科院分区：
文献类型：
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作者：
P. Gwiazda;M. Michálek;Agnieszka Świerczewska-Gwiazda
连续介质物理守恒定律系统的一个共同特征是，它们被赋予了自然的伴随定律，在这种情况下，这些伴随定律通常与热力学第二定律有关。这个观察结果很容易推广到任何对称的守恒定律系统；它们被赋予了非平凡的伴守恒定律，这些守恒定律立即被经典解所满足。不足为奇的是，弱解可能不满足伴生定律，而伴生定律通常会从相等放宽到不相等，并取代弱解的物理可接受条件的作用。我们想要回答这个问题：对于一个守恒定律的一般系统，弱解的临界规则是什么以满足一个相关的伴律作为等式？这种结果的一个典型例子是在九十年代初的Onsager猜想的背景下为不可压缩欧拉系统导出的。这个一般结果可以作为许多数学物理系统的一个简单准则，用来规定解的规则性，而解的规则性是满足一个适当的伴律所必需的。
A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such cases most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws; they are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake the role of physical admissibility conditions for weak solutions. We want to answer the question: what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality? An archetypal example of such a result was derived for the incompressible Euler system in the context of Onsager’s conjecture in the early nineties. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.