Global Weak Solutions of a Hamiltonian Regularised Burgers Equation
Global Weak Solutions of a Hamiltonian Regularised Burgers Equation
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DOI：
10.1007/s10884-022-10171-0
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发表时间：
2020-02
影响因子：
1.3
通讯作者：
Billel Guelmame;S. Junca;D. Clamond;R. Pego
中科院分区：
文献类型：
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作者：
Billel Guelmame;S. Junca;D. Clamond;R. Pego
提出并研究了无粘Burgers方程的非色散、守恒正则化。受最近由Clamond和Dutykh引入的浅水系统的相关正则化的启发，新的正则化提供了无粘Burgers方程和Hunter-Saxton方程之间的一系列伽利略不变插值。它允许弱奇异正则激波和尖点行波弱解。证明了局部光滑解的崩溃，并建立了两种类型的整体弱解的存在性，保存或耗散的能量。耗散解满足Oleinik不等式，就像无粘Burgers方程的熵解一样。由于正则化标度parameters趋于0或，耗散解的极限分别满足无粘Burgers或Hunter-Saxton方程，由一个未知的剩余项。
A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter–Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating anenergy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parametertends to 0 or, limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter–Saxton equation respectively, forced by an unknown remaining term.