Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics

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• 作者：Jan Giesselmann ; Corrado Lattanzio…
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：223
• 期：3
• 页码：1427-1484
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0673
• 卷排序：223

文摘

We consider a Euler system with dynamics generated by a potential energy functional. We propose a form for the relative energy thatexploits the variational structure and we derive a relative energy identity. When applied to specific energies, this yields relativeenergy identities for the Euler–Korteweg, the Euler–Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler–Korteweg system. For the Euler–Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier–Stokes–Korteweg system (NSK) with non-monotone pressure laws, and prove stability for the NSK system via a modified relative energy approach. We prove the continuous dependence of solutions on initial data and the convergence of solutions of a low order model to solutions of the NSK system. The last two results provide physically meaningful examples of how higher order regularization terms enable the use of the relative energy framework for models with energies which are not poly- or quasi-convex, compensated by higher-order gradients.