Abstract
In this article we study the principle of energy conservation for the Euler–Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler–Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.
Highlights
Mathematics Subject Classification 76D45 · 35G50. It is known since the works of Scheffer [28] and Shnirelmann [29] that weak solutions of the incompressible Euler equations exhibit behaviour very different to that of classical solutions
These “wild solutions”, as they are called since the seminal works of DeLellis and Székelyhidi [10,11], are often highly unphysical—for instance there is a lack of uniqueness and the principle of conservation of energy can be violated
Dissipative solutions of incompressible Euler have been extensively studied in relation to the seminal Onsager conjecture [27]
Summary
It is known since the works of Scheffer [28] and Shnirelmann [29] that weak solutions of the incompressible Euler equations exhibit behaviour very different to that of classical solutions These “wild solutions”, as they are called since the seminal works of DeLellis and Székelyhidi [10,11], are often highly unphysical—for instance there is a lack of uniqueness and the principle of conservation of energy can be violated. Dissipative solutions of incompressible Euler have been extensively studied in relation to the seminal Onsager conjecture [27]. Onsager’s conjecture was recently studied for incompressible Euler equations in bounded domains, cf [3] An overview of these results can be found in [12].
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