The normal euler methods for integrating ordinary differential equations (odes) have a major downside for the molecular dynamics simulations we are doing They do not conserve the total energy of the system very well. In this paper, we study the energy conservation of weak solutions to these systems in lebesgue space $l^ {3} (\mathbb {t}^ {3})$ and homogenous sobolev space $\dot {w}^ {1,3} (\mathbb {t}^ {3})$. The result applies to the system of quantum hydrodynamics. As remarked before this method does not require any treatment of the pressure It is an interesting open problem whether energy conservation in a general bounded domain can be proved without involving the pressure.
In this paper, we are concerned with the minimal regularity of weak solutions implying the law of balance for both energy and helicity in the incompressible euler equations. When passing to the limit as the mollification parameter tends to zero, the commutator estimates are required for treating the nonlinear terms.
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