Basically my question is why, when we go from classical solutions to weak solutions, we have to use energy inequalities instead of equalities. More preciscely, consider the Navier-Stokes equations \begin{equation} \partial _t \rho + \text{div}(\rho u) = 0 \ \ \ \text{in } I \times \Omega \end{equation} \begin{equation} \partial _t (\rho u) + \text{div}(\rho uRead more