# Energy equalities and estimates for weak solutions

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

$$$$\partial _t \rho + \text{div}(\rho u) = 0 \ \ \ \text{in } I \times \Omega$$$$

$$$$\partial _t (\rho u) + \text{div}(\rho u \otimes u) + \nabla p(\rho) - \text{div}S(\nabla u) = \rho f \ \ \ \text{in } I\times\Omega$$$$

with

$$$$u = 0 \ \ \ \text{on } \partial \Omega,$$$$

$$$$\text{div}S(\nabla u) = \mu \Delta u + (\lambda + \mu)\nabla \text{div}u$$$$

For classical solutions, the momentum equation is multiplied by $$u$$ and by the continuity equation we obtain (using integration by parts)

$$$$\int _{\Omega} \left(\frac{1}{2}\rho |u|^2 + P(\rho)\right)(t)dx + \int_I\int_{\Omega} \mu |\nabla u|^2 + (\lambda + \mu)|\text{div}u|^2dxdt = \int _{\Omega} \left(\frac{1}{2}\rho |u|^2 + P(\rho)\right)(0)dx + \int _I \int _\Omega \rho f \cdot u dxdt \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \, (*)$$$$

with $$P(\rho) = \rho \int_1^\rho \frac{p(z)}{z^2}dz.$$ Now for solutions of the weak formulation of the momentum equation, i.e.

$$$$\int_I \int _\Omega \rho u \partial _t \phi + \rho u \otimes u : \nabla \phi + p(\rho)\text{div}\phi dxdt = \int_I \int_\Omega \mu \nabla u :\nabla \phi + (\lambda + \mu) \text{div}u \text{div}\phi - \rho f \phi dxdt$$$$

for all $$\phi \in D(I\times \Omega)$$ it says we have $$(*)$$ with $$=$$ replaced by $$\leq$$ but I don't see why. It says, that the equality is lost because of the weak lower semicontinuity in the term

$$$$\int_I\int_{\Omega} \mu |\nabla u|^2 + (\lambda + \mu)|\text{div}u|^2dxdt,$$$$

so probably we have to use an approximation $$(u_n)_n\subset D(I\times \Omega)$$ to the weak solution $$u$$ as a test function and then pass to the limit, but this would only give us

$$$$\int_I \int_\Omega \mu \nabla u :\nabla u_n + (\lambda + \mu) \text{div}u \text{div}u_n$$$$

and to apply weak lower semicontinuity we would rather need something like

$$$$\lim_{n \rightarrow \infty}\int_I \int_\Omega \mu \nabla u_n :\nabla u_n + (\lambda + \mu) \text{div}u_n \text{div}u_n \geq \int_I\int_{\Omega} \mu |\nabla u|^2 + (\lambda + \mu)|\text{div}u|^2dxdt.$$$$

Thank you in advance for any hint on how to see this.