Cfrgtkky

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 u otimes u) + nabla p(rho) - text{div}S(nabla u) = rho f text{in } ItimesOmega
end{equation}

with

begin{equation}
u = 0 text{on } partial Omega,
end{equation}

begin{equation}
text{div}S(nabla u) = mu Delta u + (lambda + mu)nabla text{div}u
end{equation}

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

begin{equation}
int _{Omega} left(frac{1}{2}rho |u|^2 + P(rho)right)(t)dx + int_Iint_{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 , (*)
end{equation}

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.

begin{equation}
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
end{equation}

for all $phi in D(Itimes Omega)$ 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

begin{equation}
int_Iint_{Omega} mu |nabla u|^2 + (lambda + mu)|text{div}u|^2dxdt,
end{equation}

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

begin{equation}
int_I int_Omega mu nabla u :nabla u_n + (lambda + mu) text{div}u text{div}u_n
end{equation}

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

begin{equation}
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_Iint_{Omega} mu |nabla u|^2 + (lambda + mu)|text{div}u|^2dxdt.
end{equation}

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