sobolev spaces – Can I apply the weak formulation of a parabolic PDE to a non-test function?

Let $rho:mathbb R^dtimes(0,infty)to(0,infty)$ such that $int rho_t(x),dx=1$ for all $tgeq0,$, $rho$ is Holder-continuous (in both variables) and $rho_tin W^{1,1}(mathbb R^d)$ for a.e. $t>0$.

Let $A$ be a positive definite symmetric matrix and $bin C^2(mathbb R^d;mathbb R^d)$. Suppose that $rho$ satisfies the following weak version of a parabolic PDE:
begin{align} tag{1} & int_{mathbb R^d} varphi_t(x),rho_t(x),dx ,-, int_{mathbb R^d} varphi_0(x),rho_0(x),d x ;=\(5pt) &=, -,lim_{epsilonto0}int_epsilon^t!!int_{mathbb R^d} Big(partial_svarphi_s(x),rho_s(x) ,+, A,nabla!varphi_s(x)cdotnabla!rho_s(x) ,+, b(x)cdotnabla!varphi_s(x),rho_s(x)Big),d x,d s end{align}
for all $t>0$, for all $varphiin C(mathbb R^d!times!(0,infty))$ such that, for example, $varphi_s(x)=zeta(s),psi(x)$ with $zetain C^infty(0,infty)$ and $psiin C^infty_c(mathbb R^d),$.

My question: Is the identity (1) still true when $varphi$ is replaced by $logrho$ ? Can we prove it by some approximation argument?

I know that:
$$ int_0^t!!int_{mathbb R^d} |b(x)|^2,rho_s(x) ,d x,d s <infty ;;qquadqquadint_0^t!!int_{mathbb R^d} frac{|nablarho_s(x)|^2}{rho_s(x)} ,d x, ds <infty ;.$$

I also know from theory that there exists $partial_srho$ bounded in the dual space of $mathbb H_0^{1,1}(Btimes I)$ (*see note at the end) for every $Btimes I$ compactly contained in $mathbb R^d!times!(0,infty)$ and $||rho_epsilon-rho_0||_{L^1(mathbb R^d)}to0 $ as $epsilonto0,$.


Notice that taking $varphi=logrho$ in (1) we obtain the terms $|logrho_t|,rho_t,$, $|A,nablalogrho_scdotnablarho_s|leq||A||,frac{|nablarho_s|^2}{rho_s},$, $|bcdotnablalogrho_s|,rho_sleq|b|,|nablarho_s|,$ which all belongs to $L^1(mathbb R^d)$ (for a.e. $s>0$).
Regarding the time-derivative term on the r.h.s. of (1), I expect that:
$$ tag{2} lim_{epsilonto0}int_epsilon^t!!int_{mathbb R^d} partial_srho_s(x),d x,d s ;=, 0 ;$$

even if I’m not sure how to prove (2): I would take a ball $B_R$ of radius $R$ in $mathbb R^d$ and say
$$ int_epsilon^t!!int_{B_R} partial_srho_s(x),d x,d s ,=,
int_{B_R} int_epsilon^tpartial_srho_s(x),d s,d x ,=,
int_{B_R} big(rho_t(x) – rho_epsilon(x)big) ,d x $$

which in absolute value is bounded by
$$ int_{B_R} rho_t(x), dx ,-, int_{B_R} rho_0(x) ,d x ,+, ||rho_epsilon-rho_0||_{L^1(mathbb R^d)} ,xrightarrow(substack{epsilonto0,,Rtoinfty}){}, 1-1+0 = 0 $$
but I’m not sure this is equivalent to (2).


(*) $mathbb H_0^{1,1}(Btimes I)$ denotes the space of functions $v:Btimes Itomathbb R$ such that $v_sin W^{1,1}_0(B)$ for a.e. $sin I$ and $int_Iint_B(|v_s(x)|^p+|nabla v_s(x)|^p),d x,d s<infty$.