# 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

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.

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