Consider the following initial value problem for a parabolic PDE :
$$\begin{cases}
\textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]
u(0,x) \,=\, v(x)
\end{cases}$$
where $A$ is a constant positive-definite matrix, $b$ is a smooth vector field, $v$ is a suitable initial datum with
$$v>0\,,\ \int_{\mathbb R^d} v(x)\,d x=1\,,\ \int_{\mathbb R^d} v(x)\,|\log v(x)|\,d x\,<\infty\,. $$
I am interested in the probability solution $u$ of the problem, i.e., $u>0\,,\ \int_{\mathbb R^d} u(t,x)\,d x=1$ for all $t>0\,$. I know that it exists, is unique, belongs to $C^\infty((0,\infty)\times\mathbb R^d)$, and moreover
$$ \int_{\mathbb R^d} |b(x)|^2\,u(t,x)\,d x\,<\infty$$
for all $t\geq0\,$. I would like to find conditions on the initial datum $v$ that guarantee *continuity of the entropy at $t=0$*, i.e.,

$$\int_{\mathbb R^d} u(t,x)\,\log u(t,x)\,d x \,\xrightarrow[t\to0]{}\, \int_{\mathbb R^d} v(x)\,\log v(x)\,d x \,.$$

I've found only standard results that guarantee $$\int_{\mathbb R^d} |u(t,x)-v(x)|\,d x \,\xrightarrow[t\to0]{}\, 0 $$ which seems to be a weaker condition.