| Abstract: |
| We consider the linear transport equation $\partial_t \rho + u \cdot \nabla \rho = 0$, with unknown density $\rho$ and given divergence-free vector field $u$, together with a given initial datum $\rho(x,0) = \bar \rho(x)$. A celebrated result by DiPerna and Lions (1989) shows that if
\begin{equation*}
\bar \rho \in L^p_x, \quad u \in L^1_t W^{1,\tilde p}_x
\end{equation*}
and
\begin{equation*}
\frac{1}{p} + \frac{1}{\tilde p} \leq 1,
\end{equation*}
then the Cauchy problem admits a unique weak solution $\rho \in L^\infty_t L^p_x$. We show that the above condition is optimal: if
\begin{equation*}
\frac{1}{p} + \frac{1}{\tilde p} > 1,
\end{equation*}
then there are divergence-free vector fields $u \in C_t W^{1, \tilde p}$ and initial data $\bar \rho \in L^p_x$ for which, despite the linearity of the equation, more than one weak solution in $C_t L^p_x$ exists. The result applies also to the transport-diffusion equation $\partial_t \rho + u \cdot \nabla \rho = \Delta \rho$. |
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