IUMJ

Title: Oscillatory solutions to transport equations

Authors: Camillo De Lellis and GianLuca Crippa

Issue: Volume 55 (2006), Issue 1, 1-14

Abstract:

We show that there is no topological vector space $X \subset L^{\infty} \cap L^1_{\mathrm{loc}} (\mathbb{R} \times \mathbb{R}^n)$ that embeds compactly in $L^1_{\mathrm{loc}}$, contains $BV_{\mathrm{loc}} \cap L^{\infty}$, and enjoys the following closure property: If $f \in X^n (\mathbb{R} \times \mathbb{R}^n)$ has bounded divergence and $u_0 \in X (\mathbb{R}^n)$, then there exists $u \in X (\mathbb{R} \times \mathbb{R}^n)$ which solves \[\begin{cases}\partial_t u + \mbox{\upshape div} (uf) = 0\\ u (0, \cdot) = u_0\end{cases}\] in the sense of distributions. $X(\mathbb{R}^n)$ is defined as the set of functions $u_0 \in L^{\infty} (\mathbb{R}^n)$ such that $\tilde{u} (t,x) := u_0(x)$ belongs to $X(\mathbb{R} \times \mathbb{R}^n)$. Our proof relies on an example of N. Depauw showing an ill--posed transport equation whose vector field is "almost $BV$".