Calculus of Variations and Geometric Measure Theory

G. Crippa - C. De Lellis

Oscillatory solutions to transport equations

created by crippa on 28 Feb 2005
modified by delellis on 04 May 2011


Published Paper

Inserted: 28 feb 2005
Last Updated: 4 may 2011

Journal: Indiana Univ. Math. J.
Volume: 55
Number: 1
Pages: 1-13
Year: 2006


\documentclass{article} \begin{document} We show that there is no topological vector space $X\subset L^\infty\cap L^1_{\rm loc} (\mathbf{R}^n \times \mathbf{R}^n)$ which embeds compactly in $L^1_{\rm loc}$, contains $BV_{\rm loc}\cap L^\infty$ and enjoys the following closure property: If $f\in X^n (\mathbf{R} \times \mathbf{R}^n)$ has bounded divergence and $u_0\in X (\mathbf{R}^n)$, then there exists $u\in X (\mathbf{R} \times \mathbf{R}^n)$ which solves $$ \left\{ \begin{array}{l} \partialt u + {\rm div}\, (u f)\;=\; 0 u (0, \cdot) \;=\; u0 \end{array}\right. $$ in the sense of distributions. $X (\mathbf{R}^n)$ is defined as the set of functions $u_0\in L^\infty (\mathbf{R}^n)$ such that $U(t,x):= u_0 (x)$ belongs to $X (\mathbf{R}\times \mathbf{R}^n)$. Our proof relies on an example of N. Depauw showing an ill--posed transport equation whose vector field is ``almost $BV$''.

For the most updated version and eventual errata see the page

http:/www.math.uzh.chindex.php?id=publikationen&key1=493 \end{document}

Keywords: Transport equation, Hyperbolic systems of conservation laws