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Regularity and compactness for the DiPerna-Lions flow. (English) Zbl 1144.35365
Benzoni-Gavage, Sylvie (ed.) et al., Hyperbolic problems. Theory, numerics and applications. Proceedings of the 11th international conference on hyperbolic problems, Ecole Normale Supérieure, Lyon, France, July 17–21, 2006. Berlin: Springer (ISBN 978-3-540-75711-5/hbk). 423-430 (2008).
From the text: When \(b: [0, T]\times\mathbb{R}^d\times \mathbb{R}^d\) is a bounded smooth vector field, the flow of \(b\) is the smooth map \(X: [0, T]\times\mathbb{R}^d\to \mathbb{R}^d\) such that \[ \begin{gathered} {dX\over dt} (t, x)= b(t, X(t,x)),\quad t\in [0,T],\\ X(0,x)= x.\end{gathered}\tag{1} \] Existence and uniqueness of the flow are guaranteed by the classical Cauchy-Lipschitz theorem. The study of (1) out of the smooth context is of great importance (for instance, in view of the possible applications to conservation laws or to the theory of the motion of fluids) and has been studied by several authors. What can be said about the well-posedness of (1) when \(b\) is only in some class of weak differentiability?
This question can be, in some sense, “relaxed” (and this relaxed problem can be solved, for example, in the Sobolev or BV framework): we look for a canonical selection principle, i.e., a strategy that “selects,” for a.e. initial datum \(x\), a solution \(X(\cdot, x)\) in such a way that this selection is stable with respect to smooth approximations of \(b\). This in some sense amounts to redefine our notion of solution: we add some conditions that select a “relevant” solution of our equation.
For the entire collection see [Zbl 1126.35003].

35F10 Initial value problems for linear first-order PDEs
35G10 Initial value problems for linear higher-order PDEs
35B65 Smoothness and regularity of solutions to PDEs