Log-Lipschitz regularity and uniqueness of the flow for a field in \((W_{\text{loc}}^{n/p+1,p}(\mathbb{R}^n))^n\)
The author is interested in the uniqueness property for solutions to the initial value problem \(\dot x= b(t,x)\), \(t> 0\), \(x(0)= x_0\), with \(x= x(t)\in\mathbb{R}^n\). There are two well-known results. The theorem of Peano yields the existence of a solution, the theorem of Picard-Lindelöf implies besides existence a uniqueness result, too. One possibility to weaken the Lipschitz condition on \(b\) with respect to \(x\) is to consider \(b\) in Sobolev spaces. The author explains the situation for \(b\in (W^{s,p}_{\text{loc}}(\mathbb{R}^n))^n\) and even for \(b\in L^1(0, T;(W^{s,p}_{\text{loc}}(\mathbb{R}^n))^n)\). The critical case for uniqueness is \(s={n\over p}+1\), \(p\geq 1\). A sketch of the proof of uniqueness is given, the cases \(p=2\) and \(p=n\), \(n\geq 3\) are proved.
scientific article; zbMATH DE number 1777357