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Ordinary differential equations, transport theory and Sobolev spaces. (English) Zbl 0696.34049
Some new existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces are obtained. The outcoming point is the Cauchy-Lipschitz theorem (in global version) providing global solutions to ordinary differential equations (1) \(\dot X=b(X)\) for \(t\in {\mathbb{R}}\), \(X(0)=x\in {\mathbb{R}}^ N\) (the autonomous case is taken for simplicity). An extension to vector-fields b having bounded divergence and some Sobolev-type regularity is done. More precisely, it is shown hat if \(b\in W^{1,1}_{loc}({\mathbb{R}}^ N)\), div \(b\in L^{\infty}({\mathbb{R}}^ N)\), and \(b=b_ 1+b_ 2\), \(b_ 1\in L^ p({\mathbb{R}}^ N)\) for some \(1\leq p\leq \infty\), \(b_ 2(1+| x|)^{-1}\in L^{\infty}({\mathbb{R}}^ N),\) then there exists a unique “flow” \(X\in C({\mathbb{R}};L^ p_{loc}({\mathbb{R}}^ N))\) solving (1) and satisfying the group property \(X(t+s,.)=X(t,X(s,.))\) on \({\mathbb{R}}^ N\) for a.e. \(t,s\in {\mathbb{R}}\), and for some \(C_ 1\geq 0\) it holds \(e^{- C_ 1t}\lambda \leq \lambda_ 0X(t)\leq e^{C_ 1t}\lambda,\) for a.e. \(t\geq 0\), where \(\lambda\) is the Lebesgue measure on \({\mathbb{R}}^ N\) and \(\lambda_ 0X(t)\) denotes the image measure of \(\lambda\) by the map X(t) from \({\mathbb{R}}^ N\) into \({\mathbb{R}}^ N\), i.e. \(\int_{{\mathbb{R}}^ N}\phi d(\lambda_ 0X(t))=\int_{{\mathbb{R}}^ N}\phi (X(t,x))dx\) for all \(\phi\in {\mathcal D}({\mathbb{R}}^ N)\). To emphasize the sharpness of these results, two different types of counter-examples are presented. The first one provides for any \(p\in (1,\infty)\) a vector-field \(b\in C_ b({\mathbb{R}}^ 2)\cap W^{1,p}({\mathbb{R}}^ 2)\) with two (in fact infinitely many) distinct continuous flows, showing thus the relevance of the bound on div b. The second one gives an example of a vector-field \(b\in W^{s,1}_{loc}({\mathbb{R}}^ 2)\) for any \(s<1\) satisfying div b\(=0\) with two distinct measure-preserving \(L^ 1\)-flows, showing the sharpness of the \(W^{1,1}_{loc}\) regularity. All the obtained results can be deduced from the analysis of the associated PDE namely the following transport equation: \(\partial u/\partial t-b\nabla u=0\) in \((0,\infty)\times {\mathbb{R}}^ N\). Forthcoming applications are mentioned to kinetic Vlasov-type models, fluid mechanics and other fields.
Reviewer: I.Ginchev

34G20 Nonlinear differential equations in abstract spaces
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
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