Ordinary differential equations, transport theory and Sobolev spaces.

*(English)*Zbl 0696.34049Some 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

##### MSC:

34G20 | Nonlinear differential equations in abstract spaces |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

##### MathOverflow Questions:

Lions/diPerna type commutator estimates for differential operator in Fokker-Planck type equation##### References:

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[3] | DiPerna, R.J., Lions, P.L.: On Fokker-Planck-Boltzmann equations. Commun. Math. Phys. (1989) · Zbl 0698.45010 |

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[6] | DiPerna, R.J., Lions, P.L.: In preparation, see also in Séminaire EDP, Ecole Polytechnique, 1988-89, Palaiseau |

[7] | DiPerna, R.J., Lions, P.L.: In preparation, see also in Séminaire EDP, Ecole Polytechnique, 1988-89, Palaiseau |

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