# Ordinary differential equations, transport theory and Sobolev spaces

@article{Diperna1989OrdinaryDE, title={Ordinary differential equations, transport theory and Sobolev spaces}, author={Ronald J. Diperna and Pierre-Louis Lions}, journal={Inventiones mathematicae}, year={1989}, volume={98}, pages={511-547} }

SummaryWe obtain some new existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces. These results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.

## 1,807 Citations

### Non-uniqueness for the Transport Equation with Sobolev Vector Fields

- MathematicsAnnals of PDE
- 2018

We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields in Sobolev spaces $$W^{1, p}$$W1,p.

### Hydrodynamics and Stochastic Differential Equation with Sobolev Coefficients

- Mathematics
- 2014

In this chapter, we will explain how the Brenier’s relaxed variational principle for Euler equation makes involved the ordinary differential equations with Sobolev coefficients and how the…

### Limit Theorems for Stochastic Differential Equations with Discontinuous Coefficients

- MathematicsSIAM J. Math. Anal.
- 2011

A transference principle for stochastic differential equations (SDEs) with discontinuous coefficients is proved and the well-posedness of SDEs and Fokker–Planck equations with irregular coefficients is established.

### Support Theorem for Stochastic Differential Equations with Sobolev Coefficients

- MathematicsPotential Analysis
- 2018

In this paper we prove a support theorem for stochastic differential equations with Sobolev coefficients in the framework of DiPerna-Lions theory.

### A note on transport equation in quasiconformally invariant spaces

- Mathematics
- 2015

Abstract In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel–Lizorkin spaces.

### On two-dimensional Hamiltonian transport equations with continuous coefficients

- Mathematics
- 2001

We consider two-dimensional autonomous flows with divergence free continuous coefficients. Under a generic assumption of regularity on the set of critical points, we give a proof of uniqueness for…

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