Ordinary differential equations, transport theory and Sobolev spaces

  title={Ordinary differential equations, transport theory and Sobolev spaces},
  author={Ronald J. Diperna and Pierre-Louis Lions},
  journal={Inventiones mathematicae},
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. 

Non-uniqueness for the Transport Equation with Sobolev Vector Fields

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.

Compactness for nonlinear continuity equations

Hydrodynamics and Stochastic Differential Equation with Sobolev Coefficients

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

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.

Differential equations with singular fields

Support Theorem for Stochastic Differential Equations with Sobolev Coefficients

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

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

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



Global weak solutions of Vlasov‐Maxwell systems

We study here the Vlasov-Maxwell system in its classical and relativistic form. We prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a

On the Cauchy problem for Boltzmann equations: global existence and weak stability

We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge

On the Fokker-Planck-Boltzmann equation

We consider the Boltzmann equation perturbed by Fokker-Planck type operator. To overcome the lack of strong a priori estimates and to define a meaningful collision operator, we introduce a notion of

Uniqueness of flow solutions of differential equations

In preparation , see also C

  • R . Acad . Sci . Paris
  • 1988

In preparation, see also in S6minaire EDP, Ecole Polytechnique

  • 1988