A unified treatment for ODEs under Osgood and Sobolev type conditions

Abstract

In this paper we present a unified treatment for the ordinary differential equations under the Osgood and Sobolev type conditions, following Crippa and de Lellis's direct method. More precisely, we prove the existence, uniqueness and regularity of the DiPerna-Lions flow generated by a vector field which is "almost everywhere Osgood continuous".

The Itô SDEs and Fokker–Planck equations with Osgood and Sobolev coefficients

“…Regarding the corresponding PDE, Le Bris and Lions studied in [30] the Fokker-Planck type equations with Sobolev coefficients; using Ambrosio's commutator estimate for BV vector fields, their results was slightly extended in [32] to the case where the drift coefficient has only BV regularity. Based on a representation formula for the solutions to Fokker-Planck equations (see [22, Theorem 2.6]), the uniqueness was established in [38,33] when the coefficients are bounded and have weak spatial regularity.This paper is a continuation of the work [31], where the authors propose a unified framework for ODEs (1.2) under the mixed Osgood and Sobolev conditions on the coefficient b. The spaces consist of these kind of functions have been studied intensively in the past two decades, even in the case where the underlying space is the general metric measure space (motivated by the pioneer work [24]); see [26] for the equivalence of different Sobolev spaces and [27] for the compactness of embeddings of Sobolev type.…”

“…This paper is a continuation of the work [31], where the authors propose a unified framework for ODEs (1.2) under the mixed Osgood and Sobolev conditions on the coefficient b. The spaces consist of these kind of functions have been studied intensively in the past two decades, even in the case where the underlying space is the general metric measure space (motivated by the pioneer work [24]); see [26] for the equivalence of different Sobolev spaces and [27] for the compactness of embeddings of Sobolev type.…”

“…The spaces consist of these kind of functions have been studied intensively in the past two decades, even in the case where the underlying space is the general metric measure space (motivated by the pioneer work [24]); see [26] for the equivalence of different Sobolev spaces and [27] for the compactness of embeddings of Sobolev type. Our purpose is to extend the main results in [31] to the case of the Itô SDE (1.1). Compared to equation (1.2), a big difference in the stochastic setting is that the estimate of the Radon-Nikodym density involves the gradient of the diffusion coefficient σ (see e.g.…”

Quantitative stability estimates for Fokker–Planck equations

“…, and C d,T is a positive constant depending only on d and T . Now we consider the Fokker-Planck equation (1.1) whose coefficients σ and b satisfy the following mixed Osgood and Sobolev type condition (see [20,Example 2.4] for an example of such a function):…”