Journal ArticleDOI
Ordinary differential equations, transport theory and Sobolev spaces.
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TLDR
In this paper, the existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces were derived from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.Abstract:
We 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.read more
Citations
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Book
Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Book
Stochastic Equations in Infinite Dimensions
Giuseppe Da Prato,Jerzy Zabczyk +1 more
TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
Book
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Journal ArticleDOI
On the Cauchy problem for Boltzmann equations: global existence and weak stability
TL;DR: In this article, the authors studied the large-data Cauchy problem for Boltzmann equations with general collision kernels and proved that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution.
Book
Optimal Transport for Applied Mathematicians : Calculus of Variations, PDEs, and Modeling
TL;DR: In this paper, the primal and dual problems of one-dimensional problems are considered. But they do not consider the dual problems in L^1 and L^infinity theory.
References
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Journal ArticleDOI
On the Cauchy problem for Boltzmann equations: global existence and weak stability
TL;DR: In this article, the authors studied the large-data Cauchy problem for Boltzmann equations with general collision kernels and proved that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution.
Journal ArticleDOI
Global weak solutions of Vlasov‐Maxwell systems
TL;DR: In this paper, the Vlasov-Maxwell system in its classical and relativistic form was studied and the stability of solutions in weak topologies was proved and deduced from this stability result the global existence of a weak solution with large initial data.
Journal ArticleDOI
On the Fokker-Planck-Boltzmann equation
TL;DR: In this article, the authors considered the Boltzmann equation perturbed by Fokker-Planck type operator and introduced a notion of renormalized solution which enables them to establish stability results for sequences of solutions and global existence for the Cauchy problem with large data.