Transport equation and Cauchy problem for BV vector fields

By Luigi Ambrosio. 2004

823 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/9d44b9a07ee78d93e89dc982667756beba9569d8

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient (2004, methodology): “As it is recalled by Ambrosio in [3], these results hold true in a very general context, but we wish to consider merely the case where the jump set of the function a is (locally) a hypersurface in R .”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, background): “Finally, the uniqueness question is a tricky one: if the potential D is λ-convex then this is easy, but considerations from the DiPerna-Lions theory ([40, 4]) suggest that uniqueness should be true under weaker assumptions; nevertheless this is not really proven.”
Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, background): “In the case ε N = 0, the general setting of [1] would require K ∈ BV .”
Mean Field Games Models—A Brief Survey (2013, methodology): “Consequently, to establish uniqueness, one needs to use a approach due to Ambrosio [5, 6], as explained in detail in [20].”
Second order differentiation formula on RCD*$(K,N)$ spaces (2018, methodology): “Switching the roles of ρ0 and ρ1 we get the analogous control for t ∈ [12 , 1], whence the claim (4.”
Well posedness of Lagrangian flows and continuity equations in metric measure spaces (2014, background): “In the “Lagrangian” part we define the notion of solution to the ODE and relate well-posedness of the continuity equation to existence and uniqueness of the flow (in the same spirit as [Ambrosio 2004; 2008], where the context was Euclidean).”
Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition (2013, background): “1), also in a non-regular framework, the reader is mostly addressed to [9] and [2].”
Differential equations with singular fields (2010, background): “As such b1(Φ(tδ(t, x), x))× ∂ttδ(t, x) ∈ [1, 2], and thanks to (2.”
Estimates and regularity results for the DiPerna-Lions flow (2008, background): “Ambrosio [1] extended this result to BV vector fields with divergence in L(1).”
Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients (2010, methodology): “Such theory has also been extended to the classical Wiener space by Ambrosio and Figalli [2] (see also Fang and Luo [12]).”
The Splitting Theorem in Non-Smooth Context (2013, neutral): “where the minimum is taken among all π ∈P(C([0, 1], X)) such that (e0)]π = μ and (e1)]π = ν.”
The Pontryagin Maximum Principle in the Wasserstein Space (2017, background): “ovide ﬁrst-order optimality conditions for inﬁnite-dimensional problems arising as mean-ﬁeld limits of ﬁnite dimensional systems. Even though very general existence results a la DiPernaLions-Ambrosio [4,20] are available for Cauchy problems of the form (9), they only deal with macroscopic quantities”
Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor (2015, background): “Ambrosio in [4] in the most general case.”
Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, methodology): “Notice that the outstanding results of [30, 2] (still in the deterministic case) are concerned only with uniqueness of weak solutions.”
Characterization of absolutely continuous curves in Wasserstein spaces (2006, methodology): “This interpretation of the solution of the continuity equation, closed to the technique of Young measures, turned out to be very useful in order to study uniqueness and stability properties, in the case of low regularity of the velocity vector eld vt (see [ Amb04 ] and also [ALS05]).”
A review of the mean field limits for Vlasov equations (2014, background): “3), and later in [2] for the general case.”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, result): “The precursors to this line of research can be traced back to the analogous results in [27, 4] for ordinary differential equations.”