Renormalized Solutions to the Vlasov Equation with Coefficients of Bounded Variation

By François Bouchut. 2001

125 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9a26f0cad0582c669a41ce5300b3a54c0c5366f3

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Continuity equations and ODE flows with non-smooth velocity* (2014, background): “the adaptation of the convolution kernel to the local behaviour of the vector field, has been used at various levels of generality in [43, 85, 56] (see also [52, 53] for related results independent of this technique), until the general result [10].”
Chapter 4 – Notes on Hyperbolic Systems of Conservation Laws and Transport Equations (2007, background): “The following lemma was first proved by Bouchut in [15] and it was the starting point of Ambrosio’s original proof of his commutator estimate (see [2]).”
Differential equations with singular fields (2010, methodology): “For Vlasov equations for instance, uniqueness under a BV condition was obtained earlier and more easily in [7] ; It was even derived under slighty less than BV regularity in [25].”
Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient (2004, methodology): “The idea we use here to solve the Cauchy problem (1.1)-(1.2) is based on a previous work of Bouchut and James [8], and makes use of the so-called duality solutions.”
Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor (2015, background): “Bouchut in [9] in the kinetic context (see also M.”
Transport equation and Cauchy problem for BV vector fields (2004, methodology): “In [ 12 ] F. Bouchut extends the theory to equations of Vlasov type using the special structure of the equation and the different roles played by the time variable t, the space variable x and the velocity variable ξ (see also [29] for an extension to velocity fields with point singularities).”
A review of the mean field limits for Vlasov equations (2014, background): “This was extended to the BVloc case in [26] for second order systems like (1.”
Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations (2003, background): “Also see Bouchut [2] for a more difficult case for u ∈ BV ; see also a similar statement, not necessarily stated as such, in Carrillo [7] and Eymard-Gallouet-Herbin [15].”
Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory (2002, background): “Apart from the study of the Boltzmann equation, renormalization and velocityaveraging lemmas have become popular tools for the study of various kinetic equations [191, 375, 318, 176, 229], ordinary differential equations with nonsmooth (Sobolev-regular) coefficients [193, 97], or the reformulation o”
Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields (2009, methodology): “2) and on an anisotropic regularization procedure (first introduced by Bouchut [36] in the context of the Vlasov equation with BV field and subsequently exploited by Colombini and Lerner [60] to treat the transport equation with conormal-BV vector field).”
Hamiltonian ODEs in the Wasserstein space of probability measures (2008, background): “Coming back to the finite-dimensional case X = R2d, the results in [5] (see also [6] for special classes of Hamiltonians) ensure a kind of “gener ic” uniqueness property, or uniqueness in the flow sense, in the same spirit of DiP erna–Lions’ theory [25] (see§6 of [5] for a precise formulation).”
Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, unknown): “1), see [8], [9], [12], [32].”