On two-dimensional Hamiltonian transport equations with continuous coefficients

By François Bouchut, Laurent Desvillettes. Differential and Integral Equations, 2001

48 citations recorded by the source corpus.

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 the characteristics and for the transport equation in the framework of distributions.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2785b3d2787c2b849e7e776fa2e822c9e5bff70c

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Citations in the corpus

Uniqueness of continuous solutions for BV vector fields (2002, methodology): “fluid mechanics were recently given by B. Desjardins [ De1], [De2], [De3], F. Bouchut and L. Desvillettes [ BD ], and Bouchut and F. James [BJ].”
On two-dimensional hamiltonian transport equations with Llocp coefficients (2003, background): “Bouchut in [3] in the case when b is continuous.”
Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor (2015, background): “This can only be improved with specific additional structure, such as provided by low dimension, see [3, 11, 23, 24, 41], Hamiltonian properties [19, 44], or as a singular integral [10].”
A uniqueness result for the continuity equation in two dimensions (2014, background): “3 When b is a divergence-free autonomous vector field on the plane (d = 2), uniqueness has been proved with the BV or Sobolev regularity of b replaced by various assumptions on the direction of b ([7, 17], see also [10, 8] for the bounded divergence case).”
On the Loss of Continuity for Super-Critical Drift-Diffusion Equations (2012, background): “We give another example of this phenomenon in Section 5, where we prove that, in two dimensions, non-vanishing continuous divergence-free vector fields (not necessarily Lipschitz or even Hölder) have unique trajectories (see also [4]).”
Renormalized Solutions to the Vlasov Equation with Coefficients of Bounded Variation (2001, background): “A special case with only continuous coefficient a is also treated in [ 4 ].”
Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields (2009, result): “Previous results by Bouchut and Desvillettes [38], Hauray [100] and Colombini and Lerner ([58] and [60]) show that uniqueness holds for the transport equation relative to an autonomous bounded divergence-free vector field, under the following additional condition on the local direction of the vector”