On two-dimensional hamiltonian transport equations with Llocp coefficients

By M. Hauray. 2003

52 citations recorded by the source corpus.

Abstract We consider two-dimensional autonomous divergence free vector-fields in L loc 2 . Under a condition on direction of the flow and on the set of critical points, we prove the existence and uniqueness of a stable a.e. flow and of renormalized solutions of the associated transport equation.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/a283567d7e9137fe31702dc9823e138548a09956

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In Applied Mathematics

Citations in the corpus

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).”
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”
Differential equations with singular fields (2010, unknown): “In low dimension this structure is especially useful as illustrated in [9] where uniqueness is obtained for Vlasov equations in dimension 2 of phase space for continuous force terms; In [24] for a general hamiltonian (still in 2d) with L coefficients ; And in [14] for a continuous b in 2d but only a”
Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient (2004, unknown): “In the specific case of two-dimensional hamiltonian transport equations, (1.1) has been solved with continuous (non differentiable) coefficients by Bouchut and Desvillettes [7], and more recently with Lploc coefficients by Hauray [ 18 ].”
The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems (2011, 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”