he paper establishes a solution to the Monge problem in ℝ for a possibly asymmetric norm cost function and absolutely continuous initial measures, under the assumption that the unit ball is strictly convex-but not necessarily differentiable nor uniformly convex. The proof follows the strategy initially proposed by Sudakov in 1976, found to be incomplete in 2000; the missing step is fixed in the above case adapting a disintegration technique introduced for a variational problem. By strict convexity, mass moves along rays, and we also investigate the divergence of the vector field of rays
We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, that means transported by a flow of the associated ordinary differential equation. We work in a framework of lack of local integrability of the solution, in which the classical DiPerna-Lions theory of uniqueness and Lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator. We introduce a general principle to prove that a solution is Lagrangian: we rely on a disintegration along the unique flow and on a new directional Lipschitz extension lemma, used to construct a large class of test functions in the Lagrangian distributional formulation of the continuity equation
Schaeffer's regularity theorem for scalar conservation laws can be loosely speaking formulated as follows. Assume that the flux is uniformly convex, then for a generic smooth initial datum the admissible solution is smooth outside a locally finite number of curves in the (t, x) plane. Here the term "generic" is to be interpreted in a suitable sense, related to the Baire Category Theorem. Whereas other regularity results valid for scalar conservation laws with convex fluxes have been extended to systems of conservation laws with genuinely nonlinear characteristic fields, in this work we exhibit an explicit counterexample which rules out the possibility of extending Schaeffer's Theorem. The analysis relies on careful interaction estimates and uses fine properties of the wave front-tracking approximation.
Abstract. We prove that if t → u(t) ∈ BV(R) is the entropy solution to a N × N strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields ut + f (u)x = 0, then up to a countable set of times {tn} n∈N the function u(t) is in SBV, i.e. its distributional derivative ux is a measure with no Cantorian part.The proof is based on the decomposition of ux(t) into waves belonging to the characteristic familiesand the balance of the continuous/jump part of the measures v i in regions bounded by characteristics.To this aim, a new interaction measure µ i,jump is introduced, controlling the creation of atoms in the measure v i (t).The main argument of the proof is that for all t where the Cantorian part of v i is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure µ i,jump is positive.
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