%0 Report
%D 2017
%T A Lagrangian approach for scalar multi-d conservation laws
%A Stefano Bianchini
%A Paolo Bonicatto
%A Elio Marconi
%G en
%U http://preprints.sissa.it/handle/1963/35290
%1 35596
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-08-08T08:57:31Z
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%0 Report
%D 2017
%T Regularity estimates for scalar conservation laws in one space dimension
%A Elio Marconi
%X In this paper we deal with the regularizing effect that, in a scalar conservation laws in one space dimension, the nonlinearity of the flux function ƒ has on the entropy solution. More precisely, if the set ⟨w : ƒ " (w) ≠ 0⟩ is dense, the regularity of the solution can be expressed in terms of BV Ф spaces, where Ф depends on the nonlinearity of ƒ. If moreover the set ⟨w : ƒ " (w) = 0⟩ is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that ƒ' 0 u(t) ∈ BVloc (R) for every t > 0 and that this can be improved to SBVloc (R) regularity except an at most countable set of singular times. Finally we present some examples that shows the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.
%G en
%U http://preprints.sissa.it/handle/1963/35291
%1 35597
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-08-16T06:45:14Z
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%0 Report
%D 2016
%T On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension
%A Stefano Bianchini
%A Elio Marconi
%X We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a $C^0$-sense up to the degeneracy due to the segments where $f''=0$. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.
%I SISSA
%G en
%U http://urania.sissa.it/xmlui/handle/1963/35209
%1 35508
%2 Mathematics
%# MAT/05
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2016-09-06T09:18:03Z
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