We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function f has on the entropy solution. More precisely, if the set w : f″(w)≠0 is dense, the regularity of the solution can be expressed in terms of BVΦ spaces, where Φ depends on the nonlinearity of f. If moreover the set w : f″(w) = 0 is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that f′∘ u(t) ∈BV loc(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 show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.

%B Journal of Hyperbolic Differential Equations %V 15 %P 623-691 %G eng %U https://doi.org/10.1142/S0219891618500200 %R 10.1142/S0219891618500200 %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 No. of bitstreams: 1 main.pdf: 427188 bytes, checksum: 8ab383e6ab2a6dcbf06a22d007db5dda (MD5) %0 Journal Article %J Contemporary Mathematics. Fundamental Directions %D 2017 %T Lagrangian representations for linear and nonlinear transport %A Stefano Bianchini %A Paolo Bonicatto %A Elio Marconi %XIn this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

%B Contemporary Mathematics. Fundamental Directions %I Peoples' Friendship University of Russia %V 63 %P 418–436 %G eng %U http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=cmfd&paperid=327&option_lang=eng %R 10.22363/2413-3639-2017-63-3-418-436 %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 No. of bitstreams: 1 Regularity_preprint.pdf: 603969 bytes, checksum: 29e1eefa4da3dc5272a6e9c6119108a5 (MD5) %0 Journal Article %J Discrete & Continuous Dynamical Systems - S %D 2016 %T On the concentration of entropy for scalar conservation laws %A Stefano Bianchini %A Elio Marconi %K concentration %K Conservation laws %K entropy solutions %K Lagrangian representation %K shocks %XWe prove that the entropy for an $L^∞$-solution to a scalar conservation laws with continuous initial data is concentrated on a countably $1$-rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution.

%B Discrete & Continuous Dynamical Systems - S %V 9 %P 73 %G eng %U http://aimsciences.org//article/id/ce4eb91e-9553-4e8d-8c4c-868f07a315ae %R 10.3934/dcdss.2016.9.73 %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 %XWe 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 No. of bitstreams: 1 1608.02811v1.pdf: 591028 bytes, checksum: 069b218b01f350df6db7904e245ef701 (MD5)