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.
Regularity estimates for scalar conservation laws in one space dimension
Marconi E.
2018
Abstract
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.Pubblicazioni consigliate
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