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Elio Marconi

Publications and source records attributed to Elio Marconi.

29 records · Page 2Linked to original sources
arXiv2021

Initial-boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a "lagrangian formulation" of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the $BV$ regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a $BV$-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property.

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arXiv2021

Characterization of minimizers of Aviles-Giga functionals in special domains

We consider the singularly perturbed problem $F_\varepsilon (u,Ω):=\int_Ω\varepsilon |\nabla^2u| + \varepsilon^{-1}|1-|\nabla u|^2|^2$ on bounded domains $Ω\subset\mathbb{R}^2$. Under appropriate boundary conditions, we prove that if $Ω$ is an ellipse then the minimizers of $F_\varepsilon(\cdot,Ω)$ converge to the viscosity solution of the eikonal equation $|\nabla u|=1$ as $\varepsilon \to 0$.

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arXiv2021

Differentiability properties of the flow of 2d autonomous vector fields

We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation.

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arXiv2020

Rectifiability of entropy defect measures in a micromagnetics model

We study the fine properties of a class of weak solutions $u$ of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294-338). In particular we prove that the entropy defect measure associated to $u$ is concentrated on a 1-rectifiable set, which detects the jump-type discontinuities of $u$.

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arXiv2020

The rectifiability of the entropy defect measure for Burgers equation

We consider bounded weak solutions to the Burgers equation for which every entropy dissipation is representable by a measure and we prove that all these measures are concentrated on the graphs of countably many Lipschitz curves. The main tool is the Lagrangian representation, which is an extension of the method of characteristics to the non-smooth setting.

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arXiv2019

Local limit of nonlocal traffic models: convergence results and total variation blow-up

Consider a nonlocal conservation where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from $0$. We also exhibit a counter-example showing that, if the initial datum attains the value $0$, then there are severe obstructions to a convergence proof.

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arXiv2019

On the structure of weak solutions to scalar conservation laws with finite entropy production

We consider weak solutions with finite entropy production to the scalar conservation law \begin{equation} \partial_t u+\mathrm{div}_x F(u)=0 \quad \mbox{in }(0,T)\times \mathbb{R}^d. \end{equation} Building on the kinetic formulation we prove under suitable nonlinearity assumption on $f$ that the set of non Lebesgue points of $u$ has Hausdorff dimension at most $d$. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure.

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arXiv2019

Structure and regularity of solutions to 1d scalar conservation laws

We consider bounded entropy solutions to the scalar conservation law in one space dimension: \begin{equation*} u_t+f(u)_x=0. \end{equation*} We quantify the regularizing effect of the non linearity of the flux $f$ on the solution $u$ in terms of spaces of functions with bounded generalized variation.

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arXiv2017

Regularity estimates for scalar conservation laws in one space dimension

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 $f$ has on the entropy solution. More precisely, if the set $\{w:f''(w)\ne 0\}$ is dense, the regularity of the solution can be expressed in terms of $\mathrm{BV}^Φ$ 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 $f'\circ u(t)\in \mathrm{BV}_{\mathrm{loc}}(\mathrm{R})$ for every $t>0$ and that this can be improved to $\mathrm{SBV}_{\mathrm{loc}}(\mathbb{R})$ 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.

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arXiv2016

On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension

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.

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