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Maria Colombo

Publications and source records attributed to Maria Colombo.

At least 55 records · Page 3Linked to original sources
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.

math.AP↗
arXiv2019

Global regularity for the hyperdissipative Navier-Stokes equation below the critical order

We consider solutions of the Navier-Stokes equation with fractional dissipation of order $α\geq 1$. We show that for any divergence-free initial datum $u_0$ such that $||u_0||_{H^δ} \leq M$, where $M$ is arbitrarily large and $δ$ is arbitrarily small, there exists an explicit $ε=ε(M, δ)>0$ such that the Navier-Stokes equations with fractional order $α$ has a unique smooth solution for $α\in (\frac{5}{4}-ε, \frac{5}{4}]$. This is related to a new stability result on smooth solutions of the Navier-Stokes equations with fractional dissipation showing that the set of initial data and fractional orders giving rise to smooth solutions is open in $H^{5/4} \times (\frac 34, \frac{5}{4}]$.

math.AP↗
arXiv2019

Bounds on optimal transport maps onto log-concave measures

We consider strictly log-concave measures, whose bounds degenerate at infinity. We prove that the optimal transport map from the Gaussian onto such a measure is locally Lipschitz, and that the eigenvalues of its Jacobian have controlled growth at infinity.

math.AP↗
arXiv2019

Regularity results for rough solutions of the incompressible Euler equations via interpolation methods

Given any solution $u$ of the Euler equations which is assumed to have some regularity in space - in terms of Besov norms, natural in this context - we show by interpolation methods that it enjoys a corresponding regularity in time and that the associated pressure $p$ is twice as regular as $u$. This generalizes a recent result by Isett [16] (see also Colombo and De Rosa [8]), which covers the case of Hölder spaces.

math.AP↗
arXiv2019

On the well-posedness of branched transportation

We show in full generality the stability of optimal traffic paths in branched transport: namely we prove that any limit of optimal traffic paths is optimal as well. This solves an open problem in the field (cf. Open problem 1 in the book Optimal transportation networks, by Bernot, Caselles and Morel), which has been addressed up to now only under restrictive assumptions.

math.AP↗
arXiv2019

On the singular local limit for conservation laws with nonlocal fluxes

We give an answer to a question posed in [P. Amorim, R. Colombo, and A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.

math.AP↗
arXiv2019

On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws

We deal with the numerical investigation of the local limit of nonlocal conservation laws. Previous numerical experiments suggest convergence in the local limit. However, recent analytic results state that (i) in general convergence does not hold because one can exhibit counterexamples; (ii) convergence can be recovered provided viscosity is added to both the local and the nonlocal equations. Motivated by these analytic results, we investigate the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws. In particular, we show that the numerical viscosity of Lax-Friedrichs type schemes jeopardizes the reliability of the numerical scheme and erroneously detects convergence in cases where convergence is ruled out by analytic results. We also test Godunov type schemes, less affected by numerical viscosity, and show that in some cases they provide more reliable results.

math.AP↗
arXiv2019

Recent results on the singular local limit for nonlocal conservation laws

We provide an informal overview of recent developments concerning the singular local limit of nonlocal conservation laws. In particular, we discuss some counterexamples to convergence and we highlight the role of numerical viscosity in the numerical investigation of the nonlocal-to-local limit. We also state some open questions and describe recent related progress.

math.AP↗
arXiv2018

Regularity in time of Hölder solutions of Euler and hypodissipative Navier-Stokes equations

In this work we investigate some regularization properties of the incompressible Euler equations and of the fractional Navier-Stokes equations where the dissipative term is given by $(-Δ)^α$, for a suitable power $α\in (0,\frac{1}{2})$ (the only meaningful range for this result). Assuming that the solution $u \in L^\infty _t(C^θ_x)$ for some $θ\in (0,1)$ we prove that $u \in C^θ_{t,x}$, the pressure $p\in C^{2θ-}_{t,x}$ and the kinetic energy $e \in C^{\frac{2θ}{1-θ}}_t$. This result was obtained for the Euler equations in [Is13] with completely different arguments and we believe that our proof, based on a regularization and a commutator estimate, gives a simpler insight on the result.

math.AP↗
arXiv2018

Logarithmic estimates for continuity equations

The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper [Colombo, Crippa, and Spirito, Calc. Var. Partial Differential Equations 2015] concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coefficient. Second, we show how the ideas in [Colombo, Crippa, and Spirito, Calc. Var. Partial Differential Equations 2015] can be used to provide an alternative proof of the result in [Clop, Jiang, Mateu, and Orobitg, Calc. Var. Partial Differential Equations 2016], [Desjardins, Comm. Partial Diff. Eq. 1996], and [Mucha, J. Differential Equations 2010] where the usual requirement of boundedness of the divergence of the vector field has been relaxed to various settings of exponentially integrable functions.

math.AP↗
arXiv2018

On the asymptotic behavior of the solutions to parabolic variational inequalities

We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Lojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon (Ann. of Math. 118(3), 1983) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.

math.AP↗
arXiv2018

Stability for the mailing problem

We prove that optimal traffic plans for the mailing problem in $\mathbb{R}^d$ are stable with respect to variations of the given coupling, above the critical exponent $α=1-1/d$, thus solving an open problem stated in the book "Optimal transportation networks", by Bernot, Caselles and Morel. We apply our novel result to study some regularity properties of the minimizers of the mailing problem. In particular, we show that only finitely many connected components of an optimal traffic plan meet together at any branching point.

math.AP↗
arXiv2017

The singular set of minimal surfaces near polyhedral cones

We adapt the method of Simon [JDG '93] to prove a $C^{1,α}$-regularity theorem for minimal varifolds which resemble a cone $\bf{C}_0^2$ over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the singular set, we additionally establish $C^{1,α}$-regularity near the cone $\bf{C}_0^2 \times \mathbb R^m$. Combined with work of Allard [Ann. of Math. '72], Simon [JDG '93], Taylor [Ann. of Math. '76], and Naber-Valtorta [Ann. of Math. '17], our result implies a $C^{1,α}$-structure for the top three strata of minimizing clusters and size-minimizing currents, and a Lipschitz structure on the $(n-3)$-stratum.

math.DG↗
arXiv2017

Direct epiperimetric inequalities for the thin obstacle problem and applications

For the thin obstacle problem, we prove by a new direct method that in any dimension the Weiss' energies with frequency $\frac32$ and $2m$, for $m\in \mathbb N$, satisfy an epiperimetric inequality, in the latter case of logarithmic type. In particular, at difference from the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension $2$, we also prove the epiperimetric inequality at any free boundary point. As a first application, we improve the set of admissible frequencies for blow ups, previously known to be $λ\in \{\frac32\} \cup [2,\infty)$, and we classify the global $λ$-homogeneous minimizers, with $λ\in [\frac32,2+c]\cup\bigcup_{m\in \mathbb N}(2m-c_m^-,2m+c_m^+)$, showing as a consequence that the frequencies $\frac32$ and $2m$ are isolated. Secondly, we give a short and self-contained proof of the regularity of the free boundary previously obtained by Athanasopoulos-Caffarelli-Salsa (Amer. J. Math., 130(2) (2008), 485-498) for regular points and Garofalo-Petrosyan (Invent. Math., 177(2) (2009), 415-461) for singular points, by means of an epiperimetric inequality of logarithmic type which applies for the first time also at all singular points of thin-obstacle free boundaries. In particular we improve the $C^1$ regularity of the singular set with frequency $2m$ by an explicit logarithmic modulus of continuity.

math.AP↗
arXiv2017

Regularity for general functionals with double phase

We prove sharp regularity results for a general class of functionals of the type $$ w \mapsto \int F(x, w, Dw) \, dx\;, $$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$ w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;, $$ with $0<ν\leq b(\cdot)\leq L $. This changes its ellipticity rate according to the geometry of the level set $\{a(x)=0\}$ of the modulating coefficient $a(\cdot)$. We also present new methods and proofs, that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.

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