Publications (75)
Non-classical solutions of the -Laplace equation
Maria Colombo, Riccardo Tione
In this paper we answer Iwaniec and Sbordone's conjecture \cite{IB94} concerning very weak solutions to the -Laplace equation. Namely, on one hand we show that distributional so…
Existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters
Maria Colombo, Francesco Maggi
The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.
Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow
Maria Colombo, Massimo Gobbino
We prove that solutions of a mildly regularized Perona-Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-…
Lipschitz Changes of Variables between Perturbations of Log-concave Measures
Maria Colombo, Alessio Figalli, Yash Jhaveri
Extending a result of Caffarelli, we provide global Lipschitz changes of variables between compactly supported perturbations of log-concave measures. The result is based on a combi…
Non-uniqueness of Leray solutions to the hypodissipative Navier-Stokes equations in two dimensions
Dallas Albritton, Maria Colombo
We exhibit non-unique Leray solutions of the forced Navier-Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in \cite{albritton2021non}, the…
Rigidity of equality cases in Steiner's perimeter inequality
Filippo Cagnetti, Maria Colombo, Guido De Philippis +1
Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equalit…
An overview on the local limit of non-local conservation laws, and a new proof of a compactness estimate
Maria Colombo, Gianluca Crippa, Elio Marconi +1
Consider a non-local (i.e., involving a convolution term) conservation law: when the convolution term converges to a Dirac delta, in the limit we formally recover a classical (or "…
A convex integration scheme for the continuity equation past the Sobolev embedding threshold
Maria Colombo, Roberto Colombo, Anuj Kumar
We introduce a convex integration scheme for the continuity equation in the context of the Di Perna-Lions theory that allows to build incompressible vector fields in $C_{t}W^{1,p}_…
First order expansion in the semiclassical limit of the Levy-Lieb functional
Maria Colombo, Simone Di Marino, Federico Stra
We prove the conjectured first order expansion of the Levy-Lieb functional in the semiclassical limit, arising from Density Functional Theory (DFT). This is accomplished by interpr…
Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space
Elia Brué, Maria Colombo
For the two dimensional Euler equations, a classical result by Yudovich states that solutions are unique in the class of bounded vorticity; it is a celebrated open problem whether…
Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies
Lénaïc Chizat, Maria Colombo, Roberto Colombo +1
We study the quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancy (KMD) (also known as Maximum Mean Discrepancy (MMD)) functionals. Our setting covers…
Typicality results for weak solutions of the incompressible Navier--Stokes equations
Maria Colombo, Luigi De Rosa, Massimo Sorella
In this work we show that, in the class of distributional solutions of the incompressible Navier-Stokes system, the ones which are smooth in som…
Improved Hausdorff dimension estimate of the singular set of the supercritical surface quasigeostrophic equation
Maria Colombo, Silja Haffter
We prove that the spacetime singular set of any suitable Leray-Hopf solution of the surface quasigeostrophic equation with fractional dissipation of order has…
Convergence of drift-diffusion PDEs arising as Wasserstein gradient flows of convex functions
Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real
We study the quantitative convergence of drift-diffusion PDEs that arise as Wasserstein gradient flows of linearly convex functions over the space of probability measures on ${\mat…
Quantitative estimate on singularities in isoperimetric clusters
Maria Colombo, Luca Spolaor
We prove a quantitative estimate on the number of certain singularities in almost minimizing clusters. In particular, we consider the singular points belonging to the lowest stratu…
Optimality of integrability estimates for advection-diffusion equations
Stefano Bianchini, Maria Colombo, Gianluca Crippa +1
We discuss integrability estimates for the solution of the advection-diffusion equation , where the velocity field $b \in L^r_t L^…
A global existence result for the semigeostrophic equations in three dimensional convex domains
Luigi Ambrosio, Maria Colombo, Guido De Philippis +1
Exploiting recent regularity estimates for the Monge-Ampère equation, under some suitable assumptions on the initial data we prove global-in-time existence of Eulerian distributio…
A logarithmic epiperimetric inequality for the obstacle problem
Maria Colombo, Luca Spolaor, Bozhidar Velichkov
For the general obstacle problem, we prove by direct methods an epiperimetric inequality at regular and singular points, thus answering a question of Weiss (Invent. Math., 138 (199…
On the well-posedness of branched transportation
Maria Colombo, Antonio De Rosa, Andrea Marchese
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…
On multidimensional nonlocal conservation laws with BV kernels
Maria Colombo, Gianluca Crippa, Laura V. Spinolo
We establish local-in-time existence and uniqueness results for nonlocal conservation laws in several space dimensions under weak (that is, Sobolev or BV) differentiability assumpt…
On the asymptotic behavior of the solutions to parabolic variational inequalities
Maria Colombo, Luca Spolaor, Bozhidar Velichkov
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 ne…
On the Lagrangian structure of transport equations: the Vlasov-Poisson system
Luigi Ambrosio, Maria Colombo, Alessio Figalli
The Vlasov-Poisson system is a classical model in physics used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of…
The generalized Caffarelli-Kohn-Nirenberg Theorem for the hyperdissipative Navier-Stokes system
Maria Colombo, Camillo De Lellis, Annalisa Massaccesi
We introduce a notion of suitable weak solution of the hyperdissipative Navier-Stokes equations and we achieve a corresponding extension of the regularity theory of Caffarelli-Kohn…
On the singular local limit for conservation laws with nonlocal fluxes
Maria Colombo, Gianluca Crippa, Laura V. Spinolo
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. Con…
Regularity in time of Hölder solutions of Euler and hypodissipative Navier-Stokes equations
Maria Colombo, Luigi De Rosa
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 b…
Regularity results for rough solutions of the incompressible Euler equations via interpolation methods
Maria Colombo, Luigi De Rosa, Luigi Forcella
Given any solution 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 method…
Minimizing movements along a sequence of functionals and curves of maximal slope
Andrea Braides, Maria Colombo, Massimo Gobbino +1
We prove that a general condition introduced by Colombo and Gobbino to study limits of curves of maximal slope allows also to characterize minimizing movements along a sequence of…
Counterexamples in multimarginal optimal transport with Coulomb cost and spherically symmetric data
Maria Colombo, Federico Stra
We disprove a conjecture in Density Functional Theory, relative to multimarginal optimal transport maps with Coulomb cost. We also provide examples of maps satisfying optimality co…
Recent results on the singular local limit for nonlocal conservation laws
Maria Colombo, Gianluca Crippa, Marie Graff +1
We provide an informal overview of recent developments concerning the singular local limit of nonlocal conservation laws. In particular, we discuss some counterexamples to converge…
Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope
Luigi Ambrosio, Maria Colombo, Simone Di Marino
In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, , in metric measure spaces $(X,\sfd,\mm)$. In the fina…
Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem
Maria Colombo, Luca Spolaor, Bozhidar Velichkov
We answer a question left open in [Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184] and [Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784], by proving that the blow-up limit of minimi…
A transmission problem for -Laplacian
Maria Colombo, Sunghan Kim, Henrik Shahgholian
In this paper, we consider a double-phase problem characterised by a transmission that takes place across the zero level "surface" of the minimiser of the functional $$ J(v,Ω) = \…
Bounds on optimal transport maps onto log-concave measures
Maria Colombo, Max Fathi
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,…
Improved stability of optimal traffic paths
Maria Colombo, Antonio De Rosa, Andrea Marchese
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the ner…
Vanishing viscosity non-unique solutions to the forced 2D Euler Equations
Dallas Albritton, Maria Colombo, Giulia Mescolini
The forced 2D Euler equations exhibit non-unique solutions with vorticity in , , whereas the corresponding Navier-Stokes solutions are unique. We investigate whether th…
Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations
Maria Colombo, Camillo De Lellis, Luigi De Rosa
We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian …
Stability for the mailing problem
Maria Colombo, Antonio De Rosa, Andrea Marchese
We prove that optimal traffic plans for the mailing problem in are stable with respect to variations of the given coupling, above the critical exponent , t…
Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulence
Maria Colombo, Gianluca Crippa, Massimo Sorella
The Obukhov-Corrsin theory of scalar turbulence [Obu49, Cor51] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analog…
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
Dallas Albritton, Elia Brué, Maria Colombo
In the seminal work [39], Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with…
Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow
Lénaïc Chizat, Maria Colombo, Roberto Colombo +1
Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the m…
Renormalized solutions to the continuity equation with an integrable damping term
Maria Colombo, Gianluca Crippa, Stefano Spirito
We consider the continuity equation with a nonsmooth vector field and a damping term. In their fundamental paper, DiPerna and Lions proved that, when the damping term is bounded in…
Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field
Elia Bruè, Maria Colombo, Anuj Kumar
Given a divergence-free vector field and a nonnegative initial datum , the celebrated DiPerna--Lions theory establishe…
Optimal regularity for the fully nonlinear thin obstacle problem
Maria Colombo, Xavier Fernández-Real, Xavier Ros-Oton
In this work we establish the optimal regularity for solutions to the fully nonlinear thin obstacle problem. In particular, we show the existence of an optimal exponent such…
Ole\uınik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit
Giuseppe Maria Coclite, Maria Colombo, Gianluca Crippa +5
We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=\mathbb{1}_{(-\infty,0]}(\cdot)\exp(\cdot) \ast…
Effective Lagrangian regularity and the uniqueness threshold for random Hölder velocity fields
Maria Colombo, Elias Hess-Childs, Keefer Rowan
We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale fi…
On the lower semicontinuous envelope of functionals defined on polyhedral chains
Maria Colombo, Antonio De Rosa, Andrea Marchese +1
In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by $H \colon \mathbb…
Obstructions to regularity in the classical Monge problem
Maria Colombo, Emanuel Indrei
We provide counterexamples to regularity of optimal maps in the classical Monge problem under various assumptions on the initial data. Our construction is based on a variant of the…
Existence and uniqueness of maximal regular flows for non-smooth vector fields
Luigi Ambrosio, Maria Colombo, Alessio Figalli
In this paper we provide a complete analogy between the Cauchy-Lipschitz and the DiPerna-Lions theories for ODE's, by developing a local version of the DiPerna-Lions theory. More p…
Instability and nonuniqueness for the Euler equations in vorticity form, after M. Vishik
Dallas Albritton, Elia Brué, Maria Colombo +4
In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space, \[ \partial_t Ï+ u \cdot \nabla Ï= f…
Gluing non-unique Navier-Stokes solutions
Dallas Albritton, Elia Bruè, Maria Colombo
We construct non-unique Leray solutions of the forced Navier-Stokes equations in bounded domains via gluing methods. This demonstrates a certain locality and robustness of the non-…
Infinite-width limit of deep linear neural networks
Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real +1
This paper studies the infinite-width limit of deep linear neural networks initialized with random parameters. We obtain that, when the number of neurons diverges, the training dyn…
On the commutativity of flows of rough vector fields
Maria Colombo, Riccardo Tione
In the class of Sobolev vector fields in of bounded divergence, for which the theory of DiPerna and Lions provides a well defined notion of flow, we characterize the…
Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity
Elia Bruè, Maria Colombo, Anuj Kumar
We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $…
Stability of optimal traffic plans in the irrigation problem
Maria Colombo, Antonio de Rosa, Andrea Marchese +2
We prove the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This is the Lagrangian cou…
Slow time behavior of the semidiscrete Perona-Malik scheme in dimension one
Maria Colombo, Massimo Gobbino
We consider the long time behavior of the semidiscrete scheme for the Perona-Malik equation in dimension one. We prove that approximated solutions converge, in a slow time scale, t…
Logarithmic estimates for continuity equations
Maria Colombo, Gianluca Crippa, Stefano Spirito
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…
Essential connectedness and the rigidity problem for Gaussian symmetrization
Filippo Cagnetti, Maria Colombo, Guido De Philippis +1
We provide a geometric characterization of rigidity of equality cases in Ehrhard's symmetrization inequality for Gaussian perimeter. This condition is formulated in terms of a new…
Instability of two-dimensional Taylor-Green Vortices
Gonzalo Cao-Labora, Maria Colombo, Michele Dolce +1
For a wide class of linear Hamiltonian operators we develop a general criterion that characterizes the unstable eigenvalues as the zeros of a holomorphic function given by the dete…
Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations
Maria Colombo, Michele Dolce, Riccardo Montalto +1
In 1959, Kolmogorov proposed to study the instability of the shear flow in the vanishing viscosity regime in tori . This question was…
Estimate on the dimension of the singular set of the supercritical surface quasigeostrophic equation
Maria Colombo, Silja Haffter
We consider the SQG equation with dissipation given by a fractional Laplacian of order . We introduce a notion of suitable weak solution, which exists for every $L^…
Global regularity for the hyperdissipative Navier-Stokes equation below the critical order
Maria Colombo, Silja Haffter
We consider solutions of the Navier-Stokes equation with fractional dissipation of order . We show that for any divergence-free initial datum such that $||u_0||_{H^…
Nonlocal traffic models with general kernels: singular limit, entropy admissibility, and convergence rate
Maria Colombo, Gianluca Crippa, Elio Marconi +1
Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the mode…
On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws
Maria Colombo, Gianluca Crippa, Marie Graff +1
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 an…
The singular set of minimal surfaces near polyhedral cones
Maria Colombo, Nick Edelen, Luca Spolaor
We adapt the method of Simon [JDG '93] to prove a -regularity theorem for minimal varifolds which resemble a cone over an equiangular geodesic net. For varif…
Regularity for general functionals with double phase
Paolo Baroni, Maria Colombo, Giuseppe Mingione
We prove sharp regularity results for a general class of functionals of the type featuring non-standard growth conditions and non-uniform…
Positive solutions of transport equations and classical nonuniqueness of characteristic curves
Elia Bruè, Maria Colombo, Camillo De Lellis
The seminal work of DiPerna and Lions [Invent. Math., 98, 1989] guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suita…
Local limit of nonlocal traffic models: convergence results and total variation blow-up
Maria Colombo, Gianluca Crippa, Elio Marconi +1
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 convolu…
Lyapunov exponents, entropy and mixing for DiPerna-Lions flows
Elia Brué, Maria Colombo, Carl Johan Peter Johansson
The main goal of this work is to establish an asymptotic form of Bressan's mixing conjecture. To this end, we develop an ergodic-theoretic framework for incompressible DiPerna-Lion…
Onsager critical solutions of the forced Navier-Stokes equations
Elia Bruè, Maria Colombo, Gianluca Crippa +2
We answer positively to [BDL22, Question 2.4] by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissi…
Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case
Luigi Ambrosio, Maria Colombo, Guido De Philippis +1
In this paper we use the new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations estabilished by G.De Philippis and A.Figalli to provide a global…
Mixing and diffusion for rough shear flows
Maria Colombo, Michele Coti Zelati, Klaus Widmayer
This article addresses mixing and diffusion properties of passive scalars advected by rough () shear flows. We show that in general, one cannot expect a rough shear flow to i…
Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1
Tristan Buckmaster, Maria Colombo, Vlad Vicol
We prove non-uniqueness for a class of weak solutions to the Navier-Stokes equations which have bounded kinetic energy, integrable vorticity, and are smooth outside a fractal set o…
Direct epiperimetric inequalities for the thin obstacle problem and applications
Maria Colombo, Luca Spolaor, Bozhidar Velichkov
For the thin obstacle problem, we prove by a new direct method that in any dimension the Weiss' energies with frequency and , for , satisfy an epiperi…
Regularity estimates in transport equations via heat flow and quantitative differentiation
Elia Brué, Maria Colombo, Guido De Philippis +1
The paper establishes quantitative estimates for commutators in transport equations, showing how logarithmic Sobolev regularity propagates via heat flow and using quantitative diff…
Global regularity for the nonlinear wave equation with slightly supercritical power
Maria Colombo, Silja Haffter
We consider the defocusing nonlinear wave equation in . We prove that for any initial datum with a scaling-subcrit…