arXiv subjects

Gianluca Crippa

Publications and source records attributed to Gianluca Crippa.

At least 37 records · Page 2Linked to original sources

arXiv2022

Nonlocal traffic models with general kernels: singular limit, entropy admissibility, and convergence rate

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 modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counter-example showing that the convexity assumption is necessary for our main compactness estimate.

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arXiv2022

Growth of Sobolev norms and loss of regularity in transport equations

We consider transport of a passive scalar advected by an irregular divergence free vector field. Given any non-constant initial data $\bar ρ\in H^1_\text{loc}({\mathbb R}^d)$, $d\geq 2$, we construct a divergence free advecting velocity field $v$ (depending on $\bar ρ$) for which the unique weak solution to the transport equation does not belong to $H^1_\text{loc}({\mathbb R}^d)$ for any positive positive time. The velocity field $v$ is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space $W^{s,p}$ that does not embed into the Lipschitz class. The velocity field $v$ is constructed by pulling back and rescaling an initial data dependent sequence of sine/cosine shear flows on the torus. This loss of regularity result complements that in [Ann. PDE, 5(1):Paper No. 9, 19, 2019].

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arXiv2021

Sub-exponential mixing of generalized cellular flows with bounded palenstrophy

We study the mixing properties of a passive scalar advected by an incompressible flow. We consider a class of cellular flows (more general than the class in [Crippa-Schulze M3AS 2017]) and show that, under the constraint that the palenstrophy is bounded uniformly in time, the mixing scale of the passive scalar cannot decay exponentially.

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arXiv2021

Strong convergence of the vorticity for the 2D Euler Equations in the inviscid limit

In this paper we prove the uniform-in-time $L^p$ convergence in the inviscid limit of a family $ω^ν$ of solutions of the $2D$ Navier-Stokes equations towards a renormalized/Lagrangian solution $ω$ of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $ω^ν$ to $ω$ in $L^p$. Finally, we show that solutions of the Euler equations with $L^p$ vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.

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arXiv2021

A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space $W^{1,p}$, where $p$ is larger than the space dimension, under the assumption that the so-called "forward-backward integral curves" associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable.

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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

Smooth approximation is not a selection principle for the transport equation with rough vector field

In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation.

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arXiv2019

A note on the Lagrangian flow associated to a partially regular vector field

In this paper we derive quantitative estimates for the Lagrangian flow associated to a partially regular vector field of the form $$ b(t,x_1,x_2) = (b_1(t,x_1),b_2(t,x_1,x_2)) \in {\mathbb R}^{n_1}\times{\mathbb R}^{n_2} \,, \qquad (x_1,x_2)\in{\mathbb R}^{n_1}\times{\mathbb R}^{n_2}\,. $$ We assume that the first component $b_1$ does not depend on the second variable $x_2$, and has Sobolev $W^{1,p}$ regularity in the variable $x_1$, for some $p>1$. On the other hand, the second component $b_2$ has Sobolev $W^{1,p}$ regularity in the variable $x_2$, but only fractional Sobolev $W^{α,1}$ regularity in the variable $x_1$, for some $α>1/2$. These estimates imply well-posedness, compactness, and quantitative stability for the Lagrangian flow associated to such a vector field.

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arXiv2019

Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy

We discuss the Lagrangian property and the conservation of the kinetic energy for solutions of the 2D incompressible Euler equations. Existence of Lagrangian solutions is known when the initial vorticity is in $L^p$ with $1\leq p\leq \infty$. Moreover, if $p\geq 3/2$ all weak solutions are conservative. In this work we prove that solutions obtained via the vortex method are Lagrangian, and that they are conservative if $p>1$.

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arXiv2019

Loss of regularity for the continuity equation with non-Lipschitz velocity field

We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in $W^{r,p}$, with $r>1$, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space $W^{r,p}$ does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper "Exponential self-similar mixing by incompressible flows" (J. Amer. Math. Soc. 32 (2019), no. 2), and have been announced in "Exponential self-similar mixing and loss of regularity for continuity equations" (C. R. Math. Acad. Sci. Paris, 352 (2014), no. 11).

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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.

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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.

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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.

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arXiv2019

On smooth approximations of rough vector fields and the selection of flows

In this work we deal with the selection problem of flows of an irregular vector field. We first summarize an example from \cite{CCS} of a vector field $b$ and a smooth approximation $b_ε$ for which the sequence $X^ε$ of flows of $b_ε$ has subsequences converging to different flows of the limit vector field $b$. Furthermore, we give some heuristic ideas on the selection of a subclass of flows in our specific case.

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arXiv2019

Polynomial mixing under a certain stationary Euler flow

We study the mixing properties of a scalar $ρ$ advected by a certain incompressible velocity field $u$ on the two dimensional unit ball, which is a stationary radial solution of the Euler equation. The scalar $ρ$ solves the continuity equation with velocity field $u$ and we can measure the degree of mixedness of~$ρ$ with two different scales commonly used in this setting, namely the geometric and the functional mixing scale. We develop a physical space approach well adapted for the quantitative analysis of the decay in time of the geometric mixing scale, which turns out to be polynomial for a large class of initial data. This extends previous results for the functional mixing scale, based on the explicit expression for the solution in Fourier variable, results that are also partially recovered by our approach.

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arXiv2018

Strong continuity for the 2D Euler equations

We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second result provides uniform continuity and is restricted to Hölder continuous solutions.

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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.

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arXiv2018

Exponential self-similar mixing by incompressible flows

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case $s=1$ and $1 \leq p \leq \infty$ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.

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