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Stefano Modena

Publications and source records attributed to Stefano Modena.

14 recordsLinked to original sources
arXiv2025

On some typicality and density results for nonsmooth vector fields and the associated ODE and continuity equation

These notes address two problems. First, we investigate the question of ``how many'' are (in Baire sense) vector fields in $L^1_t L^q_x$, $q \in [1, \infty)$, for which existence and/or uniqueness of local, distributional solutions to the associated continuity equation holds. We show that, in certain regimes, existence of solutions (even locally in time, for at least one nonzero initial datum) is a meager property, whereas, on the contrary, uniqueness of solutions is a generic property. Secondly, despite the fact that non-uniqueness is a meager property, we prove that (Sobolev) counterexamples to uniqueness, both for the continuity equation and for the ODE, in the spirit of [Bru\`e, Colombo, Kumar 2024] and [Kumar 2024] respectively, form a dense subset of the natural ambient space they live in.

math.AP↗
arXiv2024

Compactly supported anomalous weak solutions for 2D Euler equations with vorticity in Hardy spaces

In a previous work (arXiv:2306.05948), we constructed by convex integration examples of energy dissipating solutions to the 2D Euler equations on $\mathbb{R}^2$ with vorticity in the real Hardy space $H^p(\mathbb{R}^2)$. In the present paper, we develop tools that significantly improve that result in two ways: Firstly, we achieve vorticities in $H^p(\mathbb{R}^2)$ in the optimal range $p\in (0,1)$ compared to $(2/3,1)$ in our previous work. Secondly, the solutions constructed here possess compact support and in particular preserve linear and angular momenta.

math.AP↗
arXiv2023

Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift

We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli and Priola. We consider periodic solutions in $\rho \in L^{\infty}_{t} L_{x}^{p}$ for divergence-free drifts $u \in L^{\infty}_{t} W_{x}^{\theta, \tilde{p}}$ for a large class of parameters. We prove local-in-time pathwise nonuniqueness and compare them to uniqueness results by Beck, Flandoli, Gubinelli and Maurelli, addressing a conjecture made by these authors, in the case of bounded-in-time drifts for a large range of spatial parameters. To this end, we use convex integration techniques to construct velocity fields $u$ for which several solutions $\rho$ exist in the classes mentioned above. The main novelty lies in the ability to construct deterministic drift coefficients, which makes it necessary to consider a convex integration scheme \textit{with a constraint}, which poses a series of technical difficulties.

math.PR↗
arXiv2022

On the failure of the chain rule for the divergence of Sobolev vector fields

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d \geq 3$, for which the chain rule problem has a negative answer. In particular, for any renormalization map $\beta$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T = {\rm div}\, h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u \in W^{1, \tilde p}$ and a density $\rho \in L^p$ for which ${\rm div}\, (\rho u) =0$ but ${\rm div}\, (\beta(\rho) u) = T$, provided $1/p + 1/\tilde p \geq 1 + 1/(d-1)$

math.AP↗
arXiv2020

Non Uniqueness of power-law flows

We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension $d\ge 2$. For the power index $q$ below the compactness threshold, i.e. $q \in (1, \frac{2d}{d+2})$, we show ill-posedness of Leray-Hopf solutions. For a wider class of indices $q \in (1, \frac{3d+2}{d+2})$ we show ill-posedness of distributional (non-Leray-Hopf) solutions, extending the seminal paper of Buckmaster and Vicol. In this wider class we also construct non-unique solutions for every datum in $L^2$.

math.AP↗
arXiv2019

Convex integration solutions to the transport equation with full dimensional concentration

We construct infinitely many incompressible Sobolev vector fields $u \in C_t W^{1,\tilde p}_x$ on the periodic domain $\mathbb{T}^d$ for which uniqueness of solutions to the transport equation fails in the class of densities $\rho \in C_t L^p_x$, provided $1/p + 1/\tilde p > 1 + 1/d$. The same result applies to the transport-diffusion equation, if, in addition $p'<d$.

math.AP↗
arXiv2017

A "forward-in-time" quadratic potential for systems of conservation laws

A quadratic interaction potential $t \mapsto \Upsilon(t)$ for hyperbolic systems of conservation laws is constructed, whose value $\Upsilon(\bar t)$ at time $\bar t$ depends only on the present and the future profiles of the solution and not on the past ones. Such potential is used to bound the change of the speed of the waves at each interaction.

math.AP↗
arXiv2014

Quadratic interaction functional for general systems of conservation laws

For the Glimm scheme approximation $u_\epsilon$ to the solution of the system of conservation laws in one space dimension \begin{equation*} u_t + f(u)_x = 0, \qquad u(0,x) = u_0(x) \in \mathbb R^n, \end{equation*} with initial data $u_0$ with small total variation, we prove a quadratic (w.r.t. $\mathrm{TV}(u_0)$) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux $f$ are made (apart smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely we obtain the following results: \newline - a new analysis of the interaction estimates of simple waves; \newline - a Lagrangian representation of the derivative of the solution, i.e. a map $\mathtt x(t,w)$ which follows the trajectory of each wave $w$ from its creation to its cancellation; \newline - the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves; \item a new functional $\mathfrak Q$ controlling the variation in speed of the waves w.r.t. time. \newline This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems. The main result is that the distribution $D_{tt} \mathtt x(t,w)$ is a measure with total mass $\leq \mathrm{const} \mathrm{TV}(u_0)^2$.

math.AP↗
arXiv2013

Quadratic interaction functional for systems of conservation laws: a case study

We prove a quadratic interaction estimate for wavefront approximate solutions to the triangular system of conservation laws \[ \begin{cases} u_t + \tilde f(u,v)_x = 0, \cr v_t - v_x = 0. \end{cases} \] This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. Our aim is to extend the analysis, done for scalar conservation laws, in the presence of transversal interactions among wavefronts of different families. The proof is based on the introduction of a quadratic functional $\mathfrak Q(t)$, decreasing at every interaction, and such that its total variation in time is bounded. %cancellations it variation is controlled by the total variation growths at most of the total variation of the solution multiplied by the amount of cancellation. The study of this particular system is a key step in the proof of the quadratic interaction estimate for general systems: it requires a deep analysis of the wave structure of the solution $(u(t,x),v(t,x))$ and the reconstruction of the past history of each wavefront involved in an interaction.

math.AP↗
arXiv2013

On a quadratic functional for scalar conservation laws

We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. The proof is based on the introduction of a quadratic functional $\mathfrak Q(t)$, decreasing at every interaction, and such that its total variation in time is bounded. Differently from other interaction potentials present in the literature, the form of this functional is the natural extension of the original Glimm functional, and coincides with it in the genuinely nonlinear case.

math.AP↗