arXiv subjects

Alessandro Violini

Publications and source records attributed to Alessandro Violini.

5 recordsLinked to original sources

arXiv2026

$L^2(\mathbb{R}^2)$ Well-Posedness and Logarithmic Lipschitz Regularity for the Density Patch Problem

We study the density patch problem for the two-dimensional inhomogeneous incompressible Navier--Stokes system with vacuum, for initial data consisting of a Lipschitz density patch and a divergence-free velocity field in $L^2(\mathbb{R}^2)$. We establish uniqueness of solutions at the natural energy level, thereby concluding the global well-posedness for $L^2$ data. Furthermore, we prove a log-Lipschitz estimate for the velocity field, extending the classical result of Chemin-Lerner to the inhomogeneous setting. As a consequence, the associated flow belongs to $L^\infty_t C_x^{1-\varepsilon}$. for any $\varepsilon \in (0,1)$, ensuring that the patch boundary remains a continuous curve of Hausdorff dimension $1$, thus preserving its initial dimension for all time.

math.AP↗

arXiv2026

On Lions' density patch problem at a critical level of regularity

In this article, we study Lions' density patch problem in two space dimensions at critical regularity. We prove global existence, uniqueness, and stability for a fluid occupying a bounded Lipschitz region surrounded by vacuum and evolving according to the incompressible Navier--Stokes equations, with initial velocity in $\dot{B}^0_{2,1}(\mathbb{R}^2)$. Moreover, we show that the Lipschitz regularity of the patch is preserved, and that its long-time dynamics is a rigid motion leading to the emergence of an asymptotic domain.

math.AP↗

arXiv2024

Regularity aspects of Leray-Hopf solutions to the 2D Inhomogeneous Navier-Stokes system and applications to weak-strong uniqueness

We characterize the Leray--Hopf solutions of the 2D inhomogeneous Navier--Stokes system that become strong for positive times. This characterization relies on the strong energy inequality and the regularity properties of the pressure. As an application, we establish a weak-strong uniqueness result and provide a unified framework for several recent advances in the field.

math.AP↗

arXiv2024

Inhomogeneous incompressible Euler with codimension $1$ singular structures

This paper is concerned with the inhomogeneous incompressible Euler system. We establish a Duchon--Robert type approximation theorem for the distribution describing the local energy flux of bounded solutions. The velocity field is assumed to have bounded variation or bounded deformation with respect to the spatial variable. The density satisfies no-vacuum condition and has $BV$ regularity in space, allowing for a system made by two immiscible fluids separated by a Lipschitz hypersurface. By means of a careful analysis of the traces along hypersufaces of bounded vector fields with measure divergence, we show that the dissipation does not give mass to hypersurfaces of codimension one, even if the velocity field, the density and the pressure have jumps. This feature is specific of incompressible models. As a consequence, we show that the dissipation measure vanishes if it is concentrated on a countably rectifiable set of space-time codimension one, even if this is dense. For instance, in the simplest case of a system made by two incompressible immiscible fluids with $BV$ velocity, the dissipation measure vanishes as soon as the interface between the two fluids has (space-time) finite perimeter.

math.AP↗

arXiv2024

Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations

We present a weak-strong uniqueness result for the inhomogeneous Navier-Stokes (INS) equations in $\mathbb{R}^d$ ($d=2,3$) for bounded initial densities that are far from vacuum. Given a strong solution within the class employed in Paicu, Zhang and Zhang (2013) and Chen, Zhang and Zhao (2016), and a Leray-Hopf weak solution, we establish that they coincide if the initial data agree. The strategy of our proof is based on the relative energy method and new $W^{-1,p}$-type stability estimates for the density. A key point lies in proving that every Leray-Hopf weak solution originating from initial densities far from vacuum remains distant from vacuum at all times.

math.AP↗