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Marco Bravin

Publications and source records attributed to Marco Bravin.

16 recordsLinked to original sources
arXiv2026

On the impact of clusters of rigid balls on the motion of a viscous fluid

We develop a new approach to the problem of the motion of a large number of rigid bodies immersed in a viscous fluid. The leading idea is the concept of cluster - a collection of individual rigid objects that may be grouped or even connected in such a way that their collective impact on the bulk motion of the system is similar to that of a single body. The applications of the new approach include: 1. Improving the critical value of the number of balls of small radius such that their cloud has no impact on the limit system represented by the incompressible Navier--Stokes equations. 2. The balls follow the fluid flow in the asymptotic limit of vanishing radius and increasing number even if a gravitational force is imposed.

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arXiv2025

On the long time behaviour of a system of several rigid bodies immersed in a viscous fluid

We consider several rigid bodies immersed in a viscous Newtonian fluid contained in a bounded domain in $R^3$. We introduce a new concept of dissipative weak solution of the problem based on a combination of the approach proposed by Judakov with a suitable form of energy inequality. We show that global--in--time dissipative solutions always exist as long as the rigid bodies are connected compact sets. In addition, in the absence of external driving forces, the system always tends to a static equilibrium as time goes to infinity. The results hold independently of possible collisions of rigid bodies and for any finite energy initial data.

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arXiv2024

Well-Posedness and Regularity of the Heat Equation with Robin Boundary Conditions in the Two-Dimensional Wedge

Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge is established in an $L^{2}$-setting of monomially weighted spaces. A mathematical framework is developed which allows to obtain arbitrarily high regularity without a smallness assumption on the opening angle of the wedge. The challenging aspect is that the resolvent problem exhibits two breakings of the scaling invariance, one in the equation and one in the boundary condition.

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arXiv2024

Well-posedness of the Stokes equations on a wedge with Navier-slip boundary conditions

We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces. The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.

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arXiv2024

On the collective effect of a large system of heavy particles immersed in a Newtonian fluid

We consider the motion of a large number of heavy particles in a Newtonian fluid occupying a bounded spatial domain. When we say "heavy", we mean a particle with a mass density that approaches infinity at an appropriate rate as its radius vanishes. We show that the collective effect of heavy particles on the fluid motion is similar to the Brinkman perturbation of the Navier-Stokes system identified in the homogenization process.

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arXiv2024

Global existence for non-homogeneous incompressible inviscid fluids in presence of Ekman pumping

In this paper, we study the global solvability of the density-dependent incompressible Euler equations, supplemented with a damping term of the form $ \mathfrak{D}_{\alpha}^{\gamma}(\rho, u) = \alpha \rho^{\gamma} u $, where $\alpha>0$ and $ \gamma \in \{0,1\} $. To some extent, this system can be seen as a simplified model describing the mean dynamics in the ocean; from this perspective, the damping term can be interpreted as a term encoding the effects of the celebrated Ekman pumping in the system. On the one hand, in the general case of space dimension $d\geq 2$, we establish global well-posedness in the Besov spaces framework, under a non-linear smallness condition involving the size of the initial velocity field $u_0$, of the initial non-homogeneity $\rho_0-1$ and of the damping coefficient $\alpha$. On the other hand, in the specific situation of planar motions and damping term with $\gamma=1$, we exhibit a second smallness condition implying global existence, which in particular yields global well-posedness for arbitrarily large initial velocity fields, provided the initial density variations $\rho_0-1$ are small enough. The formulated smallness conditions rely only on the endpoint Besov norm $B^1_{\infty,1}$ of the initial datum, whereas, as a byproduct of our analysis, we derive exponential decay of the velocity field and of the pressure gradient in the high regularity norms $B^s_{p,r}$.

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arXiv2023

Ad hoc test functions for homogenization of compressible viscous fluid with application to the obstacle problem in dimension two

In this paper we highlight a set of ad hoc test functions to study the homogenization of viscous compressible fluid in domains with tiny holes. This set of functions allows to improve previous results in dimensions two and three. As an application we show that the presence of a small obstacle does not influence the dynamics of a viscous compressible fluid in dimension two.

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arXiv2022

On the trajectory of a light small rigid body in an incompressible viscous fluid

In this paper we study the dynamics of a small rigid body in a viscous incompressible fluid in dimension two and three. More precisely we investigate the trajectory of the rigid body in the limit when the its mass and its size tend to zero. We show that the velocity of the center of mass of the rigid body coincides with the background fluid velocity in the limit. We are able to consider the case where the density of the small rigid body is uniformly bounded respect to its size.

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arXiv2022

On the velocity of a small rigid body in a viscous incompressible fluid in dimension two and three

In this paper we study the evolution of a small rigid body in a viscous incompressible fluid, in particular we show that a small particle is not accelerated by the fluid in the limit when its size converges to zero under a lower bound on its mass. This result is based on a new a priori estimate on the velocities of the centers of mass of rigid bodies that holds in the case when their masses are also allowed to decrease to zero.

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arXiv2022

Fast rotating non-homogeneous fluids in thin domains and the Ekman pumping effect

In this paper, we perform the fast rotation limit $\varepsilon\rightarrow0^+$ of the density-dependent incompressible Navier-Stokes-Coriolis system in a thin strip $Ω_\varepsilon\,:=\,\mathbb{R}^2\times\,]-\ell_\varepsilon,\ell_\varepsilon[\,$, where $\varepsilon\in\,]0,1]$ is the size of the Rossby number and $\ell_\varepsilon>0$ for any $\varepsilon>0$. By letting $\ell_\varepsilon\longrightarrow0^+$ for $\varepsilon\rightarrow0^+$ and considering Navier-slip boundary conditions at the boundary of $Ω_\varepsilon$, we give a rigorous justification of the phenomenon of the Ekman pumping in the context of non-homogeneous fluids. With respect to previous studies (performed for flows of contant density and for compressible fluids), our approach has the advantage of circumventing the complicated analysis of boundary layers. To the best of our knowledge, this is the first study dealing with the asymptotic analysis of fast rotating incompressible fluids with variable density in a $3$-D setting. In this respect, we remark that the case $\ell_\varepsilon\geq\ell>0$ for all $\varepsilon>0$ remains largely open at present.

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arXiv2021

On the existence of weak solutions for the 2D incompressible Euler equations with in-out flow and source and sink points

Well-posedness for the two dimensional Euler system with given initial vorticity is known since the works of Judovič. In this paper we show existence of solutions in the case where we allowed the fluid to enter in and exit from the boundaries and from some points of the fluid domain. In particular we derive the equations of the model as the limit when we replace the points by some small holes. To do that we extend the DiPerna-Lions theory with non-tangent velocity field on the boundary to the case of time-dependent domain, we extend the existence result for the two dimensional Euler system with in-out flow to time-dependent domain and finally we derive the system that models a fluid which is allowed to enter in and exit from the boundary and some points. The solutions are characterized by the presence of source, sink and vortex points.

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arXiv2021

On the one dimensional cubic NLS in a critical space

In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.

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arXiv2021

Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks

A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper we investigate the existence of weak solutions to this system by relying on \textit{a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a $L^p$ integrability in space, with $p $ in $[1,+\infty]$, and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where $p >\frac43$, in the renormalized sense in the case where $p >1$, and in a symmetrized sense in the case where $p =1$.

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arXiv2020

On the vanishing rigid body problem in a viscous compressible fluid

In this paper we study the interaction of a small rigid body in a viscous compressible fluid. The system occupies a bounded three dimensional domain. The object it allowed to freely move and its dynamics follows the Newton's laws. We show that as the size of the object converges to zero the system fluid plus rigid body converges to the compressible Navier-Stokes system under some mild lower bound on the mass and the inertia momentum. It is a first result of homogenization in the case of fluid-structure interaction in the compressible situation. As a corollary we slightly improved the result on the influence of a vanishing obstacle in a compressible fluid for $ γ\geq 6$ .

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arXiv2018

On the weak uniqueness of "viscous incompressible fluid + rigid body" system with Navier slip-with-friction conditions in a 2D bounded domain

The existence of weak solutions to the "viscous incompressible fluid + rigid body" system with Navier slip-with-friction conditions in a 3D bounded domain has been recently proved by Gérard-Varet and Hillairet in \cite{exi:GeH}. In 2D for a fluid alone (without any rigid body) it is well-known since Leray that weak solutions are unique, continuous in time with $ L^{2} $ regularity in space and satisfy the energy equality.In this paper we prove that these properties also hold for the 2D "viscous incompressible fluid + rigid body" system.

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