arXiv subjects

Gabriela Planas

Publications and source records attributed to Gabriela Planas.

9 recordsLinked to original sources

arXiv2026

Quantitative approximation of the Vlasov(-Fokker-Planck)-Navier-Stokes system by stochastic particle systems

This paper is concerned with a fluid-particle system given by the incompressible Navier-Stokes equations coupled with the Vlasov(-Fokker-Planck) equation through a drag force. Such a model arises naturally in the study of aerosols, sprays, and more generally two-phase flows. In dimensions $d\in \{2,3\}$, we establish a rate of convergence for a system of $N$ interacting stochastic particles coupled with a fluid, towards the Vlasov(-Fokker-Planck)-Navier-Stokes system, as $N\to \infty$. The case of particles with a noise that vanishes as $N\to \infty$ is considered and leads specifically to the Vlasov-Navier-Stokes system. More precisely, we prove that the empirical measure associated with the particle system converges to the Vlasov(-Fokker-Planck) component, while the fluid velocity converges to the Navier-Stokes component of the coupled system. The proofs combine stochastic calculus and PDE techniques to establish energy estimates and commutator estimates for both the discrete and continuous systems.

math.PR↗

arXiv2025

Dynamics of dissipative solutions to the Hardy-Sobolev parabolic equation

We study the long-time behaviour of solutions to the Hardy-Sobolev parabolic equation in critical function spaces for any spatial dimension $d \geq 5$. By employing the Fourier splitting method, we establish precise decay rates for dissipative solutions, meaning those whose critical norm vanishes as time approaches infinity. Our findings offer a deeper understanding of the asymptotic properties and dissipation mechanisms governing this equation.

math.AP↗

arXiv2025

A convergent time scheme for a chemotaxis-fluids model with potential consumption

The present work deals with a Keller-Segel-Navier-Stokes system with potential consumption, under homogeneous Neumann boundary conditions for cell density and chemical signal, and of Dirichlet type for the velocity field, over a bounded three-dimensional domain. The paper aims to develop a time discretization scheme converging to weak solutions of the system, which are uniformly bounded at infinite time. While global existence results are already known for simplified cases, either in absence of fluid flow or for linear consumption, the existence of global weak solutions for the fully coupled system with potential consumption has remained as an open problem.

math.AP↗

arXiv2025

Global dynamics for the energy-critical nonlinear heat equation

We examine the energy-critical nonlinear heat equation in critical spaces for any dimension greater or equal than three. The aim of this paper is two-fold. First, we establish a necessary and sufficient condition on initial data at or below the ground state that dichotomizes the behavior of solutions. Specifically, this criterion determines whether the solution will either exist globally with energy decaying to zero over time or blow up in finite time. Secondly, we derive the decay rate for solutions that exist globally. These results offer a comprehensive characterization of solution behavior for energy-critical conditions in higher-dimensional settings

math.AP↗

arXiv2023

Algebraic decay rates for 3D Navier-Stokes and Navier-Stokes-Coriolis equations in $ \dot{H}^{\frac{1}{2}}$

An algebraic upper bound for the decay rate of solutions to the Navier-Stokes and Navier-Stokes-Coriolis equations in the critical space $\dot{H} ^{\frac{1}{2}} (\mathbb{R} ^3)$ is derived using the Fourier Splitting Method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts.

math.AP↗

arXiv2023

Decay rates for the 4D energy-critical nonlinear heat equation

In this paper we address the decay of solutions to the four-dimen\-sional energy-critical nonlinear heat equation in the critical space $\dot{H}^1$. Recently, it was proven that the $\dot{H}^1$ norm of solutions goes to zero when time goes to infinity, but no decay rates were established. By means of the Fourier Splitting Method and using properties arising from the scale invariance, we obtain an algebraic upper bound for the decay rate of solutions.

math.AP↗

arXiv2017

Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain

In this paper we treat three problems on a two-dimensional `punctured periodic domain': we take $Ω_r=(-L,L)^2\setminus D_r$, where $D_r=B(0,r)$ is the disc of radius $r$ centred at the origin. We impose periodic boundary conditions on the boundary of the box $Ω=(-L,L)^2$, and Dirichlet boundary conditions on the circumference of the disc. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier-Stokes equations, all with a fixed forcing function $f$ (which must satisfy $\int_Ωf=0$ for the stationary problems), and examine the behaviour of solutions as $r\to0$. In all three cases we show convergence of the solutions to those of the limiting problem, i.e.\ the problem posed on all of $Ω$ with periodic boundary conditions.

math.AP↗

arXiv2017

On the "viscous incompressible fluid + rigid body" system with Navier conditions

In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole system "viscous incompressible fluid + rigid body" is assumed to occupy the full space $\R^{3}$. We start by proving the existence of global weak solutions to the Cauchy problem. Then, we exhibit several properties of these solutions. First, we show that the added-mass effect can be computed which yields better-than-expected regularity (in time) of the solid velocity-field. More precisely we prove that the solid translation and rotation velocities are in the Sobolev space $H^1$. Second, we show that the case with the body fixed can be thought as the limit of infinite inertia of this system, that is when the solid density is multiplied by a factor converging to $+\infty$. Finally we prove the convergence in the energy space of weak solutions "à la Leray" to smooth solutions of the system "inviscid incompressible fluid + rigid body" as the viscosity goes to zero, till the lifetime $T$ of the smooth solution of the inviscid system. Moreover we show that the rate of convergence is optimal with respect to the viscosity and that the solid translation and rotation velocities converge in $H^1 (0,T)$.

math.AP↗

arXiv2012

On the non-homogeneous Navier-Stokes system with Navier friction boundary conditions

We address the issue of existence of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions allowing the presence of vacuum zones and assuming rough conditions on the data. We also study the convergence, as the viscosity goes to zero, of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions to the strong solution of the Euler equations with variable density, provided that the initial data converge in $L^{2}$ to a smooth enough limit.

math.AP↗