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Davide Catania

Publications and source records attributed to Davide Catania.

10 recordsLinked to original sources
arXiv2023

Global existence for a 3D Tropical Climate Model with damping and small initial data in $\dot H^{1/2}(\mathbb{R}^3)$

We consider a 3D Tropical Climate Model with damping terms in the equation of the barotropic mode $u$ and in the equation of the first baroclinic mode $v$ of the velocity. The equation for the temperature $\theta$ is free from dampings. We prove global existence in time for this system assuming the initial data $(u_0, v_0,\theta_0)$ small, in terms of the homogeneous space $\dot H^{1/2}(\mathbb{R}^3)$.

math.AP↗
arXiv2022

A regularity criterion for a 3D tropical climate model with damping

In this paper we deal with the 3D tropical climate model with damping terms in the equation of the barotropic mode $u$ and in the equation of the first baroclinic mode $v$ of the velocity, and we establish a regularity criterion for this system thanks to which the local smooth solution $(u, v, θ)$ can actually be extended globally in time.

math.AP↗
arXiv2016

Weak stability of the plasma-vacuum interface problem

We consider the free boundary problem for the two-dimensional plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the Maxwell system for the electric and the magnetic fields. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. We study the linear stability of rectilinear plasma-vacuum interfaces by computing the Kreiss-Lopatinskii determinant of an associated linearized boundary value problem. Apart from possible resonances, we obtain that the piecewise constant plasma-vacuum interfaces are always weakly linearly stable, independently of the size of tangential velocity, magnetic and electric fields on both sides of the characteristic discontinuity. We also prove that solutions to the linearized problem obey an energy estimate with a loss of regularity with respect to the source terms, both in the interior domain and on the boundary, due to the failure of the uniform Kreiss-Lopatinskii condition, as the Kreiss-Lopatinskii determinant associated with this linearized boundary value problem has roots on the boundary of the frequency space. In the proof of the a priori estimates, a crucial part is played by the construction of symmetrizers for a reduced differential system, which has poles at which the Kreiss-Lopatinskii condition may fail simultaneously.

math.AP↗
arXiv2014

Stability of the linearized MHD-Maxwell free interface problem

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the Maxwell system for the electric and the magnetic fields, in order to investigate the well-posedness of the problem, in particular in relation with the electric field in vacuum. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. Under suitable stability conditions satisfied at each point of the plasma-vacuum interface, we derive a basic a priori estimate for solutions to the linearized problem. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method. An interesting novelty is represented by the fact that the interface is characteristic with variable multiplicity, so that the problem requires a different number of boundary conditions, depending on the direction of the front velocity (plasma expansion into vacuum or viceversa). To overcome this difficulty, we recast the vacuum equations in terms of a new variable which makes the interface characteristic of constant multiplicity. In particular, we don't assume that plasma expands into vacuum.

math.AP↗
arXiv2014

Global Attractor for the Navier-Stokes Equations with horizontal filtering

We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator. It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiflow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the flow under periodicity in all directions and discuss the possibility to introduce a solution semiflow.

math.AP↗
arXiv2012

Convergence of approximate deconvolution models to the mean Magnetohydrodynamics Equations: Analysis of two models

We consider two Large Eddy Simulation (LES) models for the approximation of large scales of the equations of Magnetohydrodynamics (MHD in the sequel). We study two $α$-models, which are obtained adapting to the MHD the approach by Stolz and Adams with van Cittert approximate deconvolution operators. First, we prove existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations. Then we study the behavior of solutions as the deconvolution parameter goes to infinity. The main result of this paper is the convergence to a solution of the filtered MHD equations. In the final section we study also the problem with filtering acting only on the velocity equation.

math.AP↗
arXiv2004

Blow Up for the Semilinear Wave Equation in Schwarzschild Metric

We study the semilinear wave equation in Schwarzschild metric (3+1 dimensional space--time). First, we establish that the problem is locally well--posed in $\cs H^σ$ for any $σ\geq 1$; then we prove the blow up of the solution for every real $p \in ]1,1+\sqrt{2}[$ and non--negative non--trivial initial data.

math-ph↗