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^\sigma for any $\sigma \geq 1$; 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.Comment: some typos are corrected and some references are adde

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.Comment: arXiv admin note: substantial text overlap with arXiv:1112.310

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.Comment: 38 page

Linear and Nonlinear Perturbed Wave Equations

We consider several Cauchy problems for the wave equation with some perturbation. First of all, we consider the wave equation with a metric perturbation, that is, we consider the d'Alembert operator in the Schwarzschild metric (which is a model for a static black hole). Because of the sign changing properties of the solution to this equation, it is not trivial to establish the global existence or the blow-up of the solution depending on the power of the nonlinearity. However, introducing suitable weighted average functions and proving some modified versions of the well-known Kato lemma, we are able to provide two blow-up results, one in the case of small data far from the black hole, and one when the initial data are close to the black hole but large (even this case is not trivial at all). In both cases, we restrict ourselves to radial solutions and power p less than 1+sqrt(2). We treat even the case of a linear wave equation with a potential-like perturbation. We consider a small electromagnetic potential depending on space and time with optimal decay properties, and null initial data. Under these assumptions, we can prove optimal dispersive estimates and in particular a 1/t decay in time. The proof exploits the gauge invariancy of the electromagnetic potential, which allows a suitable integral representation of the solution. The thesis also reviews some important known results concerning the previous problems and deals with related and open problems

Existence and Stability for the 3D Linearized Constant-Coefficient Incompressible Current-Vortex Sheets

We consider the free boundary problem for current-vortex sheets in ideal incompressible magnetohydrodynamics. The problem of current-vortex sheets arises naturally, for instance, in geophysics and astrophysics. We prove the existence of a unique solution to the constant-coefficient linearized problem and an a priori estimate with no loss of derivatives. This is a preliminary result to the study of linearized variable-coefficient current-vortex sheets, a first step to prove the existence of solutions to the nonlinear problem

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 $\alpha$-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

Video streaming over Bluetooth

In recent years, multimedia content has become more accessible to mobile phone devices increasing the demand for multimedia services. Streaming video to or from mobile phones over mobile phone operator networks is one option. In this paper we report on the result of a study which analyzes the suitability of using the Bluetooth network as a last hop network for streaming video to and from mobile phone devices. A number of studies have been reported in the literature, simulating video streaming over Bluetooth. However, few field studies have been reported fuelling the need to build an implementation infrastructure to conduct an empirical study using mobile phone devices, in video streaming applications. In our study we have implemented a testbed comprising a Linux-based Bluetooth video-streaming gateway and a Nokia mobile phone device to stream video clips and real-time video to and from the mobile phone over a Bluetooth connection, using both pre-recorded video and real-time streams from the mobile phone's onboard video camera. The testbed allows various Bluetooth network protocols and parameters to be tested in our framework. The work carried out reinforces the impor-tance of adequate packetization, which proved to be beneficial even with higher protocol layers such as the L2CAP protocol. The data throughputs achieved using Bluetooth v1.1 and Bluetooth v2.0 adapters were also compared and the effect of Wi-Fi interference proved to be detrimental to the performance of the Bluetooth network's data throughput. The use of L2CAP and RFCOMM sockets were compared, highlighting the importance of the choice of an adequate protocol. Video quality degradation at different distances when transferring video over Bluetooth was measured in terms of the mean square error metric. It was shown that the mobile phone device is indeed a resource constrained device and special care must be taken to ensure working streaming-video solutions over Blue-tooth.peer-reviewe

On the Boussinesq equations with anisotropic filter in a vertical pipe

We propose a new Large Eddy Simulation (LES) model for the Boussinesq equations. We consider the motion in a three-dimensional domain with solid walls, and in a particular geometric setting we look for solutions which are periodic in the vertical direction and satisfy homogeneous Dirichlet conditions on the lateral boundary. We are thus modeling a vertical pipe and one main difficulty is that of considering regularizations of the equation which are well behaved also in presence of a boundary. The LES model we consider is then obtained by introducing a vertical filter, which is the natural one for the setting that we are considering. The related interior closure problem is treated in a standard way with a simplified-Bardina deconvolution model. The most technical analytical point is related to the fact that anisotropic filters provide less regularity than the isotropic ones and, in principle, the density term appearing in the Boussinesq equations may behave very differently from the velocity. We are able to define an appropriate notion of regular weak solution, for which we prove existence, uniqueness, and we also show that the energy associated to the model is exactly preserved