1,721,004 research outputs found
Linear and Nonlinear Perturbed Wave Equations
No full textWe 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
Dispersive estimates for a linear wave equation with electromagnetic potential
No full textWe consider radial solutions to the Cauchy problem for a linear wave equation with a small short-range electromagnetic potential (depending on space and time) and zero initial data. We present two dispersive estimates that provide, in particular, an optimal decay rate in time for the solution. Also, we apply these estimates to obtain similar results for the linear massless Dirac equation perturbed by a potential
Existence and Stability for the 3D Linearized Constant-Coefficient Incompressible Current-Vortex Sheets
Get PDFWe 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
Remarks on Global Attractors for the 3D Navier-Stokes Equations with horizontal filtering
No full textLarge time behaviour of solutions to the semilinear wave equation in Schwarzschild metric
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