Related papers: The Initial Value Problem for Wave Equation and a …
The first part of my thesis lays the foundations to generalized Lorentz geometry. The basic algebraic structure of finite-dimensional modules over the ring of generalized numbers is investigated. The motivation for this part of my thesis…
Solutions of the Cauchy problem for the wave equation on a non-globally hyperbolic spacetime, which contains closed timelike curves (time machines) are considered. It is proved, that there exists a solution of the Cauchy problem, it is…
Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of…
The initial value problem of scalar-tensor theories of gravity (STT) is analyzed in the physical (Jordan) frame using a 3+1 decomposition of spacetime. A first order strongly hyperbolic system is obtained for which the well posedness of the…
In this manuscript, we put forth a general scheme for defining initial value problems from Einstein's equations of General Relativity constrained by homogeneous and isotropic expansion. The cosmological models arising as solutions are…
The 3D-primitive equations with only horizontal viscosity are considered on a cylindrical domain $\Omega=(-h,h) \times G$, $G\subset \mathbb{R}^2$ smooth, with the physical Dirichlet boundary conditions on the sides. Instead of considering…
We devise a new time-stepping algorithm for two-dimensional nonlinear unsteady surface and interfacial waves. The algorithm uses Cauchy's integral formula, which only requires information on the interface, to solve Laplace equation by using…
The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in…
We cast the four-dimensional field equations of the Nonsymmetric Gravitational Theory (NGT) into a form appropriate for numerical study. In doing so, we have restricted ourselves to spherically symmetric spacetimes, and we have kept only…
The aim of this thesis is to derive new gradient estimates for parabolic equations. The gradient estimates found are independent of the regularity of the initial data. This allows us to prove the existence of solutions to problems that have…
In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin-Ono equation $$\left. \begin{array}{rl} u_t+\mathcal H \Delta u +uu_x &\hspace{-2mm}=0,\qquad\qquad (x,y)\in\mathbb T^2,\; t\in\mathbb…
This paper considers to the problems of diffraction of electromagnetic waves on a half-plane, which has a finite inclusion in the form of a Lipschitz curve. The diffraction problem formulated as boundary value problem for Helmholtz…
Computational techniques which establish the stability of an evolution-boundary algorithm for a model wave equation with shift are incorporated into a well-posed version of the initial-boundary value problem for gravitational theory in…
The Cauchy problem is considered for the scalar wave equation in the Schwarzschild geometry. We derive an integral spectral representation for the solution and prove pointwise decay in time.
We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schroedinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic…
In this paper, we deal with the initial value problem for a class of fully nonlinear parabolic equations with a singular Dirichlet boundary condition in one space dimension. The interior equation includes, for example, a fully nonlinear…
We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form $[0,T] \times \Sigma$, where $\Sigma$ is a compact manifold with smooth boundaries $\partial\Sigma$. By using an appropriate…
In this paper, we study the characteristic initial value problem for a class of nonlinear wave equations with data on a conic light cone in the Minkowski space $\mathbb{R}^{1+3}$. We show the existence of local solution for a class of…
In this work, we study the initial-value problem associated with the Kuramoto-Sivashinsky equation. We show that the associated initial value problem is locally and globally well-posed in Sobolev spaces $H^s(\mathbb{R})$, where $s>1/2$. We…
Whether the 3D compressible Navier-Stokes-Poisson equations admit global classical solutions for general large initial data has long been a challenging open problem. In this paper, we provide an affirmative answer to this question under…