Related papers: Energy estimate for initial data on a characterist…

In this work a new numerical technique to prepare Cauchy data for the initial value problem (IVP) formulation of Einstein's field equations is presented. Directly inspired by the exterior asymptotic gluing (EAG) result of Corvino (2000) our…

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere $S$ with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small…

We prove the well-posedness of the initial boundary value problem for the Einstein equations with sole boundary condition the requirement that the timelike boundary is totally geodesic. This provides the first well-posedness result for this…

In Einstein theory of gravity the initial configuration of metric field and its time derivative are related to matter configuration by four equations called constraints. We use the method of conformal metrics (York Method) to solve…

In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In…

The details are presented of a new evolution algorithm for the characteristic initial-boundary value problem based upon an affine parameter rather than the areal radial coordinate used in the Bondi-Sachs formulation. The advantages over the…

We consider the Cauchy problem in R^n for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L^{1,1}(R^n) initial data by employing a simple method introduced by the first author. The obtained…

For compact, isometrically embedded Riemannian manifolds $ N \hookrightarrow \mathbb{R}^L$, we introduce a fourth-order version of the wave map equation. By energy estimates, we prove an $\textit{a priori}$ estimate for smooth local…

A combination of some weighted energy estimates is applied for the Cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions. Alternative proofs for global solutions are shown including the…

We consider the Einstein-Maxwell system as a Cauchy initial value problem taking the electric and magnetic fields as independent variables. Maxwell's equations in curved spacetimes are derived in detail using a 3+1 formalism and their…

We consider an inverse problem for the elastic wave of simultaneously reconstructing the impedance and the geometric information of the bounded body that is occupied by a homogeneous and isotropic elastic medium from the measured Cauchy…

We construct high-frequency initial data for the Einstein vacuum equations in dimension 3+1 by solving the constraint equations on $\mathbb{R}^3$. Our family of solutions $(\bar{g}_\lambda,K_\lambda)_{\lambda\in(0,1]}$ is defined through a…

We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for…

We prove a logarithmic stability estimate for the inverse problem of determining the potential in a wave equation from boundary measurements obtained by varying the first component of the initial condition. The novelty of the present work…

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that,…

It was shown recently that the constraints on the initial data for Einstein's equations may be posed as an evolutionary problem [9]. In one of the proposed two methods the constraints can be replaced by a first order symmetrizable…

Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in…

We prove a weighted a priori energy estimate for the two dimensional water-waves problem with contact points in the absence of gravity and surface tension. When the surface graph function and its time derivative have some decay near the…

We explore whether a new method to solve the constraints of Einstein's equations, which does not involve elliptic equations, can be applied to provide initial data for black holes. We show that this method can be successfully applied to a…

We consider the Cauchy problem for the system of elastodynamic equations in two dimensions. Specifically, we focus on materials characterized by a null condition imposed on the quadratic part of the nonlinearity. We can construct non-zero…