Related papers: On the global "two-sided" characteristic Cauchy pr…

We consider the classical Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to APS-boundary conditions. This is achieved by deriving suitable…

We study the Cauchy problem for classical and weak shock-forming solutions to a model quasilinear wave equation in $1+1$ dimensions arising from a convenient choice of $C^{\infty}$ initial data, which allows us to solve the equation using…

Four-dimensional spacetimes foliated by a two-parameter family of homologous two-surfaces are considered in Einstein's theory of gravity. By combining a 1+(1+2) decomposition, the canonical form of the spacetime metric and a suitable…

In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy…

In this conference published in 1997 some problems on the geodesics of a Lorentzian manifold concerning causality and infinite-dimensional variational methods, are pointed out. Even though a big progress on many of these questions have been…

We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and…

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with $(-1)$-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are…

We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier-Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically…

We consider a nonlinear, spatially-nonlocal initial value problem in one space dimension on $\mathbb{R}$ that describes the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth…

This paper is the second part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant $\Lambda$, with the data on the…

We consider nonlocal initial boundary value problems with integral boundary conditions for integro-differential first order hyperbolic systems. We prove a general regularity result stating that the $L^2$-generalized solutions become…

We equip the space of Cauchy hypersurfaces in a globally hyperbolic spacetime with a natural Hausdorff-type metric and study its properties, in particular completeness and local compactness, for Lorentzian manifolds and in more general…

Inthispaper,westudytheCauchyproblemoftheinhomogeneous Landau equation with hard potentials under the perturbation framework to global equilibrium. We prove that the solution to the Cauchy problem enjoys the analytic Gelfand-Shilov…

We give a construction that connects the Cauchy problem for Liouville elliptic equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H3, and solve both of them. We construct the only mean…

The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in $L^{p}$ framework. It is proved that the small global solutions constructed in $L^{2}$-Sobolev spaces in our preceding paper [12]…

Here we prove a global existence theorem for the solutions of the semi-linear wave equation with critical non-linearity admitting a positive definite Hamiltonian. Formulating a parametrix for the wave equation in a globally hyperbolic…

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in $H^s(M)$, $s<1/2$, where $M$ is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be…

We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem…

The paper starts by giving a motivation for this research and justifying the considered stochastic diffusion models for cosmic microwave background radiation studies. Then it derives the exact solution in terms of a series expansion to a…

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.