Related papers: Cauchy Problem for the Dirac operator on spatially…
We discuss the application of recent results on generalized solutions to the Cauchy problem for hyperbolic systems to Dirac equations with external fields. In further analysis we focus on the question of existence of associated…
For a Dirac theory of quantum gravity obtained from the refined algebraic quantization procedure, we propose a quantum notion of Cauchy surfaces. In such a theory, there is a kernel projector for the quantized scalar and momentum…
The author gives an alternative and simple proof of the global existence of smooth solutions to the Cauchy problem for wave maps from the 1+2-dimensional Minkowski space to an arbitrary compact smooth Riemannian manifold without boundary,…
This thesis is devoted to the study of hyperbolic differential operators on globally hyperbolic manifolds, linear gauge theories and their quantisation. In the first part, we treat the Cauchy problem for symmetric hyperbolic systems and…
Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form $(D+\lambda W)f=0$ are studied, where $D$ is a normal or prenormal hyperbolic differential operator on…
We consider supersymmetric quantum mechanical systems in arbitrary dimensions on curved spaces with nontrivial gauge fields. The square of the Dirac operator serves as Hamiltonian. We derive a relation between the number of supercharges…
We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u +…
We consider the Cauchy problem for a system of fully nonlinear parabolic equations. In this paper, we shall show the existence of global-in-time solutions to the problem. Our condition to ensure the global existence is specific to the fully…
This chapter is an up-to-date account of results on globally hyperbolic spacetimes, and serves several purposes. We begin with the exposition of results from a foundational level, where the main tools are order theory and general topology,…
We establish global existence and derive sharp pointwise decay estimates of solutions to cubic Dirac and Dirac-Klein-Gordon systems on a curved background, close to the Minkowski spacetime. By squaring the Dirac operator, we reduce the…
We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove that there exists…
After the heroic epoch of Causality Theory, problems concerning the smoothability of time functions and Cauchy hypersurfaces remained as unanswered folk questions. Just recently solved, our aim is to discuss the state of the art on this…
We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the…
In this paper we give a survey on how to apply recent techniques of Clifford analysis over conformally flat manifolds to deal with instationary flow problems on cylinders and tori. Solutions are represented in terms of integral operators…
For hyperbolic differential operators $P$ with non-effectively hyperbolic double characteristics, we study the relationship between the Gevrey well-posedness threshold for strong well-posedness and the associated Hamilton map and flow. In…
It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact…
In this paper, we establish a sharp $C^{2+\alpha}$-theory for stochastic partial differential equations of parabolic type in the whole space.
We propose a geometric approach to construct the Cauchy evolution operator for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic 4-manifolds. We realise the Cauchy evolution operator as the sum of two invariantly defined…
We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic H\"{o}lder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator…
We introduce a class of causal manifolds which contains the globally hyperbolic spacetimes and we prove global propagation theorems for sheaves on such manifolds. As an application, we solve globally the Cauchy problem for hyperfunction…