Related papers: Hyperbolic extensions of constrained PDEs
In this paper we consider the Cauchy problem for higher order weakly hyperbolic equations. We assume that the principal symbol depends only on one space variable and the characteristic roots $\tau_j$ verify the inequality \[\tau_j^2(x) +…
The global solutions in critical spaces to the multi-dimensional compressible viscoelastic flows are considered. The global existence of the Cauchy problem with initial data close to an equilibrium state is established in Besov spaces.…
Assuming as starting point the validity of the Einstein-Rosen metric, we study the hyperbolic system of P.D.E. to which the Einstein's field's equations can be reduced. We prove using the implicit function theorem in Banach spaces, the…
We prove the well posedness of a class of non linear and non local mixed hyperbolic-parabolic systems in bounded domains, with Dirichlet boundary conditions. In view of control problems, stability estimates on the dependence of solutions on…
We consider the Cauchy problem for first order systems. Assuming that the set of the singular points of the characteristic variety is a smooth manifold and the characteristic values are real and semi-simple we introduce a new class which is…
This review examines classical and recent results on controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions. We aim to illustrate the applicability of Carleman estimates to establish…
In this paper, we study higher order hyperbolic pseudo-differential equations with variable multiplicities. We work in arbitrary space dimension and we assume that the principal part is time-dependent only. We identify sufficient conditions…
General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the…
This is essentially a survey paper in which we solve the global Cauchy problem on causal manifolds for hyperbolic systems of linear partial differential equations in the framework of hyperfunctions. Besides the classical Cauchy-Kowalevsky…
Quantum dynamics of a particle in the vicinity of a hyperbolic point is considered. Expectation values of dynamical variables are calculated, and the singular behavior is analyzed. Exponentially fast extension of quantum dynamics is…
This paper studies the feedback stabilization of abstract Cauchy problems with unbounded output operators by finite-dimensional controllers. Both necessary conditions and sufficient conditions for feedback stabilizability are presented. The…
In this paper, we would like to consider the Cauchy problem for a multi-component weakly coupled system of semi-linear $\sigma$-evolution equations with double dissipation for any $\sigma\ge 1$. The first main purpose is to obtain the…
In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are…
We consider gauge theories from the free evolution point of view, in which initial data satisfying constraints of a theory are given. Because the constraints are compatible with the field equations they remain so. We study a model…
In this paper, we establish the existence of solutions for a particular class of degenerate hyperbolic equations. Following this, we approximate these degenerate equations by employing a sequence of uniformly hyperbolic equations. Notably,…
We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable…
The Equation Problem in finitely presented groups asks if there exists an algorithm which determines in finite amount of time whether any given equation system has a solution or not. We show that the Equation Problem in central extensions…
We consider small nonlinear perturbations of linear systems on a time scale with the phase space being finite or infinite-dimensional. For $\Delta$-differential operators, corresponding to linear dynamic systems we consider their…
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 define and study hyperbolic extensions.