The crease set of an event horizon or a Cauchy horizon is an important object which determines qualitative properties of the horizon. In particular, it determines the possible topologies of the spatial sections of the horizon. By Fermat’s principle in geometric optics, we relate the crease set and the Maxwell set of a smooth function in the context of singularity theory. We thereby give a class...

In this paper, we prove in two dimensions global identifiability of the viscosity in an incompressible fluid by making boundary measurements. The main contribution of this work is to use more natural boundary measurements, the Cauchy forces, than the Dirichlet-to-Neumann map previously considered in [7] to prove the uniqueness of the viscosity for the Stokes equations and for the Navier-Stokes ...

We prove the existence of viable solutions to the Cauchy problem x′′ ∈ F (x, x′), x(0) = x0, x′(0) = y0, where F is a set-valued map defined on a locally compact set M ⊂ R, contained in the Fréchet subdifferential of a φ-convex function of order two.

We present a formulation of the incompressible viscous Navier-Stokes equation based on a generalization of the inviscid Weber formula, in terms of a diffusive “back-to-labels” map and a virtual velocity. We derive a generalization of the inviscid Cauchy formula and obtain certain bounds for the objects introduced.

We introduce a suitable generalization of the momentum map for nonholonomic field theories and prove a covariant form of the nonholonomic momentum equation. We show that these covariant objects coincide with their counterparts in mechanics by making the transition to the Cauchy formalism.

Let F be a nonlinear Fréchet differentiable map in a real Hilbert space. Condition sufficient for existence of a solution to the equation F (u) = 0 is given, and a method (dynamical systems method, DSM) to calculate the solution as the limit of the solution to a Cauchy problem is justified under suitable assumptions.

The main results of this paper are concerned with the “bad” behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class C2 in any Sobolev space.

We consider a Cauchy problem for a Sturm-Liouville type differential inclusion involving a nonconvex set-valued map and we prove that the set of selections corresponding to the solutions of the problem considered is a retract of the space of integrable functions on unbounded interval.

In this paper we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in ${\mathbb{R}^3}$. As application study when the euclidean Gauss map has harmonic argument solve general Cauchy problem class of surfaces.

The N-extended supersymmetric self-dual Poincaré supergravity equations provide a natural local description of supermanifolds possessing hyperkähler structure. These equations admit an economical formulation in chiral superspace. A reformulation in harmonic superspace encodes self-dual supervielbeins and superconnections in a graded skew-symmetric supermatrix superfield prepotential satisfying ...