The Global Formulation of the Cauchy Problem

A Geometrical model for the global Cauchy problem, generalizing the traditional Cauchy problem is considered .The complete correspondence between the known analytical formulation and the geometrical interpretation is described, we have utilized the generalized Green's function and the open mapping theorem appropriate to the problem.


Introduction
In this paper we discuss the global formulation of the Cauchy problem (Bar, Ginoux, & Pfaffe, 2007;Minguzzi & Sánchez, 2008), and its solution for globally hyperbolic space time (Beem, Ehrlich, & Easley, 1996;O'neill, 1983).Also we discuss the role of open mapping theorem (Bär & Ginoux, 2012;Kreyszig, 1989), in our solution, because of its various properties.The open mapping theorem seems to be a good tool for investigating that for general maps between topological spaces.For the formulation of the global Cauchy problem we need to know two kinds of structure, the first is a time orientation which separates future from past (Bär & Fredenhagen, 2009), the second ingredient is that of a hyper surface Σ in which we can specify the initial values.In order to approach the global existence of solutions we assume that M is globally hyperbolic with a smooth spacelike Cauchy hyper surface Σ .

For every M p∈
we have a unique time t p with t Σ ∈ , on each t Σ (Mühlhoff, 2011), we also have a Riemannian  (Stakgold & Holst, 2011),   Let G be a domain in the (n-1)-dimensional subspace  ) ( ) For ,...., , and f , g are sufficiently smooth functions defined in G. Conditions (2),(3) are called Cauchy conditions or initial conditions f , g are called Cauchy data and the system (1), ( 2) and ( 3) is called a Cauchy problem, G is called the initial manifold.In the IVP G is the hypersurface obtained by the intersection of the n-dimensional region T and the hyperplane 0 = n x . An initial domain may not be a proper subset of the boundary, for example in 2 E consisting of point (x ,t) , the initial domain may be 0 = t or a subset of it.In general elliptic equations are associated with boundary conditions and hyperbolic and parabolic equations with initial conditions.
The D' Alembert's solution to the Cauchy problem is The solution exists, is unique and depends continuously on the data ( ) x f and ( ) . Hence the Cauchy problem for the wave equation is well-posed.

M U ⊆
be an open subset.Then U is called causal if there is a geodesically convex open subset

A Causal and Achronal Subsets
Let M A ⊆ be a subset of a time-oriented Lorentz manifold.Then A is called i.) a chronal if every timelike curve intersects A in at most one point.
ii.) a causal if every causal curve intersects A in at most one point

(Theorem) A Chronal Hyper Surfaces
Let ( ) g M , be a time-oriented Lorentz manifold and M A ⊆ a chronal.Then A is a topological hyper surface in M if and only if A does not contain any of its edge points.

Cauchy Hyper Surface
Let ( ) is called a Cauchy hyper surface if every inextensible timelike curve meets Σ in exactly one point.

Cauchy development
Let M A ⊆ be a subset.The future Cauchy development of A is the set of all those points M p∈ for which every past-inextensible causal curve through p also meets A Alogously, one defines the past Cauchy (5) the Cauchy development of A .

Globally Hyperbolic Spacetime
A time-oriented Lorentz manifold ( ) 2.10.Time Function (Baer & Strohmaier, 2015) Let ( ) g M , be a time-oriented Lorentz manifold and a continuous function.Then t is called a i.) time function if t is strictly increasing along all future directed causal curves.ii.) temporal function if t is smooth and grad t is future directed and timelike.i.) Cauchy time function if t is a time function whose level sets are Cauchy hypersurfaces.v.) Cauchy temporal function if t is a temporal function such that all level sets are Cauchy hyper surfaces.Vol. 8, No. 3;20162.11. Theorem(Baer & Strohmaier, 2015), Let ( ) g M , be a connected time-oriented Lorentz manifold.Then the following statements are equivalent: i.) ( ) g M , is globally hyperbolic.ii.)There exists a topological Cauchy hypersurface.iii.)There exists a smooth spacelike Cauchy hypersurface.
In this case there even exists a Cauchy temporal function t and ( ) of the temporal function t is a smooth spacelike Cauchy hypersurface.

Proposition
Let ( ) g M , be a time-oriented Lorentz manifold with a smooth spacelike hyper surface is a Cauchy hyper surface for U .Then there exists a unique solution and given inhomogeneity ) . in addition we have ( ) be a globally hyperbolic and let M  Σ : ι be a smooth spacelike Cauchy hypersurface with future directed normal vector field . Assume that is a solution to the wave equation 0 (11) Moreover to develop our constructing we assume that Μ is globally hyperbolic with Σ is smooth spacelike.
where γ is an at least piecewise curve joining inside .considerits Cauchy development is a nice open neighbourhood of p allowing a local fundamental solution [4].

Lemma,(Waldmann, 2012)
The function Is well defined and lower semi-continuous.

Lemma
is well-defined and lower semi-continuous, where , and Then we can use the proposition 3.1 to obtain smooth solution ,      For the past of t we already know that u w = whence in total φ is continuous.Take the result of theorem(2.5) ., )

(Theorem )
Let ( )   with the following property: for every test section


is a smooth section of E depending continuously on u and satisfying (28) yield advanced or retarded Green operator for D, respectively.ii.) Assume ± Μ G are advanced or retarded Green operator for D , respectively.Then ii.)The duals of the Green operators restrict to maps

Conclusion
The formulation of the Cauchy problem in Euclidean space with specified boundary condition is well known.In that formulation the traditional Green's function is involved in the construction of the solution.However one need a generalization of the Cauchy problem to spaces that are not Euclidean, such as Lorentzian manifolds, with pseudo-Riemannian metric .The consideration of this problem in such a geometrical Lorentzian manifold has very important impact on wave propagation with applications cosmic wave, Thus we have treated the formulation of Cauchy

(
of the function ro 2 θ and the choice of δ we see that

[
proof Green Functions and Cauchy Problem in this part we, show the Well-posedness of the Cauchy problem with respect to the usual locally convex topologies of smooth linear map.If φ is surjective then φ is an open map.As usual, a map φ is called open if the images of open subsets are again open 4 linear map (19) sending the initial conditions and the inhomogeneity to the corresponding solution of the Cauchy problem is continuous.

(
of global advanced or retarded fundamental solutions of T D at every point Μ ∈ P The Green operators have unique * weak continuous extensions to operators

(
∪ be the solution according to Proposition 3.1. 29)defines a family of advanced and retarded fundamental solutions of