Mathematical Principles of Theoretical Physics

136 CHAPTER 3. MATHEMATICAL FOUNDATIONS

2.Leray decomposition.LetΩ∈Rnbe a domain, andu∈L^2 (TΩ)be ann-dimensional
vector field. Thenucan be decomposed as

(3.3.5)

u=∇φ+v,

v·n|∂Ω= 0 , divv= 0 , φ∈H^1 (Ω),

∫

Ω

∇φ·vdx= 0.

The Leray decomposition (3.3.5) is crucial in fluid dynamics.
The decompositions (3.3.4) and (3.3.5) can be generalized to more general tensor fields
as shown in (3.3.1)-(3.3.3). Now we discuss the simplest case to illustrate the main idea.
Letu:Rn→TRnbe a given vector field. Then divuis a known function. It is known
that the Poisson equation

(3.3.6) ∆φ=divu forx∈Rn

has a weak solutionφ∈H^1 (Rn), enjoying

(3.3.7)

∫

Rn

[∇φ−u]·∇φdx= 0 ∀φ∈H^1 (Rn).

Letv=u−∇φ. Then, by (3.3.7) we have
∫

R

v·∇φdx= 0 ,

which means that divv=0. Thus we obtain the orthogonal decompositionu=∇φ+vwith
divv=0.

3.3.2 Orthogonal decomposition theorems

The aim of this subsection is to derive an orthogonal decomposition for(k,r)-tensor fields,
withk+r≥1, into divergence-free and gradient parts. This decomposition plays a crucial
role for the unified field theory coupling four fundamental interactions to be introduced in
Chapter 4 of this book.
LetMbe a closed Riemannian manifold orM=S^1 ×M ̃be a closed Minkowski mani-
fold with metric (3.2.43), andv∈L^2 (TrkM) (k+r≥ 1 ). We say thatvis divA−free, denoted
by divAv=0, if

(3.3.8)

∫

M

(∇Aψ,v)

√

−gdx= 0 ∀∇Aψ∈L^2 (TrkM).

Hereψ∈H^1 (Trk−^1 M)orH^1 (Tr−k 1 M),∇Aand divAare as in (3.2.38).
We remark that ifv∈H^1 (TrkM)satisfies (3.3.8), thenvis weakly differentiable, and
divv=0 inL^2 -sense. Ifv∈L^2 (TrkM)is not differentiable, then (3.3.8) means thatvis
divA-free in the distribution sense.

Theorem 3.17(Orthogonal Decomposition Theorem).Let A be a given vector field or cov-
ector field, and u∈L^2 (TrkM). Then the following assertions hold true: