# Weak Solution Of Yang-Mills Flow In Dimension N Greater Than Or Equal To 4.

### Overview

I show short time existence of weak solution of Yang-Mills Flow on SO(l) vector bundle eta over a closed connected Riemannian Manifold Mn with W2-4n,n2 initial data D0, when n ≥ 4. The Yang-Mills Flow is the gradient flow of Yang-Mills Functional, namely dDdt=-D*F D, 0.1 starting from some initial connection D0=D0. 0.2 The main result is the following:

Theorem 0.1. For any connection D 0 ∈ W2-4n,n2 (O¹(adeta)) on eta, there exists T > 0 and a weak solution D( t) = D1 + A( t) ...

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### Overview

I show short time existence of weak solution of Yang-Mills Flow on SO(l) vector bundle eta over a closed connected Riemannian Manifold Mn with W2-4n,n2 initial data D0, when n ≥ 4. The Yang-Mills Flow is the gradient flow of Yang-Mills Functional, namely dDdt=-D*F D, 0.1 starting from some initial connection D0=D0. 0.2 The main result is the following:

Theorem 0.1. For any connection D 0 ∈ W2-4n,n2 (O¹(adeta)) on eta, there exists T > 0 and a weak solution D( t) = D1 + A( t) to the Yang-Mills Flow (0.1) with initial condition (0.2) for 0 ≤ t ≤ T, such that A∈C0&parl0;0,T ,Ln2&parl0;W 1&parl0;adh&parr0;&parr0;&parr0;∩W1, n2&parl0;0,T ,Ln2&parl0;W 1&parl0;adh&parr0;&parr0;&parr0;, F∈C0&parl0;&parl0;0,T&sqbr0;,L n2&parl0;W2&parl0; adh&parr0;&parr0;&parr0;. Moreover D is gauge-equivalent to a smooth solution of equation (0.1) in the following sense: there is a solution Dˆ( t) = D1 + A( t) to equation (0.1) with A&d4;∈C0&parl0; 0,T,Ln2 &parl0;W1&parl0;adh&parr0;&parr0;&parr0; ∩W1,n2&parl0; 0,T,Ln2 &parl0;W1&parl0;adh&parr0;&parr0;&parr0; , and smooth for 0 < t < T, and a sequence of smooth gauge transformations S k ∈ G , such that Sk → S 0 in W1,n2 (Auteta), and S&d4;*0 (Dˆ) = D.

### Product Details

• ISBN-13: 9781244708471
• Publisher: BiblioLabsII
• Publication date: 9/30/2011
• Pages: 122
• Product dimensions: 7.44 (w) x 9.69 (h) x 0.26 (d)

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