^{1}

^{*}

^{1}

We show that Yang-Mills equation in 3 dimensions is local well-posedness in
*H*
^{s} if the norm is sufficiently. Here, we construct a solution on the quadric that is independent of the time. And we also construct a solution of the polynomial form. In the process of solving, the polynomial is used to solve the problem before solving.

This paper is concerned with the solution of the Yang-Mills equation.

We shall denote

The curvature of a connection

Here [,] denotes the Lie bracket of

We can expand this as

where

The Cauchy problem for Yang-mills equation is not well-posed because of gauge invariance (see [

and

where

The local well-posedness of the Equations (1) and (2) have already proved in [

Below we will construct the exact solution of the equation on the general quadric that denotes by

where

We bring (3) to Equation (2), because the equation is used in the two general surfaces, we define the general quadric by

Divergence terms can be

Finally, the sections of Lie bracket can be

Combining the above calculations we have

We will use the properties of polynomials to list the coefficient equations in order to solve the (3). For the cross terms and square terms coefficient, we have

First, we consider

The constant coefficient equation is

The coefficient equation of

The coefficient equation of

The coefficient equation of

Because of the (4), the coefficient equation of constant can be

Deformation by (6), we have

Simulaneous (8) and (10), we have

First, for (9) we can use mathematica to get

where

Next, from (11) we can obtain

where

We can observe the above

Because of

By two surfaces we can obtain

Similarly, we can prove that

In summary, when the Equation (2) is acting on the quadric, we have

Below we construct a polynomial solution. First, the constant must satisfy the equation so that all constant are the solutions of the Equation (1) and (2). Then we define the solution of a polynomial form on a surface by

where

Equation (12) is composed of three equations. First we consider the case of

The coefficient equation of

The coefficient equation of

The coefficient equation of

When

When

There exist 12 equations. By solving the above equations, we can obtain

Therefore

where

In summary, the solution of the polynomial form of Yang-Mills equation is expressed in the form of (13).

In this section, we mainly discuss the solution of the quadratic polynomial form of the Yang-Mills equation on the two surfaces. We define by

where

There exist 30 equations and 30 unknowns. Solving the equations we can obtain the following results

So the solution of the equation can be written

where

In summary, the solution of the quadratic polynomial form of Yang-Mills equation is (14). It obvious that (13) is equal to (14). So we conjecture that the solution of n-degree polynomial on n-sub surface is also (14). In the next section, we will proof the hypothesis.

In this section, we mainly use mathematical induction to prove the hypothesis. We define that by

where

In the front two sections, it is easy for us to conclude that when

First, we assume that when

where

Now when

To further simplify (15), we have

To bring into the equation, we have

where

On the number of

And the number of more than

In summary, we can get the solution of the polynomial type of Yang-Mills equation by mathematical induction is

where

Zhu, P. and Ding, L.Y. (2017) The Solution of Yang-Mills Equations on the Surface. Applied Mathematics, 8, 35-43. http://dx.doi.org/10.4236/am.2017.81004