Proof of Local Existence and Uniqueness of Solutions to Non-commutative Yang-Mills Equations in Sobolev Spaces

This paper investigates the local existence and uniqueness of solutions to thenon-commutative Yang-Mills equations in Sobolev spaces. By introducing a mod?ified energy functional, gauge covariant artificial viscosity terms, and fractionalSobolev spaces, the energy estimates of non-commutative terms are optimized.Combined with the renormalization group equation and matrix model extension,the limitations of the small θ assumption and the adaptability of the Kato-Laitheory are addressed. Theoretical proof shows that in the Hsspace (s > 3/2),the equation satisfies the local Lipschitz condition, and the local existence anduniqueness of solutions are strictly derived through the Picard-Lindel¨of theorem.Furthermore, numerical simulations verify the stability of solutions under differentnon-commutative parameters θ, laying the foundation for the subsequent researchon global solutions and mass gaps.