Local Existence and Uniqueness of Solutions to
Yang-Mills Heat Flow Problem

Duong Giao Ky

doi:https://doi.org/10.14445/22315373/IJMTT-V68I4P514

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 4 | Year 2022 | Article Id. IJMTT-V68I4P514 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I4P514

Local Existence and Uniqueness of Solutions to Yang-Mills Heat Flow Problem

Duong Giao Ky

Received	Revised	Accepted
18 Mar 2022	20 Apr 2022	30 Apr 2022

Abstract

In this paper, we consider the following nonlinear equation ∂u/∂t = ∂2ru+(d+1)/r ∂ru -3(d-2)u2-(d-2)r2u3 where u:(r,t) ɛ ℝ2 + |->ℝ, d ɛ N*. T his equation has been investigated by Grotowskiin 2001 in studying the Yang-Mills heat flow connections on Riemann manifolds. In the paper, we prove the local Cauchy problem for above equation that is well-posed in L∞1+r2( ℝ+). More precisely, for any initial data u0 ɛ L∞1+r2( ℝ+), there exists such that the above equation has a unique solution u(t) ɛ L∞1+r2( ℝ+) for all t ε[0, T(u0)].

Keywords

Local Cauchy problem, Local existence and uniqueness problem, Yang-Mills heat flows, Yang-Mills connections, Geometric flows.

References

[1] A. Actor, Classical Solutions of So(2)-Yang-Mills Theories, Rev. Mod. Phys. 51 (1979) 461-525.
[2] P. Biernat and Y. Seki, Type Ii Blow-Up Mechanism for Supercritical Harmonic Map Heat Flow, Int. Math. Res. Not. Imrn. 2 (2019) 407-456.
[3] R. Donninger, Stable Self-Similar Blowup In Energy Supercritical Yang-Mills Theory, Mathematischezeitschrift. 278 (2012) 1005-1032.
[4] R. Donninger, and B.Schörkhuber, Stable Blowup for the Supercritical Yang-Mills Heat Flow, Journal Differential Geometry. 113(1) (2019) 55-94.
[5] A. Gastel, Singularities of the First Kind In the Harmonic Map and Yang-Mills Heat Flows, Mathematischezeitschrift. 242(1) (2002) 47-62.
[6] J. F. Grotowski, Finite Time Blow-Up for the Yang-Mills Heat Flow In Higher Dimensions, Mathematischezeitschrift. 237(2) (2001) 321-333.
[7] S. Klainerman, and M. Machedon,Finite Energy Solutions of the Yang-Mills Equations In 3 1+, Annals of Mathematics. 142(1) (1995) 39-119.
[8] S. J. Oh,Gauge. Choice for the Yang-Mills Equations Using the Yang-Mills Heat Flow and Local Well-Posednessin 1 H ,Journal of Hyperbolic Differential Equations. 11(1) (2014) 1-108.
9] P. Quittner and P. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Birkhäuserverlag. Basel, (2007).
[10] T. Tao, Local Well-Posedness of the Yang-Mills Equation In the Temporal Gauge Below the Energy Norm, Journal Differential Equation. 189(2) (2003) 366-382.
[11] G. 'Thooft, 50 Years of Yang-Mills Theory, World Scientific Publishing Co., Pte. Ltd. Hackensack, (2005).
[12] O, Dumitrascu, Equivariant Solutions of the Yang-Mills Equations, Stud. Cerc. Mat. 34(4) (1982) 329-333.
[13] H. Kozono, Y. Maeda, and H. Naito, Global Solution for the Yang-Mills Gradient Flow on 4-Manifolds,Nagoya Mathematical Journal. 139 (1995) 93-128.
[14] J. Rade, on the Yang-Mills Heat Equation in Two and Three Dimensions, Journal Für Die Reine Und Angewandtemathematik. 431 (1992) 123-163.
[15] A. Schlatter, Global Existence of the Yang-Mills Flow in Four Dimensions, Journal Fürdiereine Und Angewandtemathematik.479 (1996) 133-148.
[16] M. Struwe,the Yang-Mills Flow In Four Dimensions, Calculus of Variations and Partial Differential Equations. 2(2) (1994) 123-150.
[17] B. Weinkove, Singularity Formation In the Yang-Mills Flow, Calculus of Variations and Partial Differential Equations. 19(2) (2004) 211-220.
[18] S.K. Donaldson, and P.B. Kronheimer, the Geometry of Four-Manifolds, Oxford Mathematical Monographs, the Clarendon Press Oxford University Press. New York, (1990).
[19] M.C. Hong, G. Tian, Asymptotical Behaviour of the Yang-Mills Flow and Singular Yang-Mills Connections, Math. Ann. 330 (2004) 441-472.
[20] A. Schlatter, M. Struwe, and A. Shaditahvildar-Zadeh. Global Existence of the Equivariant Yang-Mills Heat Flow in Four Space Dimensions, American Journal of Mathematics. 120(1) (1998) 117–128.
[21] A. Schlatter, Long-Time Behaviour of the Yang-Mills Flow In Four Dimensions, Annals of Global and Geometry. 15(1) (1997) 1-25.
[22] C. Kelleher, and J. Streets, Entropy, Stability, and Yang–Mills Flow, Communications in Contemporary Mathematics. 18(2) (2016) 1550032.
[23] P. Bizon, Formation of Singularities in Yang-Mills Equations, Actaphysicapolonica B. 33(7) (2002) 1893-1922.
[24] Z. Chen, and Y. Zhang, Stabilities of Homothetically Shrinking Yang-Mills Solitons, Trans. Amer. Math. Soc. 367(7) (2015) 5015–5041.
[25] C. Kelleher, and J. Streets, Singularity Formation of the Yang-Mills Flow, Ann. Inst. H. Poincaré Anal. Non Linéaire,. 35(6) (2018) 1655–1686.
[26] G. Tian, Gauge Theory and Calibrated Geometry I, Annals of Mathematics. 151 (2000) 193–268.
[27] Chen, Y., Shen, C.-L.: Monotonicity Formula and Small Action Regularity for Yang-Mills Flows in Higher Dimensions. Calc. Var. Pdes. 2 (1994) 389– 403.
[28] M. Atiyah, and R. Bott, the Yang-Mills Equations Over Riemann Surfaces, Philosophical Transactions of the Royal Society A. 308 (1982) 524–615

Citation :

Duong Giao Ky, "Local Existence and Uniqueness of Solutions to Yang-Mills Heat Flow Problem," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 4, pp. 87-93, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I4P514