REFINED CONSERVATION LAW FOR AN EVEN ORDER ELLIPTIC SYSTEM WITH ANTISYMMETRIC POTENTIAL

doi:10.1007/s10473-024-0604-2

Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (6): 2111-2124.doi: 10.1007/s10473-024-0604-2

Previous Articles Next Articles

REFINED CONSERVATION LAW FOR AN EVEN ORDER ELLIPTIC SYSTEM WITH ANTISYMMETRIC POTENTIAL

Changyu GUO^1,2, Changlin XIANG^3,†, Gaofeng ZHENG⁴

1. Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao 266237, China;
2. Department of Physics and Mathematics, University of Eastern Finland, Joensuu 80101, Finland;
3. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China;
4. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2023-10-08 Revised:2024-06-03 Published:2024-12-06
Contact: † Changlin XIANG, E-mail : changlin.xiang@ctgu.edu.cn
About author:Changyu GUO, E-mail : changyu.guo@sdu.edu.cn; Gaofeng ZHENG, E-mail : gfzheng@ccnu.edu.cn
Supported by:
Guo's research was supported by the Young Scientist Program of the Ministry of Science and Technology of China (2021YFA1002200), the National Natural Science Foundation of China (12101362) and the Natural Science Foundation of Shandong Province (ZR2021QA003). Xiang's research was supported by the National Natural Science Foundation of China (12271296) and the Natural Science Foundation of Hubei Province (2024AFA061). Zheng's research was supported by the National Natural Science Foundation of China (11571131) and the Open Research Fund of Key Laboratory of Nonlinear Analysis & Applications (Central China Normal University), Ministry of Education, P. R. China.

Abstract

CLC Number:

58E20

Changyu GUO, Changlin XIANG, Gaofeng ZHENG. REFINED CONSERVATION LAW FOR AN EVEN ORDER ELLIPTIC SYSTEM WITH ANTISYMMETRIC POTENTIAL[J].Acta mathematica scientia,Series B, 2024, 44(6): 2111-2124.

Trendmd

[1] Adams R C, Fournier J F. Sobolev Spaces. Amsterdam: Elsevier/Academic Press, 2003
[2] Bauer W, Isralowitz J. Compactness characterization of operators in the Toeplitz algebra of the Fock space. J Funct Anal, 2012, 263: 1323-1355
[3] Chang S Y A, Wang L, Yang P C. A regularity theory of biharmonic maps. Commun Pure Appl Math, 1999, 52: 1113-1137
[4] Coifman R, Lions P L, Meyer Y, Semmes S. Compensated compactness and Hardy spaces. J Math Pures Appl, 1993, 72: 247-286
[5] de Longueville F L, Gastel A. Conservation laws for even order systems of polyharmonic map type. Calc Var Partial Differential Equations, 2021, 60: Art 138
[6] Garrido-Atienza M J, Marín-Rubio P. Navier Stokes equations with delays on unbounded domains. Nonlinear Anal, 2006, 64: 1100-1118
[7] Gastel A, Scheven C. Regularity of polyharmonic maps in the critical dimension. Comm Anal Geom, 2009, 17: 185-226
[8] Guo C Y, Wang C, Xiang C L. $L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions. Calc Var Partial Differential Equations, 2023, 62: Art 31
[9] Guo C Y, Xiang C L. Regularity of solutions for a fourth order linear system via conservation law. J Lond Math Soc, 2020, 101: 907-922
[10] Guo C Y, Xiang C L. Regularity of weak solutions to higher order elliptic systems in critical dimensions. Tran Amer Math Soc, 2021, 374: 3579-3602
[11] Guo C Y, Xiang C L. Conservation law for harmonic mappings in supercritical dimensions. Comptes Rendus Mathématique, 2024, 362(G7): 769-773
[12] Guo C Y, Xiang C L, Zheng G F. The Lamm-Riviere system I: $L^p$ regularity theory. Calc Var Partial Differential Equations, 2021, 60: Art 213
[13] Guo C Y, Xiang C L, Zheng G F. $L^p$ regularity theory for even order elliptic systems with antisymmetric first order potentials. J Math Pure Appl, 2022, 165: 286-324
[14] Goldstein P, Strzelecki P, Zatorska-Goldstein A. On polyharmonic maps into spheres in the critical dimension. Ann Inst H Poincaré Anal Non Linéaire, 2009, 26: 1387-1405
[15] Hélein F. Harmonic Maps, Conservation Laws and Moving Frames. Cambridge: Cambridge University Press, 2002
[16] Hörter J, Lamm T. Conservation laws for even order elliptic systems in the critical dimensions-a new approach. Calc Var Partial Differential Equations, 2021, 60: Art 125
[17] Lamm T, Rivière T. Conservation laws for fourth order systems in four dimensions. Comm Partial Differential Equations, 2008, 33: 245-262
[18] Lamm T, Wang C. Boundary regularity for polyharmonic maps in the critical dimension. Adv Calc Var, 2009, 2: 1-16
[19] Málek J, Nečas J, Rokyta M, Ružička M. Weak and Measure-valued Solutions to Evolutionary PDE. New York: Champman-Hall, 1996
[20] Rivière T. Everywhere discontinuous harmonic maps into spheres. Acta Math, 1995, 175: 197-226
[21] Rivière T. Conservation laws for conformally invariant variational problems. Invent Math, 2007, 168: 1-22
[22] Rivière T. Analysis aspects of Willmore surfaces. Invent Math, 2008, 174: 1-45
[23] Rivière T. The role of integrability by compensation in conformal geometric analysis. Analytic aspects of problems in Riemannian geometry: elliptic PDEs, solitons and computer imaging. Soc Math France, Paris, Sémin Congr, 2011, 22: 93-127
[24] Rivière T. Conformally invariant variational problems. Lecture notes at ETH Zurich, available at https://people.math.ethz.ch/riviere/lecture-notes, 2012
[25] Rivière T, Struwe M. Partial regularity for harmonic maps and related problems. Comm Pure Appl Math, 2008, 61: 451-463
[26] Schwarz G. Hodge Decomposition--a Method for Solving Boundary Value Problems. Berlin: Springer-Verlag, 1995
[27] Sharp B, Topping P. Decay estimates for Rivière's equation, with applications to regularity and compactness. Trans Amer Math Soc, 2013, 365: 2317-2339
[28] Shatah J. Weak solutions and development of singularities of the SU(2) $\sigma$-model. Commun Pure Appl Math, 1988, 41: 459-469
[29] Struwe M. Partial regularity for biharmonic maps, revisited. Calc Var Partial Differential Equations, 2008, 33: 249-262
[30] Uhlenbeck K. Connections with $L^p$ bounds on curvature. Comm Math Phys, 1982, 83: 31-42
[31] Wang C Y. Biharmonic maps from $R^4$ into a Riemannian manifold. Math Z, 2004, 247: 65-87
[32] Wang C Y. Stationary biharmonic maps from $R^m$ into a Riemannian manifold. Comm Pure Appl Math, 2004, 57: 419-444
[33] Wente H C. An existence theorem for surfaces of constant mean curvature. J Math Anal Appl, 1969, 26: 318-344
[34] Xiang C L, Zheng G F. Sharpe Meorry regularity theory for a fourth order geometrical equation. Acta Math Sci, 2024, 44B(2): 420-430
[35] Ziemer W P. Weakly Differentiable Functions. New York: Springer-Verlag, 1989

REFINED CONSERVATION LAW FOR AN EVEN ORDER ELLIPTIC SYSTEM WITH ANTISYMMETRIC POTENTIAL

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Zihou ZHANG, Jing ZHOU. THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS [J]. Acta mathematica scientia,Series B, 2024, 44(2): 445-454.
[2]	Fen HE, Zhen WANG, Tingting CHEN. THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗ [J]. Acta mathematica scientia,Series B, 2023, 43(4): 1717-1734.
[3]	Fei WU, Zejun WANG, Fangqi CHEN. THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗ [J]. Acta mathematica scientia,Series B, 2023, 43(4): 1668-1674.
[4]	Jianli LIU, Jie XUE. GLOBAL SIMPLE WAVE SOLUTIONS TO A KIND OF TWO DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS [J]. Acta mathematica scientia,Series B, 2020, 40(1): 261-271.
[5]	Ting ZHANG, Wancheng SHENG. GLOBAL SOLUTIONS OF THE PERTURBED RIEMANN PROBLEM FOR THE CHROMATOGRAPHY EQUATIONS [J]. Acta mathematica scientia,Series B, 2019, 39(1): 57-82.
[6]	Zejun WANG, Qi ZHANG. FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA [J]. Acta mathematica scientia,Series B, 2019, 39(1): 83-93.
[7]	Shujun LIU, Fangqi CHEN, Zejun WANG. EXISTENCE OF GLOBAL L^∞ SOLUTIONS TO A GENERALIZED n×n HYPERBOLIC SYSTEM OF LEROUX TYPE [J]. Acta mathematica scientia,Series B, 2018, 38(3): 889-897.
[8]	Chunyan LIU, Zihou ZHANG, Yu ZHOU. A NOTE IN APPROXIMATIVE COMPACTNESS AND MIDPOINT LOCALLY K-UNIFORM ROTUNDITY IN BANACH SPACES [J]. Acta mathematica scientia,Series B, 2018, 38(2): 643-650.
[9]	Corrado MASCIA. TWENTY-EIGHT YEARS WITH “HYPERBOLIC CONSERVATION LAWS WITH RELAXATION” [J]. Acta mathematica scientia,Series B, 2015, 35(4): 807-831.
[10]	Debora AMADORI, Paolo BAITI, Andrea CORLI, Edda DAL SANTO. GLOBAL EXISTENCE OF SOLUTIONS FOR A MULTI-PHASE FLOW: A BUBBLE IN A LIQUID TUBE AND RELATED CASES [J]. Acta mathematica scientia,Series B, 2015, 35(4): 832-854.
[11]	Gaowei CAO, Kai HU, Xiaozhou YANG. FORMULA OF GLOBAL SMOOTH SOLUTION FOR NON-HOMOGENEOUS M-D CONSERVATION LAW WITH UNBOUNDED INITIAL VALUE [J]. Acta mathematica scientia,Series B, 2015, 35(2): 508-526.
[12]	Adil JHANGEER. CONSERVATION LAWS FOR THE (1 + 2)-DIMENSIONAL WAVE EQUATION IN BIOLOGICAL ENVIRONMENT [J]. Acta mathematica scientia,Series B, 2013, 33(5): 1255-1268.
[13]	Rinaldo M. Colombo, Magali L′ecureux-Mercier. NONLOCAL CROWD DYNAMICS MODELS FOR SEVERAL POPULATIONS [J]. Acta mathematica scientia,Series B, 2012, 32(1): 177-196.
[14]	Debora Amadori, M. Di Francesco. THE ONE-DIMENSIONAL HUGHES MODEL FOR EDESTRIAN FLOW: RIEMANN-TYPE SOLUTIONS [J]. Acta mathematica scientia,Series B, 2012, 32(1): 259-280.
[15]	Geng Chen, Robin Young. THE VACUUM IN NONISENTROPIC GAS DYNAMICS [J]. Acta mathematica scientia,Series B, 2012, 32(1): 339-351.