关于Hamilton-Jacobi方程解的正则性和全局结构的注记——纪念李国平院士吴新谋教授诞辰100周年

Abstract

Abstract:

The paper is concerned with the Cauchy problem for the Hamilton-Jacobi equations of multi-dimensional space variables. We prove the sufficient and necessary condition for that a characteristic emanating from a given point never touches the set of singularity points is that the initial function attains its minimum at this point. Finally, we prove there exists one-to-one correspondence between the path connected components of the set of singularity points and the path connected components of the set on which the initial function does not attain
its minimum. Each path connected component of the set of the singularity points never terminates at a finite time. In particularly, it is worth pointing out that our results are obtained without assuming that the gradient of the initial function tends to zero at infinity under which the important proposition 2.7 and theorem 3.3, one of the main results in [12] are obtained.

Key words: Hamilton-Jacobi equations, Characteristic, Singularity point, Global structure

CLC Number:

35L60

WANG Jing-Hua, ZHAO Yin-Chuan, WEN Hai-Rui. Remarks on the Regularity and Global Structure of Solutions[J].Acta mathematica scientia,Series A, 2010, 30(5): 1283-1287.

Trendmd

References

[1] Barron E N, Cannasa P, Jensen R, Sinestrari C. Regularity of Hamilton-Jacobi equations when forward is backward. Indiana Univ Math J, 1999, 48: 395--409

[2] Bardi M, Evans L C. On Hopf's formulas for solutions of Hamilton-Jacobi equations. Nonlinear Anal, Theory, Methods \& Applications, 1984, 8: 1373--1381

[3] Conway E, Hopf E. Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation. J Math Mech, 1964, 13: 939--986

[4] Evans L C. Partial Differential Equations. Providence, RI: American Mathematical Society, 1998

[5] Hopf E. Generalized solutions of nonlinear equations of first order. J Math Mech, 1965, 14: 951--973

[6] Hoang N. The regularity of generalized solutions of Hamilton-Jacobi equations. Nonlinear Anal, 2004, 59: 745--757

[7] Kruzkov S N. Generalized solutions of nonlinear first order equations with several variables. II Math Sbornik, 1967, 1: 93--116

[8] Lions P L. Generalized Solutions of Hamilton-Jacobi Equations. London: Pitman Advance Publishing Program, 1982

[9] Van T D, Hoang N, Thai Son N D. Explicit global Lipschitz solutions to first-order nonlinear partial differential equations. Viet J Math, 1999, 27(2): 93--114

[10] Van T D, Hoang N, Tsuji M. On Hopf's formula for Lipschitz solutions of the Cauchy problem for Hamilton-Jacobi equations. Nonlinear Anal, 1997, 29(10): 1145--1159

[11] Van T D, Tsuji M, Thai Son N D. The characteristic method and its generalizations for first order nonlinear PDEs. London/Boca Raton, Fl: Chapman & Hall/CRC, 2000

[12] Zhao Y, Tang T, Wang J. Regularity and global structure of solutions to Hamilton-Jacobi equations I. Convex Hamiltonian. J Hyperbol Differ Eq, 2008, 5(3): 663--680

[13] Tang T, Wang J H, Zhao Y. Regularity and global structure of solutions to Hamilton-Jacobi equations II. Convex initial data. J Hyperbol Differ Eq, 2009, 6(4): 709--723

[1]	Kuang Jie,Wang Zejun. The Large Time Behavior of Inhomogeneous Burgers Equation with Periodic Initial Data [J]. Acta mathematica scientia,Series A, 2015, 35(1): 1-14.
[2]	LI Xue-Quan, TAN Wei-Ping, CAI Hai-Tao. A Fundamental Inequality in the Theory of Meromorphic Functions [J]. Acta mathematica scientia,Series A, 2010, 30(5): 1288-1291.
[3]	LI Wei-Feng, DU Jin-Yuan. Characteristic Singular Integral Equations on the Real Axis [J]. Acta mathematica scientia,Series A, 2010, 30(2): 405-410.
[4]	KONG Ling-Hua, ZHAO Li-Zhong, ZHENG Shi-Ning. Asymptotic Analysis to Parabolic Equations with Absorptions and Coupled Localized Sources [J]. Acta mathematica scientia,Series A, 2010, 30(1): 267-277.
[5]	Yuan Yirang. The Modified Characteristic Finite Difference Fractional Steps Method for Nonlinear Coupled System of Multilayer Fluid Dynamics in Porous Media [J]. Acta mathematica scientia,Series A, 2009, 29(4): 858-872.
[6]	Wu Zhaojun; Sun Daochun. On a New Singular Direction of Algebroidal Function [J]. Acta mathematica scientia,Series A, 2008, 28(5): 879-885.
[7]	Chen Chuanjun; Yuan Yirang. Quadratic Finite Volume Element Method Along Characteristics for the Semiconductor Device of Heat Conduction: H¹Error Estimates [J]. Acta mathematica scientia,Series A, 2008, 28(3): 412-423.
[8]	Sun Jingxian ;Li Hongyu. Global Structure of Positive Solutions of Singular Nonlinear Sturm-Liouville Problems [J]. Acta mathematica scientia,Series A, 2008, 28(3): 424-433.
[9]	Jang Wei. Distribution of Characteristic Roots and Exponential Stability of Singular Differential Delay Systems [J]. Acta mathematica scientia,Series A, 2007, 27(6): 1006-1012.
[10]	Wu Tianjiao. On Applications of Wu's Method in Bilevel-Programming Problems [J]. Acta mathematica scientia,Series A, 2007, 27(1): 176-183.
[11]	YAO Wei-Hong, SHANG De-Sheng, YU Min-Jie. Global Structure Analysis of the Plane Five Order System with the Stellar Node [J]. Acta mathematica scientia,Series A, 2003, 23(2): 193-207.
[12]	Chao Lu. Wuwen-tsun-Differential Charateristic Algorithm of Symmetry Vectors of Partial Differential Equations [J]. Acta mathematica scientia,Series A, 1999, 19(3): 326-332.
[13]	Yuan Yirang. The alternating birection schemes and numerical analysis for three dimensional transport accumulation (oil and water) [J]. Acta mathematica scientia,Series A, 1999, 19(2): 196-205.
[14]	Sun Daochun. The Random Dirichlet Series on the Right Half-Plane [J]. Acta mathematica scientia,Series A, 1999, 19(1): 107-112.
[15]	Shi Jinlin . On the Characteristic Exponent of Two Dimensional Linear Differential Equations with Periodic Coefficient [J]. Acta mathematica scientia,Series A, 1999, 19(1): 67-72.

Remarks on the Regularity and Global Structure of Solutions

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0