THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS

doi:10.1016/S0252-9602(11)60398-6

Acta mathematica scientia,Series B ›› 2011, Vol. 31 ›› Issue (6): 2265-2277.doi: 10.1016/S0252-9602(11)60398-6

• Articles • Previous Articles Next Articles

THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS

Wang Jinghua¹, Wen Hairui², Zhao Yinchuan³

1.Academy of Mathematics and System Sciences, The Chinese Academy of Sciences, Beijing 100190, China；2.Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China；3.Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Received:2011-09-20 Online:2011-11-20 Published:2011-11-20
Contact: Wen Hairui,wenhr@bit.edu.cn E-mail:jwang@amss.ac.cn；wenhr@bit.edu.cn；zyc747250@hotmail.com
Supported by:
This research was supported by National Natural Science Foundation of China (10871133, 11071246 and 11101143), Fundamental Research Funds of the Central Universities (09QL48). †Corresponding author: Wen Hairui.

Abstract

Abstract:

We use Hopf-Lax formula to study local regularity of solution to Hamilton-Jacobi (HJ) equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T > 0, which depends only on the Hamiltonian and initial datum, for t > T the solution of the IVP (1.1) is smooth except for a smooth n-dimensional hypersurface, across which Du(x, t) is discontinuous. And we show that the hypersurface tends asymptotically to a given hypersurface with rate t^−1/4 .

Key words: Hopf-Lax formula, Hamilton-Jacobi equations, local regularity, large time generic form

CLC Number:

49L25

Wang Jinghua, Wen Hairui, Zhao Yinchuan. THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS[J].Acta mathematica scientia,Series B, 2011, 31(6): 2265-2277.

Trendmd

References

[1] Aubin J P, Frankowska H. Set-valued Analysis. Boston: Birkhauser, 1990

[2] Bardi M, Evans L C. On Hopf’s formulas for solutions of Hamilton-Jacobi equations. Nonlinear Anal TMA, 1984, 8: 1373–1381

[3] 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

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

[5] Evans L C. Partial Differential Equations. Amer Math Soc, 1998

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

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

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

[9] Lax P D. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957, 10: 537–566

[10] Li B, Wang J. The global qualitative study of solutions to a conservation law (I). Sci Special Math Issue, 1979: 12–24 (in Chinese)

[11] Li B, Wang J. The global qualitative study of solutions of a conservation law (II). Sci Special Math Issue, 1979: 25–38 (in Chinese)

[12] Lin C T, Tadmor E. L1 Stability and error estimates for approximate Hamilton-Jacobi solutions. Numer Math, 2001, 87: 701–735

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

[14] Schaeffer D G. A regularity theorem for conservation laws. Adv Math, 1973, 11: 368–386

[15] Tang T. Wang J, Zhao Y. On the piecewise smoothness of entropy solutions to scalar conservation laws for a larger class of initial data. J Hyperbol Differ Equ, 2007, 4: 369–389

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

[17] Van T D, Tsuji M, Son N D T. The Characteristic Method and Its Generalizations for First Order Nonlinear Partial Differential Equations. London, Boca Raton: Chapman & Hall/CRC, 2000

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

THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 3

Metrics

Comments

Recommended 0

[1]	Jae-Myoung KIM. LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS [J]. Acta mathematica scientia,Series B, 2017, 37(4): 1033-1047.
[2]	GAO Hong-ya, GUO Jing, ZUO Ya-Li, CHU Yu-Ming. LOCAL REGULARITY RESULT IN OBSTACLE PROBLEMS [J]. Acta mathematica scientia,Series B, 2010, 30(1): 208-214.
[3]	GAO Gong-E, TIAN Hui-Yang. LOCAL REGULARITY RESULT FOR SOLUTIONS OF OBSTACLE PROBLEMS [J]. Acta mathematica scientia,Series B, 2004, 24(1): 71-74.