# zbMATH — the first resource for mathematics

Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations. (English) Zbl 1317.49050
Summary: We show that in one space dimension, Lipschitz solutions of generalized extremal surface equations are equivalent to entropy solutions in $$L^{\infty}(\mathbb{R})$$ of a non-strictly hyperbolic system of conservation laws. We obtain an explicit representation formula and the uniqueness of the entropy solutions to the Cauchy problem of the system. By using this formula, we also obtain the convergence and convergence rates as $$t \to +\infty$$ of the entropy solutions to explicit traveling waves in the $$L^1(\mathbb{R})$$ norm. Moreover, when initial data are constants outside of a finite space interval, the entropy solutions become the explicit traveling waves after a finite time. Finally, we prove $$L^{1}$$ stabilities of the entropy solutions.
 [1] Boillat, G., Nonlinear Hyperbolic Fields and Waves, 1640, 1-47, (1996), Springer: Springer, New York · Zbl 0877.35080 [2] Boillat, G.; Ruggeri, T., Energy momentum, wave velocities and characteristic shocks in Euler’s variational equations with application to the Born-Infeld theory, J. Math. Phys., 45, 3468, (2004) · Zbl 1071.83019 [3] Bouchut, F.; Perthame, B., Kruzkov’s estimates for scalar conservation laws revisited, Trans. Am. Math. Soc., 350, 7, 2847, (1998) · Zbl 0955.65069 [4] Brenier, Y., Some geometric PDEs related to hydrodynamics and electrodynamics, Proceedings of the International Congress of Mathematicians, 3, 761-772, (2002), Higher Education Press: Higher Education Press, Beijing · Zbl 1136.37355 [5] Brenier, Y., Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rat. Mech. Anal., 172, 65, (2004) · Zbl 1055.78003 [6] Brenier, Y., A note on deformations of 2D fluid motions using 3D Born-Infeld equations, Monatsh. Math., 142, 113, (2004) · Zbl 1063.35130 [7] Bressan, A., Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem, 20, (2000), Oxford University Press: Oxford University Press, Oxford · Zbl 0997.35002 [8] Chae, D.; Huh, H., Global existence for small initial data in the Born-Infeld equations, J. Math. Phys., 44, 6132, (2004) · Zbl 1063.81042 [9] Chen, G. Q., The method of quasidecoupling for discontinuous solutions to conservation laws, Arch. Rational Mech. Anal., 172, 131, (1992) · Zbl 0797.35112 [10] Gibbons, G. W., Aspects of Born-Infeld theory and string/M-theory, . [11] Gibbons, G. W.; Herdeiro, C. A. R., Born-Infeld theory and stringy causality, Phys. Rev., D63, 064006, (2001) · Zbl 0980.83020 [12] Glimm, J.; Lax, P. D., Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws, 101, (1970), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI · Zbl 0204.11304 [13] John, F., Nonlinear Waves Equations, Formation of Singularities, 2, (1990), American Mathematical Society: American Mathematical Society, Providence, RI [14] Kong, D. X.; Sun, Q. Y.; Zhou, Y., The equation for time-like extremal surfaces in Minkowski space \documentclass[12pt]minimal\begindocument$${\bb R}^{2+n}$$\enddocument, J. Math. Phys., 47, 1, 013503-1, (2006) · Zbl 1111.53053 [15] Kong, D. X.; Yang, T., Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Commun. Partial. Differ. Equ., 28, 1203, (2003) · Zbl 1024.35068 [16] Kruzkov, S. N., First order quasilinear equations in several independent variables, Mater. Sbornik (N.S.), 81, 228, (1970) · Zbl 0191.39703 [17] Lax, P. D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5, 611, (1964) · Zbl 0135.15101 [18] Li, T. T., Global Classical Solutions for Quasilinear Hyperbolic Systems, 32, (1994), Masson: Masson, Paris · Zbl 0841.35064 [19] Li, T. T.; Peng, Y. J.; Ruiz, J., Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl., 91, 553, (2009) · Zbl 1177.37043 [20] Lindblad, H., A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time, Proc. Am. Math. Soc., 132, 1095, (2004) · Zbl 1061.35053 [21] Liu, J. L.; Zhou, Y., Asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Models Meth. Appl. Sci., 30, 479, (2007) · Zbl 1128.35023 [22] Liu, T. P., Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Commun. Pure Appl. Math., 30, 767, (1977) · Zbl 0358.35014 [23] Peng, Y. J., Explicit solutions for 2 × 2 linearly degenerate systems, Appl. Math. Lett., 11, 75, (1998) · Zbl 0941.35047 [24] Peng, Y. J., Euler-Lagrange change of variables in conservation laws and applications, Nonlinearity, 20, 1927, (2007) · Zbl 1132.35060 [25] Polchinski, J., String Theory, I, (1998), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1006.81521 [26] Serre, D., Richness and the classification of quasilinear hyperbolic systems, Multidimensional Hyperbolic Problems and Computations, (1989), Minneapolis: Minneapolis, MN; Serre, D., Richness and the classification of quasilinear hyperbolic systems, Multidimensional Hyperbolic Problems and Computations, (1989), Minneapolis: Minneapolis, MN; · Zbl 0760.35028 [27] Serre, D., Systèmes de Lois de Conservation I-II, (1996), Diderot: Diderot, Paris [28] Serre, D., Hyperbolicity of the nonlinear models of Maxwell’s equations, Arch. Ration. Mech. Anal., 172, 309, (2004) · Zbl 1065.78005 [29] Wagner, D. H., Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Diff. Equations, 68, 118, (1987) · Zbl 0647.76049