Abstract
In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law ut + F(u)x = 0. First, we prove a simple but useful property of Lax-Oleinik formula (Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)−qF′(q)+ L(F′(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T*. Meanwhile, we can give the equation of the shock and an explicit value of T*.
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References
Lax P. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm Pure Appl Math, 1954, 7: 159–193
Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer, 1994
Liu H X, Pan T. Pointwise convergence rate of vanishing viscosity approximations for scalar conservation laws with boundary. Acta Math Sci, 2009, 29B(1): 111–128
Chen J, Xu X W. Existence of global smooth solution for scalar conservation laws with degenerate viscosity in 2-dimensional space. Acta Math Sci, 2007, 27B(2): 430–436
Kruzkov N. First-order quasilinear equations in several indenedent variables. Mat Sb, 1970, 123: 217–273
Ladyzenskaya O. On the construction of discontinuous solutions of quasilinear hyperbolic equations as a limit of solutions of the corresponding parabolic equations when the “viscosity coefficient” tends to zero. Dokl Adad Nauk SSSR, 1956, 111: 291–294 (in Russian)
Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18: 697–715
Chen G, Lu Y. A study on the applications of the theory of compensated compactness. Chinese Science Bulletin, 1988, 33: 641–644
Oleinik O. Discontinuous solutions of nonlinear differential equations. Usp Mat Nauk (NS), 1957, 12: 3–73
Hopf E. The partial differential equation u t + uu x = µu xx. Comm Pure Appl Math, 1950, 3: 201–230
Lax P. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957, 10: 537–566
Evans L C. Partial Differential Equations. Amer Math Society, 1997
Serre D. Systems of Conservaton Laws I. Cambridge University Press, 1999
Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Berlin: Springer-Verlag, 2010
Dafermos C M. Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Math J, 1977, 26: 1097–1119
Dafermos C M. Large time behaviour of solutions of hyperbolic balance laws. Bull Greek Math Soc, 1984, 25: 15–29
Dafermos C M. Generalized characteristics in hyperbolic systems of conservation laws. Arch Ration Mech Anal, 1989, 107: 127–155
Fan H, Jack K H. Large time behavior in inhomogeneous conservation laws. Arch Ration Mech Anal, 1993, 125: 201–216
Lyberopoulos A N. Asymptotic oscillations of solutions of scalar conservation laws with convexity under the action of a linear excitation. Quart Appl Math, 1990, 48: 755–765
Shearer M, Dafermos C M. Finite time emergence of a shock wave for scalar conservation laws. J Hyperbolic Differential Equations, 2010, 1: 107–116
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Zejun Wang’s research was supported in part by NSFC (11671193) and the Fundamental Research Funds for the Central Universities (NE2015005). Qi Zhang’s research was supported in part by NSFC (11271182 and 11501290).
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Wang, Z., Zhang, Q. Finite Time Emergence of A Shock Wave for Scalar Conservation Laws Via Lax-Oleinik Formula. Acta Math Sci 39, 83–93 (2019). https://doi.org/10.1007/s10473-019-0107-8
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DOI: https://doi.org/10.1007/s10473-019-0107-8