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*.
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
.
DOI: 10.1007/s10473-019-0107-8
[1] Lax P. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm Pure Appl Math, 1954, 7:159-193
[2] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York:Springer, 1994
[3] 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
[4] 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
[5] Kruzkov N. First-order quasilinear equations in several indenedent variables. Mat Sb, 1970, 123:217-273
[6] 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)
[7] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18:697-715
[8] Chen G, Lu Y. A study on the applications of the theory of compensated compactness. Chinese Science Bulletin, 1988, 33:641-644
[9] Oleinik O. Discontinuous solutions of nonlinear differential equations. Usp Mat Nauk (NS), 1957, 12:3-73
[10] Hopf E. The partial differential equation ut + uux=μuxx. Comm Pure Appl Math, 1950, 3:201-230
[11] Lax P. Hyperbolic systems of conservation laws Ⅱ. Comm Pure Appl Math, 1957, 10:537-566
[12] Evans L C. Partial Differential Equations. Amer Math Society, 1997
[13] Serre D. Systems of Conservaton Laws I. Cambridge University Press, 1999
[14] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Berlin:Springer-Verlag, 2010
[15] Dafermos C M. Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Math J, 1977, 26:1097-1119
[16] Dafermos C M. Large time behaviour of solutions of hyperbolic balance laws. Bull Greek Math Soc, 1984, 25:15-29
[17] Dafermos C M. Generalized characteristics in hyperbolic systems of conservation laws. Arch Ration Mech Anal, 1989, 107:127-155
[18] Fan H, Jack K H. Large time behavior in inhomogeneous conservation laws. Arch Ration Mech Anal, 1993, 125:201-216
[19] 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
[20] Shearer M, Dafermos C M. Finite time emergence of a shock wave for scalar conservation laws. J Hyperbolic Differential Equations, 2010, 1:107-116
