# Finite Time Emergence of A Shock Wave for Scalar Conservation Laws Via Lax-Oleinik Formula

## Abstract

In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law *u*_{t} + *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**.

## Key words

scalar conservation law Lax-Oleinik formula Riemann problem asymptotic behavior## 2010 MR Subject Classification

35B40 35L65 76Y05## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Lax P. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm Pure Appl Math, 1954,
**7**: 159–193MathSciNetCrossRefzbMATHGoogle Scholar - [2]Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer, 1994CrossRefzbMATHGoogle Scholar
- [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): 111–128MathSciNetzbMATHGoogle Scholar - [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): 430–436MathSciNetCrossRefzbMATHGoogle Scholar - [5]Kruzkov N. First-order quasilinear equations in several indenedent variables. Mat Sb, 1970,
**123**: 217–273CrossRefGoogle Scholar - [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)Google Scholar - [7]Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965,
**18**: 697–715MathSciNetCrossRefzbMATHGoogle Scholar - [8]Chen G, Lu Y. A study on the applications of the theory of compensated compactness. Chinese Science Bulletin, 1988,
**33**: 641–644Google Scholar - [9]Oleinik O. Discontinuous solutions of nonlinear differential equations. Usp Mat Nauk (NS), 1957,
**12**: 3–73Google Scholar - [10]Hopf E. The partial differential equation
*u*_{t}+*uu*_{x}=*µu*_{xx}. Comm Pure Appl Math, 1950,**3**: 201–230MathSciNetCrossRefGoogle Scholar - [11]Lax P. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957,
**10**: 537–566MathSciNetCrossRefzbMATHGoogle Scholar - [12]Evans L C. Partial Differential Equations. Amer Math Society, 1997Google Scholar
- [13]Serre D. Systems of Conservaton Laws I. Cambridge University Press, 1999CrossRefGoogle Scholar
- [14]Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Berlin: Springer-Verlag, 2010CrossRefzbMATHGoogle Scholar
- [15]Dafermos C M. Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Math J, 1977,
**26**: 1097–1119MathSciNetCrossRefzbMATHGoogle Scholar - [16]Dafermos C M. Large time behaviour of solutions of hyperbolic balance laws. Bull Greek Math Soc, 1984,
**25**: 15–29zbMATHGoogle Scholar - [17]Dafermos C M. Generalized characteristics in hyperbolic systems of conservation laws. Arch Ration Mech Anal, 1989,
**107**: 127–155MathSciNetCrossRefzbMATHGoogle Scholar - [18]Fan H, Jack K H. Large time behavior in inhomogeneous conservation laws. Arch Ration Mech Anal, 1993,
**125**: 201–216MathSciNetCrossRefzbMATHGoogle Scholar - [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**: 755–765MathSciNetCrossRefGoogle Scholar - [20]Shearer M, Dafermos C M. Finite time emergence of a shock wave for scalar conservation laws. J Hyperbolic Differential Equations, 2010,
**1**: 107–116MathSciNetCrossRefzbMATHGoogle Scholar