Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



No $ L\sb 1$-contractive metrics for systems of conservation laws

Author: Blake Temple
Journal: Trans. Amer. Math. Soc. 288 (1985), 471-480
MSC: Primary 35L65; Secondary 76L05, 76N10
MathSciNet review: 776388
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (\ast)$

$\displaystyle \quad {u_t} + F{(u)_x} = 0$

be any $ 2 \times 2$ system of conservation laws satisfying certain generic assumptions on $ F$ in a neighborhood $ \mathcal{N}$ of $ u$-space. We prove that for every nondegenerate metric $ D$ on $ u$-space there exists states $ {u_1}$ and $ {u_2}$ in $ \mathcal{N}$ such that $ \int_{ - \infty }^\infty {D(u(x,t),{u_1})\;dx} $ is a strictly increasing function of $ t$ in a neighborhood of $ t = 0$, where $ u$ is the admissible solution of $ ( \ast )$ with initial data

$\displaystyle u(x,0) = \left\{ {\begin{array}{*{20}{c}} {{u_1},} \hfill & {x \l... ...,} \hfill \\ {{u_1},} \hfill & {x \geqslant 1.} \hfill \\ \end{array} } \right.$

This contrasts with the case of a scalar equation in which $ \int_{ - \infty }^\infty {D(u(x,t),v(x,t))\;dx} $ is a decreasing function of $ t$ for all admissible solution pairs $ u$ and $ v$ when $ D$ is taken to be the absolute value norm.

References [Enhancements On Off] (What's this?)

  • [1] Neal R. Amundson, Mathematical methods in chemical engineering, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. Volume 1: Matrices and their application; Prentice-Hall International Series in the Physical and Chemical Engineering Sciences. MR 0371183
  • [2] R. Courant and K. O. Friedricks, Supersonic flow and shock waves, Wiley, New York, 1948.
  • [3] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 0194770,
  • [4] E. Isaacson, Global solution of a Riemann problem for a non-strictly hyperbolic system of conservation laws arising in enhanced oil recovery, J. Comp. Phys. (to appear).
  • [5] F. Helfferich and G. Klein, Multicomponent chromatography, Dekker, New York, 1970.
  • [6] Barbara Keyfitz Quinn, Solutions with shocks: An example of an 𝐿₁-contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125–132. MR 0271545,
  • [7] Barbara L. Keyfitz and Herbert C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1979/80), no. 3, 219–241. MR 549642,
  • [8] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653,
  • [9] Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603–634. MR 0393870
  • [10] T. P. Liu and C. H. Wang, On a hyperbolic system of conservation laws which is not strictly hyperbolic, MRC Technical Summary Report # 2184, December 29, 1980.
  • [11] D. W. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier North-Holland, New York, 1977.
  • [12] H. Rhee, R. Aris and N. R. Amundson, On the theory of multicomponent chromatography, Philos. Trans. Roy. Soc. London Ser. A 267 (1970), 419.
  • [13] Blake Temple, Global solution of the Cauchy problem for a class of 2×2 nonstrictly hyperbolic conservation laws, Adv. in Appl. Math. 3 (1982), no. 3, 335–375. MR 673246,
  • [14] -, Systems of conservation laws with invariant submanifolds, Proc. Amer. Math. Soc. (to appear).
  • [15] M. Walsh, S. Bryant, R. Schechter and L. Lake, Precipitation and dissolution of solids attending flow through porous media, University of Texas Preprint, 1982.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L65, 76L05, 76N10

Retrieve articles in all journals with MSC: 35L65, 76L05, 76N10

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society