## No $L_ 1$-contractive metrics for systems of conservation laws

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- by Blake Temple
- Trans. Amer. Math. Soc.
**288**(1985), 471-480 - DOI: https://doi.org/10.1090/S0002-9947-1985-0776388-5
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## Abstract:

Let $(\ast )$ \[ \quad {u_t} + F{(u)_x} = 0\] 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 \[ u(x,0) = \left \{ {\begin {array}{*{20}{c}} {{u_1},} \hfill & {x \leqslant 0,} \hfill \\ {{u_2},} \hfill & {0 < x < 1,} \hfill \\ {{u_1},} \hfill & {x \geqslant 1.} \hfill \\ \end {array} } \right .\] 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

- Neal R. Amundson,
*Mathematical methods in chemical engineering*, Prentice-Hall International Series in the Physical and Chemical Engineering Sciences, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. Volume 1: Matrices and their application. MR**0371183**
R. Courant and K. O. Friedricks, - James Glimm,
*Solutions in the large for nonlinear hyperbolic systems of equations*, Comm. Pure Appl. Math.**18**(1965), 697–715. MR**194770**, DOI 10.1002/cpa.3160180408
E. Isaacson, - Barbara Keyfitz Quinn,
*Solutions with shocks: An example of an $L_{1}$-contractive semigroup*, Comm. Pure Appl. Math.**24**(1971), 125–132. MR**271545**, DOI 10.1002/cpa.3160240203 - 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**, DOI 10.1007/BF00281590 - P. D. Lax,
*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**93653**, DOI 10.1002/cpa.3160100406 - Peter Lax,
*Shock waves and entropy*, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Publ. Math. Res. Center Univ. Wisconsin, No. 27, Academic Press, New York, 1971, pp. 603–634. MR**0393870**
T. P. Liu and C. H. Wang, - Blake Temple,
*Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws*, Adv. in Appl. Math.**3**(1982), no. 3, 335–375. MR**673246**, DOI 10.1016/S0196-8858(82)80010-9
—,

*Supersonic flow and shock waves*, Wiley, New York, 1948.

*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). F. Helfferich and G. Klein,

*Multicomponent chromatography*, Dekker, New York, 1970.

*On a hyperbolic system of conservation laws which is not strictly hyperbolic*, MRC Technical Summary Report # 2184, December 29, 1980. D. W. Peaceman,

*Fundamentals of numerical reservoir simulation*, Elsevier North-Holland, New York, 1977. H. Rhee, R. Aris and N. R. Amundson,

*On the theory of multicomponent chromatography*, Philos. Trans. Roy. Soc. London Ser. A

**267**(1970), 419.

*Systems of conservation laws with invariant submanifolds*, Proc. Amer. Math. Soc. (to appear). M. Walsh, S. Bryant, R. Schechter and L. Lake,

*Precipitation and dissolution of solids attending flow through porous media*, University of Texas Preprint, 1982.

## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**288**(1985), 471-480 - MSC: Primary 35L65; Secondary 76L05, 76N10
- DOI: https://doi.org/10.1090/S0002-9947-1985-0776388-5
- MathSciNet review: 776388