Electron. J. Diff. Eqns., Vol. 1995(1995), No. 09, pp. 1-15.

Singularity Formation in Systems of Non-strictly Hyperbolic Equations

R. Saxton & V. Vinod

Abstract:
We analyze finite time singularity formation for two systems of hyperbolic equations. Our results extend previous proofs of breakdown concerning $2\times 2$ non-strictly hyperbolic systems to $n \times n$ systems, and to a situation where, additionally, the condition of genuine nonlinearity is violated throughout phase space. The systems we consider include as special cases those examined by Keyfitz and Kranzer and by Serre. They take the form
$$ u_{t} + (\phi(u)u)_{x} = 0, $$
where $\phi$ is a scalar-valued function of the n-dimensional vector $u$, and
$$ u_{t}+\Lambda(u)u_{x} = 0, $$
under the assumption $\Lambda = {\rm diag}\, \{\lambda^{1},\ldots,\lambda^{n}\}$ with $\lambda^{i}=\lambda^{i}(u-u^{i})$, where $u-u^{i}\equiv\{u^{1},\ldots,u^{i-1},u^{i+1},\ldots,u^{n}\}$.

Submitted June 12, 1995. Published: June 28, 1995.
Math Subject Classification: 35L45, 35L65, 35L67, 35L80.
Key Words: Finite time breakdown, non-strict hyperbolicity, linear degeneracy.

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R. Saxton (rsaxton@math.uno.edu)
V. Vinod (vvinod@math.uno.edu)
Department of Mathematics, University of New Orleans, New Orleans, LA 70148, USA
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