Abstract.
We construct weak solutions of 3×3 conservation laws which blow up in finite time. The system is strictly hyperbolic at every state in the solution, and the data can be chosen to have arbitrarily small total variation. This is thus an example where Glimm's existence theorem fails to apply, and it implies the necessity of uniform hyperbolicity in Glimm's theorem. Because our system is very simple, we can carry out explicit calculations and understand the global geometry of wave curves.
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References
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 697–715 (1965)
Kuczma, M., Choczewski, B., Ger, R.: Iterative functional equations. Encyclopedia of Mathematics, Cambridge University Press, 1990
Lax, P.D.: Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math. 10, 537–566 (1957)
Smoller, J.: Shock waves and reaction-diffusion equations. Springer-Verlag, New York, 1982
Young, R.: Sup-norm stability for Glimm's scheme. Comm. Pure Appl. Math. 46, 903–948 (1993)
Young, R.: Exact solutions to degenerate conservation laws. SIAM J. Math. Anal. 30, 537–558 (1999)
Young, R.: Blowup in hyperbolic conservation laws. Contemp. Math. 327, 379–387 (2003)
Young, R.: Blowup of solutions and boundary instabilities in nonlinear hyperbolic equations. Comm. Math. Sci. 2, 269–292 (2003)
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Communicated by C.M. Dafermos
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Young, R., Szeliga, W. Blowup with Small BV Data in Hyperbolic Conservation Laws. Arch. Rational Mech. Anal. 179, 31–54 (2006). https://doi.org/10.1007/s00205-005-0370-9
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DOI: https://doi.org/10.1007/s00205-005-0370-9