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Let say I have a hyperbolic system of conservation law. How do I show there is a blow up in finite time? For a single conservation law, I think, I could just show that there is collision of characteristics given a certain velocity for example $a(u)=u$ in Burgers' equation.

I would also like to know good books or any other references on this topic (blow up in finite time for hyperbolic conservation law). I have with me books by Lax, Evans, and Leveque (Finite Volume ...). Thanks.

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  • $\begingroup$ Please specify what you mean by blowup, because for example solutions to Burger's equation remain bounded and $L^1$ if they are so initially. See Lax, Hyperbolic Partial Differential Equations. $\endgroup$
    – Bob Terrell
    Commented Dec 13, 2016 at 22:56
  • $\begingroup$ What I meant was the situation where the smooth solution stops to exist, shock develops etc. I am quite new with this topic and I think the term blow up that I use is correct. Anyway, I have little experience with numerical computing in which the term blow up is used when some values become very large. @BobTerrell $\endgroup$
    – mohd
    Commented Dec 14, 2016 at 0:11

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For general hyperbolic systems in one space and on time dimension, the result is treated in

John, F.
Formation of singularities in one-dimensional nonlinear wave propagation
Comm. Pure Appl. Math., 1974, 27, 377-405

The higher dimensional case is not completely understood at present. There's a lot of work by Alinhac on what he terms "geometric blowup", which is the direct analogue of intersections of characteristics in the one spatial dimension case.

Alinhac, S.
Blowup for nonlinear hyperbolic equations
Birkhäuser Boston Inc., 1995, xiv+113

Alinhac, S.
A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations
Journées ``Équations aux Dérivées Partielles'' (Forges-les-Eaux, 2002), Univ. Nantes, 2002, Exp. No. I, 33

One of the reasons that higher dimensional hyperbolic systems are complicated is that there the equation is dispersive. Dispersion gives a decay mechanism that can compete against the self-resonance (take a derivative of the Burger's equation you get Riccati) driving blow-up. And sometimes you win, and sometimes you lose.

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