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Blowup for systems of conservation laws. (English) Zbl 0969.35091
The author considers an initial value problem for a system of conservation laws $$U_t+ F(U)_x= 0$$, $$U(x,0)= U_0(x)$$, where $$U= U(x,t)\in \mathbb{R}^3$$, $$F: \mathbb{R}^3\to \mathbb{R}^3$$ is smooth and strictly hyperbolic. It is presented a class of $$3\times 3$$-systems for which one can prescribe initial data such that the solution blows up in finite time in the sense of sup-norm and total variation. The main statement is that there exist blow-up solutions $$U(x,t)$$ of the system under consideration for a special form of the flux function $$F(u,v,w)= (ua(v)+ w; \Gamma(v); u(\lambda^2_0- a^2(v))- wa(v))^t$$, where $$\Gamma$$ is strictly convex, $$-\lambda_0< \Gamma'(v)< \lambda_0$$, $$\Gamma(0)= 0$$ and $$\Gamma(-v)= \Gamma(v)$$ for all $$v\in\mathbb{R}$$. The blow-up is expressed by $$\lim_{t\to T^-}\|U(.,t)\|_\infty= +\infty$$ and $$\lim_{t\to T^-} T.V.[U(., t)]= +\infty$$, while $$\|U(.,t)\|_\infty< C$$ for all $$t< T$$ $$(C> 0)$$.

##### MSC:
 35L65 Hyperbolic conservation laws 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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