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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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