Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm
This paper deals with the Cauchy problem to the following quasilinear first-order strictly hyperbolic system NEWLINE\[NEWLINE{\partial u\over\partial t}+ A(u){\partial u\over\partial x}= 0,\quad u(0,x)= f(\varepsilon, x),\quad x\in\mathbb{R}^1,\quad t\geq 0,\tag{1}NEWLINE\]NEWLINE where \(f(0,x)= 0\), \(f\) is sufficiently smooth with respect to \(x\in\mathbb{R}^1\) and \(0<\varepsilon\ll 1\). Put NEWLINE\[NEWLINE\psi(x)= {\partial f\over\partial\varepsilon}(0, x)\in (C^1(\mathbb{R}))^n.NEWLINE\]NEWLINE The author proves that if the system (1) is not weakly linearly degenerate, \(JA(u)\) is suitably smooth near \(u= 0\), \(\| f\|_{L_1}+\| f_x'\|_{L_1}\leq C.\varepsilon\) and \(\psi^\alpha_{i_0} \psi_{i_0}'(x_0)\) has appropriate sign for some integers \(\alpha\), \(i_0\) and \(x_0\in\mathbb{R}\), then there exists \(\varepsilon_0> 0\) and such that for each fixed \(\varepsilon\in (0,\varepsilon_0)\) the first-order derivatives of the \(C^1\) solution \(u(t,x)\) of (1) must blow-up in finite time. Moreover, there exists a sharp limit formula for the life span \(\widetilde T(\varepsilon)\) of \(u\), namely \(\lim_{\varepsilon\to 0} (\varepsilon^{\alpha+1}\widetilde T(\varepsilon))^{-1}\) is expressed by \(\sup_x (A(\alpha)\psi^\alpha_i(x) \psi_i'(\alpha))\), \(A(\alpha)\) being constant independent of \(x\).
scientific article; zbMATH DE number 6072011