Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary
The author considers the quasilinear initial-boundary-value symmetric hyperbolic first-order system \[ \alpha_0(x,u)\partial_{t}u + \sum_{i=1}^n{\alpha_i(x,u)\partial_{x_i}u} = f(x,u), \] \(x\in\mathbb R^{n}\), \(u\in\mathbb R^N\), \((t,x)\in[0,h]\times\Omega\subset\mathbb R^{n+1}\), \(f\) and \(\partial\Omega\) are sufficiently smooth. Initial condition is homogeneous (without loss of generality), boundary condition is \(Mu=0\). Other assumptions deal with properties of \(\alpha_i\) and the boundary. Existence of the unique smooth time-local solution is proved. The technique extends the theory of linear symmetric hyperbolic systems to the nonlinear case. Applications to gas dynamics are described.
existence and uniqueness of smooth solution
scientific article; zbMATH DE number 5790128
scientific article