Global classical solution to partially dissipative quasilinear hyperbolic systems (Q764913)

The paper is devoted to the global existence of solutions to the Cauchy problem for one-dimensional quasilinear strictly hyperbolic systems \[ \partial_t U+ A(U)\partial_x U= F(U),\quad U(0,x)= U_0(x). \] The initial datum is supposed to be small in a weighted \(C^1\) space. The authors divide the index set \(\{1,\dots, n\}\) into the two subsets \({\mathcal P}\) and \({\mathcal L}\). For the indices belonging to \({\mathcal P}\) the authors suppose a strict dissipation condition for the matrix \(L(U)\nabla F(U)R(U)\), where \(L(U)\), \(R(U)\) denotes the matrix of left, right eigenvectors, respectively. For the indices belonging to \({\mathcal L}\), the authors assume a weak linear degeneracy condition. The main result is the unique global existence of a classical small data solution. For the proof the authors transform the system into a new one with normalized coordinates, use weighted formulas of wave decomposition and energy estimates coupled with a continuation argument. Moreover, decay behavior of the waves is discussed. Finally, some applications to gas dynamics are given.