The Hölderian solution and its singularity propagation for a class of system of quasi-linear partial differential equations (Q1902245)

We study the following system of quasilinear PDEs: \[ \partial_t U+ (U\cdot \nabla) U= F(t, x, U),\tag{1} \] where \(U\) and \(F\) are the vector- valued functions which are defined on \(\mathbb{R}^+\times \mathbb{R}^N\) and \(\mathbb{R}^+\times \mathbb{R}^N\times \mathbb{R}^N\), respectively, \(U= (u_1,\dots, u_N)\), \(F= (f_1,\dots, f_N)\), \(\nabla= (\partial_{x_1},\dots, \partial_x)_N\). Moreover, \(F\) is smooth enough, and its derivatives of all orders are bounded on \(\mathbb{R}\times \mathbb{K}\). \(\mathbb{K}\) is any compact set in \(\mathbb{R}^{2N}\). Obviously, the system (1) can be regarded as one kind of generalizations of Burger's equation and also an approximation of a class of fluid equations. For the convenience of notations, we take \(N= 2\), and the conclusions of the note are still true for all \(N\geq 2\). We first discuss the existence of the \(C^\rho\) solution for (1), and then study the propagation of Hölderian singularity for the solution -- both the microlocal and the conormal types.