Weakly stable multidimensional shocks
The author considers the system of \(N\) conservation laws in \(\mathbb R \times \mathbb R ^d\) (time-space) \[ \sum\limits_{j=0}^{d}\partial_jf_j(u)=0, \] where \(x_0\equiv t\) is the time variable, \(x\equiv (x_1,x_2,\dots,x_d)\) is the space variable and \(\partial_j\equiv \partial /\partial x_j\). The fluxes \(f_0,f_1,\dots,f_d\) are \(C^{\infty }\) functions defined on an open set \(U\subset \mathbb R ^N\) with values in \(\mathbb R ^N\). The author studies the linear stability of multidimensional shock waves for the system under consideration in the case where Majda's uniform stability condition is violated. An energy estimate is obtained. It is shown that the solutions to the linearized problem have singularities localized along bicharacteristic curves originating from the boundary. The application to isentropic gas dynamics is discussed.
scientific article; zbMATH DE number 2117889
scientific article