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Weakly stable multidimensional shocks. (English) Zbl 1072.35120
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.

##### MSC:
 35L67 Shocks and singularities for hyperbolic equations 35L50 Initial-boundary value problems for first-order hyperbolic systems 35L65 Hyperbolic conservation laws 76L05 Shock waves and blast waves in fluid mechanics 35L35 Initial-boundary value problems for higher-order hyperbolic equations 76N15 Gas dynamics (general theory)
##### Keywords:
symmetrizers; linear stability; isentropic gas dynamics
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##### References:
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