High frequency asymptotics for 2d viscous shocks (Q2733837)

This paper is devoted to the study of the asymptotic behaviour of solutions \(u\) of the linearized equations desribing a two-dimensional isentropic slightly viscous gas, that is NEWLINE\[NEWLINEu_t+ \sum^2_{j=1} A_j(x) u_{x_j}- \delta\Delta_x u=0,\quad t>0,\quad x= (x_1, x_2)\in \mathbb{R}^2,NEWLINE\]NEWLINE based on highly oscillatory initial perturbation of a planar stationary mean shock wave centered at \(x_1=0\). The author presents a rigorous geometric optics expansion NEWLINE\[NEWLINEu= e^{if/\varepsilon}(u_0+ \varepsilon u_1)NEWLINE\]NEWLINE with a detailed description of the principal term \(e^{if(x,t)/\varepsilon} u_0(x,t)= O(1)\) and its refraction pattern, caused by the viscous shock for different initial phase function \(if(x,0)/\varepsilon\) when the wave length \(\varepsilon\) is large compared to the width, \(\delta\), of the shock. The estimate of the remainder, NEWLINE\[NEWLINE\|e^{if(\cdot,t)/\varepsilon} \varepsilon\cdot u_1(\cdot, t)\|_{L^2(\mathbb{R}^2)}\ll 1,NEWLINE\]NEWLINE holds for \(t\in [0,1]\).