In this paper, we study the nonlinear stability of travelling wave solutions with shock profile for a relaxation model with a nonconvex flux, which is proposed by Jin and Xin [<i>Comm. Pure Appl. Math.</i>, 48 (1995), pp. 555--563] to approximate an original hyperbolic system numerically under the subcharacteristic condition introduced by T. P. Liu [<i>Comm. Math. Phys.</i>, 108 (1987), pp. 153--175]. The travelling wave solutions with strong shock profile are shown to be asymptotically stable under small disturbances with integral zero using an elementary but technical energy method. Proofs involve detailed study of the error equation for disturbances using the same weight function introduced in [<i>Comm. Math. Phys.</i>, 165 (1994), pp. 83--96].

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In this paper we consider the well-posedness of the compressible Prandtl boundary layer for the Navier–Stokes–Poisson equations with nonslip boundary condition and obtain the linear stability of the approximate solution constructed by boundary layer expansion.

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The zero dissipation limit of the compressible heat-conducting NavierStokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient and the viscosity coefficient satisfy = O (), = O (), c&gt; 0, as 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the NavierStokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of . The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].