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The pressure gradient system. (English) Zbl 1227.35208

The authors of this very interesting paper consider the two-dimensional compressible Euler system
\[ \begin{aligned} \rho_t&+\nabla\cdot (\rho {\mathbf u})=0,\\ (\rho{\mathbf u})_t&+\nabla\cdot (\rho{\mathbf u}\times{\mathbf u}+pI)=0,\\ (\rho E)_t&+\nabla\cdot (\rho E{\mathbf u}+ p{\mathbf u})=0, \end{aligned} \]
where \(\rho \) is the density, \({\mathbf u}\) is the velocity vector, \(p\) is the pressure, \(E=|{\mathbf u}|^2/2+p\rho^{-1}/(\gamma -1)\) is the total energy density per unit mass, and \(\gamma >1\) is the gas constant. The so-called pressure gradient system is a subsystem of the compressible Euler system \[ u_t+p_x=0, \quad v_t+p_y=0, \quad E_t+(up)_x+(vp)_y=0, \]
where \(E=p+(u^2+v^2)/2\). The authors provide numerical simulations, basic characteristic analysis and physical considerations for the Riemann problems to the considered model to find out appropriate internal conditions at the origin. This study reveals subtle structures of the velocity. Both components exhibit discontinuities at the origin. Comparing to the roll-up of shear waves or vortex-sheets of the Euler system, it turns out that the singularities are mild and occur only along rays from the origin. Numerical examples illustrate the theory, and sets of central contour lines from the pressure graph are shown in a very precise form.

MSC:

35L65 Hyperbolic conservation laws
35Q31 Euler equations
35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
76N15 Gas dynamics (general theory)
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