Numerical proof of stability of viscous shock profiles. (English) Zbl 1356.35170
Summary: We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number \(\approx 1.86\), below which profiles are known by energy estimates to be stable, to the upper stability boundary of \(\approx 1669\), above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 65G30 | Interval and finite arithmetic |
| 76N15 | Gas dynamics (general theory) |
| 35C07 | Traveling wave solutions |
| 35B35 | Stability in context of PDEs |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 68T15 | Theorem proving (deduction, resolution, etc.) (MSC2010) |
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