Global well-posedness of shock front solutions to one-dimensional piston problem for combustion Euler flows

TL;DR

This paper proves the global well-posedness and stability of shock front solutions in one-dimensional combustion Euler flows with ignition, using wave front tracking and Lyapunov functionals.

Contribution

It establishes the first well-posedness results for inviscid reacting Euler flows with ignition temperature, addressing large shocks and characteristic boundaries.

Findings

Global existence of entropy solutions for combustion Euler flows.

Stability of ZND detonation waves supported by a piston.

Development of a modified Glimm functional and weighted Lyapunov functional.

Abstract

This paper is devoted to the well-posedness theory of piston problem for compressible {combustion} Euler flows with physical ignition condition. A significant combustion phenomena called detonation will occur provided the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of $BV\cap L^1$ space. The main difficulties here stem from the discontinuous source term without uniform dissipation structure, and from the characteristic-boundary associated with degenerate characteristic field. In dealing with the obstacles caused by ignition temperature, we develop a modified Glimm-type functional to control the oscillation growth of combustion waves, even if the exothermic source fails to uniformly decay. As to the characteristic boundary, the degeneracy of contact discontinuity is fully employed to get elegant stability estimates near the piston boundary. Meanwhile, we devise a weighted Lyapunov functional to balance the nonlinear effects arising from large shock, characteristic boundary and exothermic reaction, then obtain the $L^1-$stability of combustion wave solutions. Our results reveal that one dimensional \emph{ZND} detonation waves {supported} by a forward piston are indeed nonlinearly stable under small perturbation in $BV$ sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.