Global Well-Posedness of Shock Front Solutions to One-Dimensional Piston Problem for Combustion Euler Flows

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 phenomenon called detonation will occur provided that 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 the present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of 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 and then obtain the -stability of combustion wave solutions. Our results reveal that one-dimensional Zeldovich–von Neumann–Döring detonation waves supported by a forward piston are indeed nonlinearly stable under small perturbation in sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.