On the Free Surface Motion of Highly Subsonic Heat-conducting Inviscid Flows

TL;DR

This paper establishes a priori Sobolev norm estimates for highly subsonic heat-conducting inviscid flows with free surfaces, addressing complex couplings and geometric properties to advance understanding of low Mach number free surface dynamics.

Contribution

It extends geometric analysis techniques to heat-conducting inviscid flows, handling strong couplings and derivative losses, which were not addressed in prior work by Christodoulou and Lindblad.

Findings

Sobolev norm estimates under Taylor sign condition

Geometric quantities like second fundamental form estimated

Analysis of coupling effects in free surface flows

Abstract

For a free surface problem of a highly subsonic heat-conducting inviscid flow, motivated by a geometric approach developed by Christodoulou and Lindblad in the study of the free surface problem of incompressible inviscid flows, the a priori estimates of Sobolev norms in 2-D and 3-D are proved under the Taylor sign condition by identifying a suitable higher order energy functional. The estimates for some geometric quantities such as the second fundamental form and the injectivity radius of the normal exponential map of the free surface are also given. The novelty in our analysis includes dealing with the strong coupling of large variation of temperature, heat-conduction, compressibility of fluids and the evolution of free surface, loss of symmetries of equations, and loss of derivatives in closing the argument which is a key feature compared with Christodoulou and Lindblad's work. The motivation of this paper is to contribute to the program of understanding the role played by the heat-conductivity to free surface motions of inviscid compressible flows and the behavior of such motions when the Mach number is small.