Wave propagation of a generic non--conservative compressible two--fluid model

TL;DR

This paper establishes the wave propagation theory for a complex non-conservative two-fluid model, overcoming significant analytical challenges with novel estimates and exploiting the model's special structure.

Contribution

It develops a new framework for nonlinear coupling and decay estimates in non-conservative fluid models, extending the theory beyond conservative systems.

Findings

Established the generalized Huygens principle for the non-conservative model.

Developed sharp convolution estimates for Riesz wave-IV and Huygens wave interactions.

Achieved enhanced decay rates for pressure terms, enabling nonlinear analysis.

Abstract

The generalized Huygens principle for the Cauchy problem of a generic non-conservative compressible two-fluid model in R3 was established. This work fills a key gap in the theory, as previous results were confined to systems with full conservation laws or ``equivalent" conservative structures from specific compensatory cancellations in Green's function. Indeed, the genuinely non-conservative model studied here falls outside these categories and presents two major analytical challenges. First, its inherent non-conservative structure blocks the direct use of techniques (e.g., variable reformulation) effective for conservative systems. Second, its Green's function contains a -1-order Riesz operator associated with the fraction densities, which generates a so-called Riesz wave-IV exhibiting both slower temporal decay and poorer spatial integrability compared to the standard heat kernel, necessitating novel sharp convolution estimates with the Huygens wave. To overcome these difficulties, we develop a framework for precise nonlinear coupling, including interaction of Riesz wave-IV and Huygens wave. A pivotal step is extracting enhanced decay rates for the non-conservative pressure terms. By reformulating these terms into a product involving the fraction densities and the specific combination of fractional densities, and then proving this combination decays faster than the individual densities, we meet the minimal requirements for the crucial convolution estimates. This allows us to close the nonlinear ansatz by constructing essentially new nonlinear estimates. The success of our analysis stems from the model's special structure, particularly the equal-pressure condition. More broadly, the sharp nonlinear estimates developed herein is applicable to a wide range of non-conservative compressible fluid models.