# Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves

## TL;DR

This paper investigates double phase transonic flow problems involving the Baouendi-Grushin operator, establishing existence of stationary waves with variable growth and mixed elliptic-hyperbolic behavior.

## Contribution

It introduces a novel analysis of Euler equations driven by the Baouendi-Grushin operator with variable coefficients, including weighted inequalities and existence results.

## Key findings

- Established weighted inequalities for the Baouendi-Grushin operator
- Proved existence of stationary waves under perturbations
- Analyzed nonlinear patterns and stationary waves in transonic flows

## Abstract

In this paper we are concerned with a class of double phase energy functionals arising in the theory of transonic flows. Their main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. After establishing a weighted inequality for the Baouendi-Grushin operator and a related compactness property, we establish the existence of stationary waves under arbitrary perturbations of the reaction.