Convergence Rate of the Hypersonic Similarity for Two-Dimensional Steady Potential Flows with Large Data

TL;DR

This paper proves the optimal convergence rate of hypersonic similarity for 2D steady potential flows with large data, using Riemann semigroup techniques and $L^1$ estimates, under specific bounded variation conditions.

Contribution

It establishes the optimal convergence rate for hypersonic similarity in large data regimes for 2D steady flows, extending previous results to more general conditions.

Findings

Proved the optimal convergence rate of hypersonic similarity.

Constructed an example demonstrating the optimality of the convergence rate.

Developed $L^1$ estimates for solutions using Riemann semigroup methods.

Abstract

We establish the optimal convergence rate of the hypersonic similarity for two-dimensional steady potential flows with {\it large data} past over a straight wedge in the $BV\cap L^1$ framework, provided that the total variation of the large data multiplied by $\gamma-1+\frac{a_{\infty}^2}{M_\infty^2}$ is uniformly bounded with respect to the adiabatic exponent $\gamma>1$, the Mach number $M_\infty$ of the incoming steady flow, and the hypersonic similarity parameter $a_\infty$. Our main approach in this paper is first to establish the Standard Riemann Semigroup of the initial-boundary value problem for the isothermal hypersonic small disturbance equations with large data and then to compare the Riemann solutions between two systems with boundary locally case by case. Based on them, we derive the global $L^1$--estimate between the two solutions by employing the Standard Riemann Semigroup and the local $L^1$--estimates. We further construct an example to show that the convergence rate is optimal.