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We establish the optimal convergence rate to the hypersonic similarity law,which is also called the Mach number independence principle, for steadycompressible full Euler flows over two-dimensional slender Lipschitz wedges.The problem can be formulated as the comparison of the entropy solutions in$BV\cap L^{1}$ between the two initial-boundary value problems for thecompressible full Euler equations with parameter $\tau>0$ and the hypersonicsmall-disturbance equations with curved characteristic boundaries. We establishthe $L^1$--convergence estimate of these two solutions with the optimalconvergence rate, which justifies Van Dyke's similarity theory rigorously forthe compressible full Euler flows. This is the first mathematical result on thecomparison of two solutions of the compressible Euler equations withcharacteristic boundary conditions. To achieve this, we first employ thespecial structures of the two systems and establish the global existence andthe $L^1$--stability of the entropy solutions under the smallness assumptionson the total variation of both the initial data and the tangential slope of thewedge boundary. Based on the $L^1$--stability properties of the approximatesolutions to the scaled equations with parameter $\tau$, a uniform Lipschtizcontinuous map with respect to the initial data and the wedge boundary isobtained. Next, we compare the solutions given by the Riemann solvers of thetwo systems by taking the boundary perturbations into account case by case.Then, for a given fixed hypersonic similarity parameter, as the Mach numbertends to infinity, we establish the desired $L^1$--convergence estimate withthe optimal convergence rate. Finally, we show the optimality of theconvergence rate by investigating a special solution.

Hypersonic Similarity for Steady Compressible Full Euler Flows Over Two-Dimensional Lipschitz Wedges