On Two-Dimensional Steady Hypersonic-Limit Euler Flows Passing Ramps and Radon Measure Solutions of Compressible Euler Equations

Abstract

We proposed rigorous definitions of Radon measure solutions for boundary value problems of steady compressible Euler equations which modeling hypersonic-limit inviscid flows passing two-dimensional ramps, and their interactions with still gas and pressureless jets. We proved the Newton-Busemann pressure law of drags of body in hypersonic flow, and constructed various physically interesting measure solutions with density containing Dirac measures supported on curves, also exhibited examples of blow up of certain measure solutions. This established a mathematical foundation for applications in engineering and further studies of measure solutions of compressible Euler equations.

Convergence Rate of Hypersonic Similarity for Steady Potential Flows Over Two-Dimensional Lipschitz Wedge

“…Finally, letting ν → +∞, using decay property of U w for x > ℓ w , and applying Theorem 1.1 on the L 1 difference estimate between U (τ ) w (x, •) and P (τ ) * (x − ℓ w )(U w (x, •)) for 0 < x < ℓ w , we can complete the proof of Theorem 1.2. Recently, the authors in [17,18,19,20] systematically studied the hypersonic limit, which is a different problem from ours since there is no hypersonic similarity structure. The reason is that the wedge angle (or cone angle) θ in [17,18,19,20] is fixed such that the similarity parameter K tends to the infinity as M ∞ → ∞.…”

“…Recently, the authors in [17,18,19,20] systematically studied the hypersonic limit, which is a different problem from ours since there is no hypersonic similarity structure. The reason is that the wedge angle (or cone angle) θ in [17,18,19,20] is fixed such that the similarity parameter K tends to the infinity as M ∞ → ∞. There are also many literatures on the BV solutions for the steady supersonic compressible Euler flows with free boundaries of small data such that steady supersonic flow past a Lipschitz wedge or moving over a Lipschitz bending wall (see [6,7,10,23,24] for more details) which involving the stabilities of the shock wave and rarefaction wave.…”