Convergence rate of the hypersonic similarity for two-dimensional steady potential flows with large data

Abstract We establish the optimal convergence rate of the hypersonic similarity for two-dimensional steady potential flows with large data past a straight wedge in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>V</mml:mi> <mml:mo>∩</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> framework, provided that the total variation of the large data multiplied by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mfrac> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mfrac> </mml:mrow> </mml:math> is uniformly bounded with respect to the adiabatic exponent γ &gt; 1, the Mach number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> </mml:mrow> </mml:math> of the incoming steady flow, and the hypersonic similarity parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> </mml:mrow> </mml:math> . Our main approach in this paper is first to establish the well-posedness and the Lipschitz continuous map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> </mml:mrow> </mml:math> that has the properties similar to 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 Lipschitz continuous map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">P</mml:mi> </mml:mrow> </mml:mrow> </mml:math> and the local L 1 –estimates. We further construct an example to show that the convergence rate is optimal.