On the irrotational approximation to the 2-dimensional isothermal Euler system

Abstract

The author mainly studies the difference of the weak solutions generated by a wave front tracking algorithm to the steady Euler system and the isothermal Euler system. Under the hypothesis that the initial data are of sufficiently small total variation, it is proved that the difference between the solutions of the steady Euler system and the system of isothermal supersonic flow can be bounded by the cube of the total variation of the initial perturbation.

Keywords

Irrotational approximation / Isothermal Euler system / Semigroup

References

[1] Chen G Q, Christoforou C, Zhang Y Q. Continuous dependence of entropy solutions to the Euler equations on the adiabatic exponent and Mach number. Arch. Ration. Mech. Anal., 2008, 189(1): 97-130

[2] Bressan A. The unique limit of the Glimm scheme. Arch. Rational Mech. Anal., 1995, 130: 105-230

[3] Bressan A. Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem, Oxford Lecture Series in Mathematics and Its Applications, Vol. 20, 2000, Oxford: Oxford University Press

[4] Saint-Raymond L. Isentropic approximation of the compressible Euler system in one space dimension. Arch. Rational Mech. Anal., 2000, 155: 171-199

[5] Bianchini S, Colombo R M. On the stability of the standard Riemann semigroup. Proc. American Math. Soc., 2002, 130: 1961-1973

[6] Zhang Y Q. On the irrotational approximation to steady supersonic flow. Z. Angew. Math. Phys., 2007, 58(2): 209-223

[7] Liu C, Zhang Y Q. Isentropic approximation of the steady Euler system in two space dimensions. Commun. Pure Appl. Anal., 2008, 7(2): 277-291

[8] Lax P D. Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math., 1957, 10: 537-566

[9] Smoller J. Shock Waves and Reaction-Diffusion Equations, 1983, New York: Springer-Verlag

[10] Zhang Y Q. Steady supersonic flow past an almost straight wedge with large vertex angle. J. Diff. Eq., 2003, 192: 1-46

[11] Chen G Q, Zhang Y Q, Zhu D W. Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch. Rational Mech. Anal., 2006, 181: 261-310

[12] Zhang Y Q. Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary. SIAM J. Math. Anal., 1999, 31: 166-183

[13] Corli A, Sable-Tougeron M. Stability of contact discontinuities under perturbations of bounded variation. Rend. Sem. Mat. Univ. Padova, 1997, 97: 35-60

[14] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics, 2000, Berlin: Springer-Verlag

[15] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 1965, 18: 95-105

[16] Serre D. Systems of Conservation Laws, 1999, Cambridge: Cambridge University Press

[17] Kong D, Yang T. A note on well-posedness theory for hyperbolic conservation laws. Applied Mathematics Letters, 2003, 16: 143-146

[18] Bressan A, Liu T P, Yang T. L ¹ stability estimates for n × n conservation laws. Arch. Rational Mech. Anal., 1999, 149: 1-22