Global Stability of Multi-wave Configurations for the Compressible Non-isentropic Euler System

Abstract

This paper is contributed to the structural stability of multi-wave configurations to Cauchy problem for the compressible non-isentropic Euler system with adiabatic exponent γ ∈ (1, 3]. Given some small BV perturbations of the initial state, the author employs a modified wave front tracking method, constructs a new Glimm functional, and proves its monotone decreasing based on the possible local wave interaction estimates, then establishes the global stability of the multi-wave configurations, consisting of a strong 1-shock wave, a strong 2-contact discontinuity, and a strong 3-shock wave, without restrictions on their strengths.

Keywords

Structural stability / Multi-wave configuration / Shock / Contact discontinuity / Compressible non-isentropic Euler system / Wave front tracking method

References

[1] Bressan A. Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, 2000, New York: Oxford University Press

[2] Bressan A, Colombo R M. Unique solutions of 2 × 2 conservation laws with large data. Indiana Univ. Math. J., 1995, 44(3): 677-725

[3] Chen G Q, Rigby M. Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. Acta Math. Sci. Ser. B (Engl. Ed.), 2018, 38(5): 1485-1514

[4] Chern I-L. Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities. Comm. Pure Appl. Math., 1989, 42(6): 815-844

[5] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics, 2010 Third edition Berlin: Springer-Verlag

[6] Ding M. Stability of contact discontinuities to 1-D piston problem for the compressible Euler equations. J. Differential Equations, 2018, 264(6): 3836-3863

[7] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math., 1965, 18: 697-715

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

[9] Lewicka M. L ¹ stability of patterns of non-interacting large shock waves. Indiana Univ. Math. J., 2000, 49: 1515-1537

[10] Lewicka M. Stability conditions for patterns of non-interacting large shock waves. SIAM J. Math. Anal., 2001, 32(5): 1094-1116

[11] Lewicka M. Well-posedness for hyperbolic systems of conservation laws with large BV data. Arch. Rational Mech. Anal., 2004, 173(3): 415-445

[12] Lewicka M, Trivisa K. On the L ¹ well-posedness of systems of conservation laws near solutions containing two large shocks. J. Differential Equations, 2002, 179(1): 133-177

[13] Lewicka M. Lyapunov functional for solutions of systems of conservation laws containing a strong rarefaction. SIAM J. Math. Anal., 2005, 36(5): 1371-1399

[14] Lewicka M. Stability conditions for strong rarefaction waves. SIAM J. Math. Anal., 2005, 36(4): 1346-1369

[15] Li T-T, Qin T. Physics and Parital Differential Equations, 2014, Beijing: Higher Education Press

[16] Schochet S. Sufficient conditions for local existence via Glimm’s scheme for large BV data. J. Differential Equations, 1991, 89: 317-354

[17] Smoller J. Shock Waves and Reaction-Diffusion Equations, 1983 2nd ed. New York-Berlin: Springer-Verlag