EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY

doi:10.1016/S0252-9602(10)60058-6

摘要/Abstract

摘要：

For the two-dimensional Navier--Stokes equations of isentropic magnetohydrodynamics (MHD) with γ-law gas equation of state, γ≥1, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes.
In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompressive, or undercompressive shock profiles. Considered as three-dimensional solutions,
undercompressive shocks are Lax-type (Alfven) waves. For the monatomic and diatomic cases γ=5/3 and γ= 7/5,
with standard viscosity ratio for a nonmagnetic gas, we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes, with no undercompressive shocks occurring. We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable, both with respect to two- and three-dimensional perturbations. For the same gas constants, but different viscosity ratios, we investigate also cases for which undercompressive shocks appear; these are seen numerically to be stable as well, both with respect to two-dimensional and (in the neutral sense of convergence to nearby Riemann solutions) three-dimensional perturbations.

关键词: stability of viscous shock waves, Evans function, magnetohydrodynamics

Abstract:

Key words: stability of viscous shock waves, Evans function, magnetohydrodynamics

中图分类号:

35B35

Blake Barker, Olivier Lafitte, Kevin Zumbrun. EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY[J]. 数学物理学报(英文版), 2010, 30(2): 447-498.

Blake Barker, Olivier Lafitte, Kevin Zumbrun. EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY[J]. Acta mathematica scientia,Series B, 2010, 30(2): 447-498.

参考文献

[1] Alexander J, Gardner R, Jones C. A topological invariant arising in the stability analysis of travelling waves.
J Reine Angew Math, 1990, 410: 167--212

[2] Azevedo A, Marchesin D, Plohr B, Zumbrun K. Nonuniqueness of solutions of Riemann problems. Z Angew Math Phys, 1996, 47(6): 977--998

[3] Alexander J C, Sachs R. Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation. Nonlinear World, 1995, 2(4): 471--507

[4] Anderson J E. Magnetohydrodynamic Shock Waves. MIT Press, 1963

[5] Barker B, Humpherys J, Rudd K, Zumbrun K. Stability of viscous shocks in isentropic gas dynamics. Comm Math Phys, 2008, 281(1): 231--249

[6] Barker B, Humpherys J, Zumbrun K. One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics. Preprint, 2007

[7] Beck M, Sandstede B, Zumbrun K. Nonlinear stability of time-periodic shock waves. To appear, Arch Rat Mechanics Anal

[8] Beyn W -J. The numerical computation of connecting orbits in dynamical systems. IMA J Numer Analysis, 1990, 9: 379--405

[9] Beyn W -J. Zur stabilit at von differenenverfahren f\"ur systeme linearer gewöhnlicher randwertaufgaben.
Numer Math, 1978, 29: 209--226

[10] Cabannes H. Theoretical Magnetofluiddynamics. New York: Academic Press, 1970

[11] Batchelor G K. An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press, 1999

[12] Blokhin A, Trakhinin Y. Stability of strong discontinuities in fluids and MHD//Handbook of mathematical fluid dynamics, Vol I. Amsterdam: North-Holland, 2002: 245--652

[13] Bridges T J, Derks G, Gottwald G. Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Phys D, 2002, 172(1-4): 190--216

[14] Brin L Q. Numerical testing of the stability of viscous shock waves. PhD thesis, Indiana University, oomington, 1998

[15] Brin L Q. Numerical testing of the stability of viscous shock waves. Math Comp, 2001, 70(235): 1071--1088

[16] Brin L Q, Zumbrun K. Analytically varying eigenvectors and the stability of viscous shock waves. Mat Contemp, 2002, 22: 19--32. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001)

[17] Conley C C, Smoller J. On the structure of magnetohydrodynamic shock waves. Comm Pure Appl Math, 1974, 27: 367--375

[18] Conley C C, Smoller J. On the structure of magnetohydrodynamic shock waves II. J Math Pures Appl, 1975, 9(4): 429--443

[19] Costanzino N, Humpherys J, Nguyen T, Zumbrun K. Spectral stability of noncharacteristic boundary layers of
isentropic Navier--Stokes equations. Arch Ration Mech Anal, to appear, 2008

[20] Diehl D, Rohde C. On the Structure of MHD Shock Waves in Diffusive-Dispersive Media. Preprint 34, Albert-Ludwigs-Universitt Freiburg, Fakultt fr Mathematik und Physik, Mathematisches Institut, 2002

[21] Evans J W, Feroe J A. Traveling waves of infinitely many pulses in nerve equations. Math Biosci, 1977, 37: 23--50

[22] Freistühler H. Dynamical stability and vanishing viscosity: a case study of a non-strictly hyperbolic system.
Comm Pure Appl Math, 1992, 45(5): 561--582

[23] Freistühler H, Rohde D. The bifurcation analysis of the MHD Rankine-Hugoniot equations for a perfect gas.
Phys D, 2003, 185(2): 78--96

[24] Freistühler H. Rohde C. Numerical computation of viscous profiles for hyperbolic conservation laws. Math Comp, 2002, 71(239): 1021--1042

[25] Freistühler H, Szmolyan P. Existence and bifurcation of viscous profiles for all intermediate netohydrodynamic shock waves. SIAM J Math Anal, 1995, 26(1): 112--128

[26] Freistühler H, Trakhinin Y. On the viscous and inviscid stability of magnetohydrodynamic shock waves.
Physica D: Nonlinear Phenomena, 2008, 237(23): 3030--3037

[27] Freistühler H, Zumbrun K. Examples of unstable viscous shock waves. Unpublished research report, University of Aachen, Germany, 1998

[28] Gardner R A, Zumbrun K. The gap lemma and geometric criteria for instability of viscous shock profiles.
Comm Pure Appl Math,1998, 51(7): 797--855

[29] Gardner R, Jones C K R T. A stability index for steady state solutions of boundary value problems for parabolic systems. J Diff Eqs, 1991, 91(2): 181--203

[30] Gardner R, Jones C K R T. Traveling waves of a perturbed diffusion equation arising in a phase field model.
Indiana Univ Math J, 1989, 38(4): 1197--1222

[31] Germain P. Contribution \`a la th\'eorie des ondes de choc en magn\'etodynamique des fluides. ONERA Publ No 97, Office Nat tudes et Recherche Arospatiales, Ch\^atillon, 1959

[32] Gilbarg D. The existence and limit behavior of the one-dimensional shock layer. Amer J Math, 1951, 73: 256--274

[33] Guès O, M\'etivier G, Williams M, Zumbrun K. Navier--Stokes regularization of multidimensional Euler shocks.
Ann Sci École Norm Sup, 2006, 39(4): 75--175

[34] Gues O, M{\'e}tivier G, Williams M, Zumbrun K. Viscous boundary value problems for symmetric systems with variable multiplicities. J Differential Equations, 2008, 244(2): 309--387

[35] Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics. Berlin: Springer--Verlag, 1981

[36] Howard P, Raoofi M. Pointwise asymptotic behavior of perturbed viscous shock profiles. Adv Differential Equations, 2006, 11(9): 1031--1080

[37] Howard P, Raoofi M, Zumbrun K. Sharp pointwise bounds for perturbed viscous shock waves. J Hyperbolic Differ Equ, 2006, 3(2): 297--373

[38] Howard P, Zumbrun K. Staiblity of undercompressive shocks. J Differential Equations, 2006, 225(1): 308--360

[39] Humpherys J, Zumbrun K. Spectral stability of small amplitude shock profiles for dissipative symmetric hyperbolic--parabolic systems. Z Angew Math Phys, 2002, 53: 20--34

[40] Humpherys J, Zumbrun K. An efficient shooting algorithm for evans function calculations in large systems.
Physica D, 2006, 220(2): 116--126

[41] Humpherys J, Lafitte O, Zumbrun K. Stability of viscous shock profiles in the high Mach number limit. Comm Math Phys, to appear, 2009

[42] Humpherys J, Lyng G, Zumbrun K. Spectral stability of ideal-gas shock layers. Arch Ration Mech Anal, to appear, 2009

[43] Humpherys J, Lyng G, Zumbrun K. Multidimensional spectral stability of large-amplitude navier-stokes shocks. In preparation, 2009

[44] Howard P. Nonlinear stability of degenerate shock profiles. Differential Integral Equations, 2007, 20(5): 515--560

[45] Howard P, Zumbrun K. The Evans function and stability criteria for degenerate viscous shock waves.
Discrete Contin Dyn Syst, 2004, 10(4): 837--855

[46] Hale N, Moore D R. A sixth-order extension to the matlab package bvp4c of j. kierzenka and l. shampine.
Technical Report NA-08/04, Oxford University Computing Laboratory, May 2008

[47] Jeffrey A. Magnetohydrodynamics. University Mathematical Texts, No 33. Oliver \& Boyd, Edinburgh, 1966

[48] Kato T. Perturbation Theory for Linear Operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995

[49] Kawashima S. Systems of a hyperbolic--parabolic composite type, with applications to the equations of magnetohydrodynamics
[D]. Kyoto University, 1983

[50] Kierzenka J, Shampine L F. A BVP solver that controls residual and error. J Numer Anal Ind Appl Math, 2008, 3(1/2): 27--41

[51] Mètivier G, Zumbrun K. Hyperbolic boundary value problems for symmetric systems with variable multiplicities. J Differ Equ, 2005, 211(1): 61--134

[52] Majda A. The stability of multidimensional shock fronts. Mem Amer Math Soc, 1983, 275

[53] Majda A. The existence of multidimensional shock fronts. Mem Amer Math Soc, 1983, 281

[54] Majda A, Pego R. Stable viscosity matrices for systems of conservation laws. J Diff Eqs, 1985, 56: 229--262

[55] Mascia C, Zumbrun K. Pointwise {G}reen function bounds for shock profiles of systems with real viscosity.
Arch Ration Mech Anal, 2003, 169(3): 177--263

[56] Mascia C, Zumbrun K. Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems.
Arch Ration Mech Anal, 2004, 172(1): 93--131

[57] Métivier G, Zumbrun K. Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems.
Mem Amer Math Soc, 2005, 175(826)

[58] Oh M, Zumbrun K. Stability of periodic solutions of viscous conservation laws: analysis of the Evans function.
Arch Ration Mech Anal, 2002

[59] Pego R L. Stable viscosities and shock profiles for systems of conservation laws. Trans Amer Math Soc, 1984, 282(2): 749--763

[60] Pego R L, Smereka P, Weinstein M I. Oscillatory instability of traveling waves for a KdV-Burgers equation.
Phys D, 1993, 67(1/3): 45--65

[61] Plaza R, Zumbrun K. An Evans function approach to spectral stability of small-amplitude shock profiles.
J Disc and Cont Dyn Sys, 2004, 10: 885--924

[62] Raoofi M. L^p asymptotic behavior of perturbed viscous shock profiles. J Hyperbolic Differ Equ, 2005, 2(3): 595--644

[63] Raoofi M, Zumbrun K. Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems.
J Differential Equations, 2009, 246(4): 1539--1567

[64] Shampine L F, Gladwell I, Thompson S. Solving ODEs with MATLAB. Cambridge: Cambridge University Press, 2003

[65] Texier B, Zumbrun K. Hopf bifurcation of viscous shock waves in compressible gas dynamics and MHD.
Arch Ration Mech Anal, 2008, 190(1): 107--140

[66] Trakhinin Y. A complete 2D stability analysis of fast MHD shocks in an ideal gas. Comm Math Phys, 2003, 236(1): 65--92

[67] Zumbrun K. Stability of large-amplitude shock waves of compressible Navier-Stokes equations//Handbook of Mathematical Fluid Dynamics. Vol III. Amsterdam: North-Holland, 2004: 311--533

[68] Zumbrun K. Multidimensional stability of planar viscous shock waves//Advances in the Theory of Shock Waves, Progr Nonlinear Differential Equations Appl, Vol 47. Boston, MA: Birkhäuser Boston, 2001: 307--516

[69] Zumbrun K. A local greedy algorithm and higher order extensions for global numerical continuation of analytically varying subspaces. To appear, Quart Appl Math

[70] Zumbrun K. Numerical error analysis for evans function computations: a numerical gap lemma, centered-coordinate methods, and the unreasonable effectiveness of continuous orthogonalization. Preprint, 2009

[71] Zumbrun K. Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD.
Preprint, 2009

[72] Zumbrun K. The refined inviscid stability condition and cellular instability of viscous shock waves. Preprint, 2009

[73] Zumbrun K. Stability of noncharacteristic boundary layers in the standing shock limit. To appear, Trans Amer Math Soc

[74] Zumbrun K. Stability of viscous detonations in the ZND limit. Preprint, 2009

[75] Zumbrun K, Howard P. Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ Math J, 1998, 47(3): 741--871

[76] Zumbrun K, Serre D. Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ Math J, 1999, 48(3): 937--992